infinity infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 50 pemanfaatan web khan academy dalam pembelajaran matematika oleh: joko soebagyo stt wastukancana, purwakarta jokosoebagyo@student.upi.edu abstrak artikel ini bertujuan untuk mengobservasi pemanfaatan web khan academy siswa smk swasta di jakarta utara. observasi dilakukan untuk mengamati aktivitas siswa dalam menggunakan web khan academy selama proses pembelajaran matematika. pemanfaatan teknologi dalam proses pembelajaran matematika sudah sepatutnya dilakukan oleh pihak-pihak yang ikut bertanggungjawab atas keberhasilan belajar matematika perserta didik. teknologi merupakan hal yang tidak bisa dipisahkan dalam kehidupan sehari-hari dan salah satu bentuk teknologi yang sering dimanfaatkan saat ini adalah website dimana akitifitas pekerjaan, pendidikan, hiburan dan hubungan sosial, semuanya dapat dialami melalui website. dibutuhkan sebuah website yang bias mengakomodir siswa dalam proses pembelajaran matematika baik di dalam kelas maupun di luar kelas. kriteria web yang baik haruslah memberikan kemudahan dalam pemanfaatannya, salah satu web tersebut adalah web khan academy. kata kunci : pembelajaran matematika, web khan academy abstract this paper aims to observe the use of web khan academy private vocational students in north jakarta. observations carried out to observe the activities of the students in using web khan academy during the learning process of mathematics. the use of technology in the learning process of mathematics has been duly carried out by parties who share responsibility for the success of students studying mathematics participants. technology is something that can not be separated in everyday life and one form of technology that is often used today is a website where the activity, employment, education, entertainment and social relationships, all of which can be experienced through the website. it takes a biased website to accommodate students in mathematics learning process both in the classroom and outside the classroom. criteria for good web should provide ease of use, one of the web is a web khan academy. keywords: learning mathematics, web khan academy i. pendahuluan pembelajaran matematika sampai saat ini masih menjadi momok bagi mayoritas siswa di seluruh dunia tidak terkecuali indonesia. rendahnya kemampuan matematika menjadi indikator bagi para guru dan para penggiat pembelajaran matematika yang dapat dilihat dari hasil the programme for international student assessment (pisa) tahun 2009 dan 2013. kita tengah menghadapi siswa yang hidup dalam era digital (y generation), sesuai pendapat (hirsch, martin, hopfensperger, & zbiek, 2013) bahwa teknologi merupakan hal yang tidak bisa dipisahkan dalam kehidupan sehari-hari.menurut (brckalorenz, haeger, nailos, & rabourn, 2013) salah satu bentuk teknologi yang sering dimanfaatkan saat ini adalah website infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 51 dimana akitifitas pekerjaan, pendidikan, hiburan dan hubungan sosial, semuanya dapat dialami melalui website. a. pentingnya teknlogi menurut (goos & bennison, 2008), banyak peneliti dari berbagai negara yang menyatakan bahwa teknologi memainkan peranan penting dalam pembelajaran matematika di dalam kelas yang meliputi: (a) keterampilan dan pengalaman menggunakan teknologi; (b) waktu dan kesempatan untuk belajar; (c) pengetahuan tentang bagaimana teknologi terintegrasi ke dalam pembelajaran matematika; (d) belief tentang teknologi dalam pembelajaran; (e) belief tentang matematika dan bagaimana mempelajarinya. perkembangan internet menyediakan kekayaan informasi bagi guru matematika dan siswa di semua tingkat dan sangat berlimpah dengan sebuah keberlebihan sumber daya yang sebelumnya tidak tersedia. melalui media ini, menurut (cherkas & welder, 2011) berbagai bahan berbasis web yang bertujuan untuk meningkatkan pengajaran dan pembelajaran matematika terus-menerus dan sedang dikembangkan. dengan demikian dalam proses pembelajaran matematika, sudah saatnya guru matematika memanfaatkan web guna meningkatkan kemampuan matematis siswa. b. kriteri web yang baik ketika kita searching di google dengan keyword “web pembelajaran matematika” akan muncul 239.000 hasil pencarian. semua hasil tersebut dapat saja digunakan sebagai sumber pembelajaran matematika. tetapi menurut (sunil & saini, 2013) sebuah web layak dijadikan sebagai sumber pembelajaran jika memenuhi beberapa kriteria yaitu: (1) choosing an appropriate learning approach depending on the style of the learner; (2) choosing content depending on the learning style and approach choosing the appropriate content; (3) choosing learning modules that can create a learning activity path; (4) the knowledge that the learner has acquired and needs to acquire are then mapped appropriately to the learning activity mechanism that controls the generation of learning content in the recommender based learning management system. salah satu website yang memenuhi kriteria tersebut adalah www.khanacademy.org. ii. pembahasan a. khan academy khan academy adalah organisasi nirlaba dengan misi memberikan pendidikan bagi siapa sajasecara gratis, berkelas dunia,dapat digunakan kapan saja dan di mana saja. berdasarkan observasi yang dilakukan oleh penulis terhadap web khan academy, banyak didapati soalsoal matematika seperti pada soal matematika di pisa dan un sehingga siswa diharapkan mendapat wawasan matematika lebih banyak dari sumber belajar tersebut. http://www.khanacademy.org/ infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 52 gambar 1. logo khan academy keuntungan bagi siswa ketika siswa belajar matematika di khan academy khususnya mengerjakan soal-soal matematika dan menjawab dengan benar maka ia akan mendapati jawaban tercentang (lihat gambar 2). gambar 2. menjawab soal dengan benar namun ketika mengalami kesulitan menjawabnya, ia dapat melihat petunjuk baik dengan menonton video ataupun jawaban yang tersedia tetapi ia akan mendapati jawaban dengan tanda silang (lihat gambar 3). infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 53 gambar 3. menjawab soal dengan salah dan masih banyak keuntungan lain yang diperoleh siswa ketika belajar matematika di khan academy. penulis hanya memberikan beberapa contoh di atas. keuntungan bagi guru dan orang tua bagi guru dan orangtua pasti ingin mengetahui apakah anak/siswanya belajar matematika pada hari ini? materi apakah yang dipelajari? sejauh mana kemajuannya? dan sebagainya. belajar matematika di khan academy memungkinkan semua hal tersebut terwujud, dimana jika dengan pembelajaran matematika konvensional tidak dapat dilakukan secara detail. sebagai contoh, abdul fatah seorang siswa di sebuah smk swasta sudah mempelajari 132 skills matematika seperti terlihat pada gambar 4. gambar 4. skill matematika yang dipelajari seorang siswa infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 54 penjelasan pada gambar 4 adalah salah satu keuntungan bagi guru atau orangtua tentang kondisi anak/siswa-nya. masih banyak penjelasan lain yang lebih rinci dan detail dalam pembelajaran matematika dengan memanfaatkan khan academy. kekurangan khan academy setiap alat/media pembelajaran pasti memiliki kelemahan dan kekurangan, demikian juga dengan khan academy. pembelajaran matematika di khan academy memerlukan penguasaan bahasa inggris yang baik, komputer/laptop/gadget untuk mengakses dan koneksi internet yang baik walaupun ada versi offline-nya. tanpa hal-hal tersebut, sulit rasanya untuk melakukan pembelajaran matematika di khan academy. pada saat penulis melakukan penelitian dengan memanfaatkan web khan academy, masalah bahasa dapat ditanggulangi dengan menggunakan fasilitas google translate. nah, yang paling dirasa sulit adalah menyediakan komputer, laptop atau gadget seperti android atau smartphone yang lain dan koneksi internet pada saat belajar di dalam kelas. di beberapa sekolah di jakarta, pembelajaran matematika dengan memanfaatkan khan academy mungkin dapat dilakukan. tetapi, kenyataan di lapangan mengatakan mayoritas sekolah tidak dapat melakukan pembelajaran matematika dengan memanfaatkan khan academy. b. beberapa penelitian yang relevan menurut (gönül & solano, 2013) menyatakan “khan experience does not significantly raise the exam score but does increase time in exam. a diligent or a conscientious student may achieve proficiency in more skill-sets and also spend more time completing the exam, perhaps double-checking before clicking the “submit” button”. dengan kata lain, khan academy tidak meningkatkan skor secara signifikan tetapi meningkat dalam waktu pengerjaan. dan untuk siswa yang rajin dapat meningkatkan keterampilan matematis lebih banyak lagi. iii. kesimpulan hasil observasi yang dilakukan penullis, memberikan kesimpulan antara lain bahwa aktivitas siswa yang memperoleh pembelajaran matematika dengan pemanfaatan web khan academy secara keseluruhan semakin baik setelah beberapa kali pertemuan. hal ini terlihat selama proses pembelajaran matematika, siswa terlihat lebih semangat dalam belajar matematika, lebih menyenangkan, dana adanya interaksi antar teman, eksplorasi, mengamati, serta menikmati dalam mengerjakan soal-soal di web khan academy. daftar pustaka brckalorenz, a., haeger, h., nailos, j., & rabourn, k. (2013). student perspectives on the importance and use of technology in learning. california: indiana university. cherkas, b., & welder, r. m. (2011). interactive web-based tools for learning mathematics: best practices. in a. a. al.], teaching mathematics online: emergent technologies and methodologies (p. 275). united states of america: information science reference. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 55 gönül, f. f., & solano, r. a. (2013). innovative teaching: an empirical study of computeraided instruction in quantitative business courses. journal of statistics education, 123. goos, m., & bennison, a. (2008). surveying the technology landscape: teachers’ use of technology in secondary mathematics classrooms. mathematics education research journal, 103. hirsch, c. r., martin, w. g., hopfensperger, p. w., & zbiek, r. m. (2013). core math tools and its affordances for mathematics teacher educators and for prospective teachers. amte conference (p. 4). orlando, florida: nctm. sunil, l., & saini, d. k. (2013). design of a recommender system for web based learning. world congress on engineering (p. 1). london: world congress on engineering. infinity infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 32 pembelajaran matematika menggunakan pendekatan problem posing berbasis komputer pada siswa sma kelas x oleh: 1) muhammad win afgani, 2) bagus ardi saputro, 3) jero budi darmayasa 1,2,3) mahasiswa sekolah pascasarjana universitas pendidikan indonesia 1 afgani@student.upi.edu, 2 bagusardisaputro@student.upi.edu, 3 jeromath@student.upi.edu abstrak penelitian ini bertujuan untuk mengetahui bagaimana untuk menghasilkan materi pembelajaran matematika problem possing berbasis komputer dan untuk mengetahui bagaimana kuantitas dan kualitas pertanyaan siswa dari masalah pendekatan problem possing berbasis komputer. pertanyaan dan respon siswa dikumpulkan dari 35 siswa kelas satu sma di bandung. data dianalisis dengan deskriptif dengan menggunakan rubrik leung dan taksonomi bloom. untuk menghasilkan materi problem posing berbasis komputer, pertama, guru harus memilih konsep yang diharapkan dapat mengembangkan kemampuan siswa. setelah itu, guru mencari konteks yang sesuai dengan konsep. setelah konteks yang dipilih dan cocok tersebut, guru harus memilih software untuk melakukan ide dalam bentuk yang dinamis. hasil penelitian ini menunjukkan bahwa dari 240 pertanyaan yang diberikan oleh siswa, hanya 35% yang masuk akal dan cukup masalah matematika. dari 35% pertanyaan tersebut menunjukkan bahwa 75% siswa di tingkat pemahaman berdasarkan bloom taksonomi. dari 75% siswa yang merespon tersebut menunjukkan bahwa mereka senang terhadap materi yang menggunakan pendekatan problem possing matematika berbasis komputer. kata kunci : pembelajaran matematika, problem possing, komputer abstract this study aims to know how to produce mathematics problem posing material based on computer and to know how the quantity and quality of students’ question from mathematics problem posing based on computer. students’ questions and respond is collected from 35 first grade students of senior high school in bandung. the data is analysed with descriptively by using leung’s rubric and bloom taxonomy. to produce problem posing material based on computer, first, teacher must choose a concept that wish to be gifted to students. after that, the teacher searchs a context that according to the concept. after the context is selected and match with it, the teacher must choose a software to perform the idea in dynamic form. the result of this study shows that there is 240 questions that pose by students, only 35% is plausible and sufficient mathematics problem. from 35% questions, it shows that 75% students is in understanding level based on bloom taxonomy. from the questioner, 75% students’ respond shows that they are happy toward material presentation by mathematics problem posing approach based on computer. keywords: learning mathematics. problem posing, computer. i. pendahuluan inovasi dalam pembelajaran matematika sangat dibutuhkan untuk saat ini. inovasi dapat berupa pendekatan pembelajaran ataupun penilaian hasil belajar. salah satu inovasi yang dapat dilakukan dalam pendekatan pembelajaran adalah dengan menerapkan pendekatan problem posing. polya (1994, dalam sumarmo 2015) menyatakan bahwa pengajuan masalah mailto:afgani@student.upi.edu mailto:bagusardisaputro@student.upi.edu infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 33 atau pertanyaan dalam memecahkan masalah matematik merupakan hal yang penting. problem posing adalah proses personal siswa dalam mengkonstruksi pertanyaan-pertanyaan matematika berdasarkan pengalaman matematika mereka (stoyanova & ellerton, 1996). duncker (1945, dalam sumarmo 2015) menyatakan bahwa problem posing adalah aktivitas mengkonstruksi problem baru atau masalah baru dari suatu konteks matematika atau memformulasi ulang masalah yang diberikan. dari kedua pendapat tersebut ada dua pendekatan yang mungkin dilakukan dalam pembelajaran matematika, yaitu 1) menyajikan konteks matematika atau siswa diminta untuk mengajukkan pertanyaan-pertanyaan berdasarkan pengalaman mereka, 2) mengajukkan masalah kemudian siswa membuat pertanyaan-pertanyaan baru yang mendukung proses pemecahan masalah. permasalahan yang kedua sudah dijelaskan oleh leung (1993). melalui dua pendekatan itu, guru akan menyediakan beragam materi dalam proses pembelajaran berbasis problem posing. namun, pendekatan problem posing yang diterapkan di kelas cenderung bersifat konvensional. sementara, dengan memperhatikan perkembangan teknologi informasi dan komunikasi (komputer) memungkinkan dilakukan inovasi dalam pembelajaran matematika menggunakan problem posing. inovasi yang dimaksud yaitu penggunaan berbagai aplikasi komputer dalam pembelajaran matemakomputera. inovasi itu muncul dikarenakan siswa tidak mempunyai beragam sudut pandang yang dinamis, sehingga hanya menyajikkan sedikit masalah yang baru (fukuda dan kakihana, 2009). penggunaan komputer dalam pembelajaran matematika dengan pendekatan problem posing didukung oleh brown & walter (2005) yang menyatakan bahwa komputer dapat membantu siswa mengeksplorasi dan membuat dugaan yang lebih baik serta memungkinkan komputer memberikan banyak contoh yang positif, dalam hal ini memberikan dugaan yang baru. pembelajaran matematika berbasis komputer dengan pendekatan problem posing dapat menggunakan berbagai macam software seperti geogebra, speadsheet/ microsoft office exel, maple, cabri dan lain sebagainya. materi yang didisain dalam penelitian ini menggunakan geogebra. berdasarkan uraian tersebut di atas maka akan dibahas 1) bagaimana menghasilkan materi problem posing berbasis komputer, 2) bagaimana kuantitas dan kualitas pertanyaan dari materi problem posing berbasis komputer. ii. kajian pustaka a. problem possing matematika stoyanova & ellerton (1996) mengemukakan bahwa problem posing matematika didefinisikan sebagai proses yang didasarkan pada pengalaman matematika siswa dimana dia mengkonstruk interpretasi secara individual terhadap situasi nyata dan memformulasinya sebagai permasalahan matematika yang bermakna. definisi tersebut diadopsi dalam penelitian herawati, siroj & basir (2010) yang menyatakan pembelajaran dengan pendekatan problem posing adalah pembelajaran yang menekankan pada siswa untuk membentuk/mengajukan soal berdasarkan informasi atau situasi yang diberikan. informasi yang ada diolah dalam pikiran dan setelah dipahami maka peserta didik akan bisa mengajukan pertanyaan. sedangkan haji (2011) mendefinisikan pendekatan problem posing adalah memberikan kesempatan kepada siswa untuk menyampaikan (merumuskan) suatu soal matematika yang lebih sederhana dalam rangka menyelesaikan suatu soal yang kompleks (rumit). kedua definisi tersebut semuanya sesuai dengan definisi dari penelitian susanti, sukestiyarno & sugiharti (2012) yang merujuk infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 34 definisi dari sheikhzade (2009) yang menyatakan bahwa membuat masalah dengan melibatkan dan menghasilkan masalah baru dan pertanyaan untuk mengeksplorasi tentang situasi tertentu, serta merumuskan masalah selama pemecahan masalah untuk berhubungan dengan itu. selanjutnya, susanti, sukestiyarno & sugiharti, e. (2012) dan robiyana (2013) menyatakan bahwa prestasi belajar siswa smp yang belajar dengan model/metode problem posing lebih baik dari pada prestasi belajar siswa mdengan metode konvensional. lebih khusus dari itu, menurut herawati, siroj & basir (2010) kemampuan pemahaman konsep matematis siswa pada pembelajaran problem posing lebih tinggi dari pada pembelajaran konvensional. walaupun begitu, pembelajaran problem posing lebih berpengaruh terhadap siswa berkemampuan tinggi dari pada siswa berkemampuan sedang ataupun rendah. kemudian, haji (2011) memberikan problem yang kompleks kepada siswa, kemudian mereka menggunakan pendekatan problem posing dalam menyelesaikannya. hasil penelitian menunjukkan hasil belajar matematika siswa, tingkat pemahaman soal siswa, variasi cara menyelesaikan soal siswa lebih baik dari pada teman mereka pada kelas konvensional. b. tahap pembelajaran problem posing berbasis komputer beberapa tahap pembelajaran problem posing menggunakan komputer 1) pertama, guru memberikan masalah dalam bentuk aktivitas, kemudian siswa diminta menyusun pertanyaan. pada tahap kedua, guru memvisualisasikan permasalahan dalam komputer atau siswa merekonstrusksi permasalahan menggunakan software aplikasi, kemudian siswa diminta menyusun pertanyaan kembali. guru memberikan komentar pada pertanyaan yang diajukan siswa. siswa memperbaiki pertanyaan sehingga cukup informasi dan dapat diselesaikan. selanjutnya, siswa diberi kesempatan untuk memilih dan menjawab pertanyaan yang telah dikumpulkan. penelitian yang menerapkan tahap pembelajaran ini dilakukan oleh ferrigo software yang digunakan adalah microsoft excel dan geogebra. hasil penelitiannya menunjukkan bahwa siswa memberikan respon yang baik terhadap pembelajaran. kemudian, fukuda dan kakihana melaporkan bahwa siswa memahami masalah lebih dalam dengan merekonstruksi ekspresi dalam masalah; siswa mendapatkan beragam intuisi, dan eksplorasi yang luas; siswa dapat menemukan dua atau lebih metode penyajian masalah dan untuk mengkonfirmasi penyelesaian yang diperkirakan melalui percobaan; siswa dapat mengembangkan suatu sudut pandang dinamis dimana masalah dapat diubah dalam situasi khusus; dan siswa dapat membuat aspek masalah yang diekspresikan secara simbolis atau membuat generalisasi. 2) guru memberikan masalah dalam bentuk simulasi komputer, kemudian siswa diminta menyusun pertanyaan. guru memberikan komentar pada pertanyaan yang diajukan siswa. siswa memperbaiki pertanyaan sehingga cukup informasi dan dapat diselesaikan. selanjutnya, siswa diberi kesempatan untuk memilih dan menjawab pertanyaan yang telah dikumpulkan. tahap pembelajaran ini merupakan modifikasi dari tahap pembelajaran yang pertama. 3) guru merancang permasalahan dalam komputer, kemudian siswa diminta menyusun pertanyaan. pertanyaan tersebut dihimpun dalam sistem komputer, melalui jaringan online atau offline, pertanyaan disebarluaskan ke siswa lain sehingga siswa dapat.memilih pertanyaan untuk diselesaikan. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 35 pembelajaran secara online dilakukan dalam penelitian beal & cohen (2012). mereka mengembangkan aplikasi website yang mendukung aktivitas siswa dalam belajar. aplikasi tersebut untuk memfasilitasi siswa yang mungkin baru menemukan dan membuat pertanyaan di tempat yang berbeda pada waktu diluar jam pelajaran reguler, kemudian dia ingin membaginya dengan temannya dan ingin melihat serta mencoba menjawab pertanyaan yang dibuat temannya yang relatif terbatas pada aktivitas kelas tradisional. pembelajaran secara offline dirancang oleh chang, et al. (2011). pembelajaran dibuat dalam suatu sistem komputer dimana sistem ini memberikan para siswa kesempatan untuk merefleksi problem posing yang dibuatnya dengan mengujinya sehingga meningkatkan kemampuan problem posing mereka. siswa dapat kembali berulang-ulang untuk menyelesaikan masalah dari problem posing sehingga memperbaiki skornya. melalui komentar dan umpan balik dari guru yang tersimpan dalam sistem, siswa dapat memperbaiki kemampuan problem posing-nya. studi ini menerapkan langkah-langkah pembelajaran yang kedua, tetapi hanya sampai pada siswa menyusun pertanyaan. pertanyaan-pertanyaan yang terkumpul dianalisa mengenai kualitasnya. merujuk studi yang dilakukan zakaria, et. al. (2014), ada tiga tahapan, yaitu 1) mengidentifikasi masalah apakah dapat diselesaikan, 2) mengidentifikasi konten masalah masuk dalam kategori apa, dan 3) memberikan skor berdasarkan kreatifitas siswa, sedangkan studi ini hanya sampai tahap ke-2 dan mengidentifikasi jenjang kognitif taksonomi bloom terhadap pertanyaan-pertanyaan yang diajukan siswa. (widodo & pujiastuti, 2006). iii. metode penelitian penelitian ini adalah penelitian deskriptif. adapun subjek penelitiannya yaitu siswa sma kelas x di salah satu sekolah kota bandung sebanyak 35 orang. teknik pengumpulan data menggunakan dokumentasi dan angket terbuka. data dokumentasi berupa hasil problem posing yang diajukan siswa. data tersebut dianalisis menggunakan rubrik leung (2012) dan taksonomi bloom. data angket terbuka digunakan untuk mengetahui respon siswa terhadap materi yang disajikan. kedua data tersebut dianalisis secara deskriptif. iv. hasil penelitian dan pembahasan disain materi diawali dengan memilih konsep yang ingin disampaikan, kemudian mencari konteks yang sesuai dengan konsep tersebut. setelah konteks terpilih dan bersesuaian maka dipilihlah software yang tepat untuk merepresentasikan gagasan kedalam bentuk lain yang dinamis. untuk lebih jelasnya, lihat gambar berikut: konteks software geometri mpp berbasis komputer spreadsheet aljabar statistika konsep infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 36 gambar 1. skema disain materi mpp berbasis komputer disain mpp berbasis komputer yang sudah dihasilkan disajikan dalam gambar 2 berikut ini: gambar 2. problem sehari-hari yang disajikan secara problem posing. setelah materi ini diuji coba, didapat pertanyaan–pertanyaan yang dialisis untuk melihat kuantitas dan kualitas pertanyaan tersebut. analisis merujuk kepada rubrik leung (2012). hasil yang didapat menunjukkan bahwa terdapat 240 pertanyaan yang dibuat oleh 35 siswa. 35% pertanyaan merupakan masalah matematika yang cukup informasi. permasalahan dalam kategori ini dapat digunakan untuk mengantar siswa untuk belajar materi fungsi, sedangkan sisanya merupakan pertanyaan yang tidak dapat digunakan dalam pembelajaran.\ tabel 1. contoh hasil kategorisasi problem posing yang diajukan siswa no kategori problem posing yang diajukan siswa 1 bukan suatu masalah apakah penumpang bisa mengendarai motor gojek tersebut? 2 bukan masalah matematika apakah dengan mengganti tarif harga gojek menandakan gojek sudah sukses meraup pelanggan? 3 masalah matematika yang tidak mungkin diselesaikan apa keuntungan dan kerugian untuk perusahaan gojek dan pelanggan apabila tarif gojek dihitung perkilometer? 4 masalah matematika yang tidak cukup informasi apakah kecepatan perpindahan dari satu titik ke titik selanjutnya konstan? 5 masalah matematika yang cukup informasi jika kita memiliki uang rp. 20.500,berapa kilometer yang bisa kita capai? infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 37 hasil keseluruhan kategorisasi problem posing yang diajukan siswa dapat dilihat pada diagram di bawah ini. gambar 3. diagram persentase kategorisasi problem posing siswa hasil analisis jenjang kognitif taksonomi bloom terhadap 35% pertanyaan yang merupakan masalah matematika dan cukup informasi yang diajukan siswa menunjukkan bahwa 75% masih dalam tingkat pemahaman, artinya kebanyakan siswa masih terbiasa menyelesaikan soal-soal yang bersifat pemahaman. berikut hasil keseluruhan jenjang kognitif taksonomi bloom terhadap problem posing kategori 5 yang diajukan siswa dapat dilihat pada diagram di bawah ini. gambar 4. diagram jenjang kognitif taksonomi bloom terhadap problem posing siswa kategori 5 0% 5% 10% 15% 20% 25% 30% 35% 40% kategori 1 kategori 2 kategori 3 kategori 4 kategori 5 1% 75% 0% 15% 7% 1% 0% 10% 20% 30% 40% 50% 60% 70% 80% pengetahuan pemahamaan penerapan analisis sintesis evaluasi infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 38 berikut pertanyaan siswa yang termasuk jenjang kognitif evaluasi, yaitu: “apakah semakin jauh jaraknya akan semakin mahal?” pertanyaan-pertanyaan siswa yang termasuk jenjang kognitif sintesis, yaitu: “berapa perbandingan antara jarak dan tarif?” “berapa perbandingan antara jarak max dan jarak min?” “berapa perbandingan antara tarif max dengan tarif min?” “berapa perbandingan antara tarif max dengan jarak max?” “berapa perbandingan antara tarif min dengan tarif max?” pertanyaan-pertanyaan siswa yang termasuk jenjang kognitif analisis, yaitu: “bagaimana cara membuat grafik seperti itu?” “untuk apa grafik tersebut dibuat?” “grafik ini menjelaskan tentang apa?” “bagaimana sistem tarif gojek disetiap jarak?” “berapa selisih antara tarif max dengan jarak max?” “berapa selisih antara tarif min dengan jarak min?” “apakah harga ini terus berlanjut ketika x  ? atau berubah pada satu titik?”, beberapa pertanyaan-pertanyaan siswa yang termasuk jenjang kognitif pemahaman, yaitu: “berapa tarif gojek setiap 1 km?” “6 km = 15000, 7 km = 17.500, apakah setiap 1 km = 2500?” “bagaimana tarif yang dikenakan terhadap pelanggan gojek yang menempuh jarak kurang dari satu kilometer?” “berapa kenaikan tarif setiap 1 km?” “berapa tarif gojek per kilometer?” “berapa tarif jika menempuh jarak 100 km?” “jika kita memiliki uang rp 20.500, berapa kilo kita bisa capai?” “bagaimana jika jarak tempuh (km) bukan bilangan bulat, misal 1,7 km; 2,3 km, berapa harga jasanya?” terakhir, pertanyaan siswa yang termasuk jenjang kognitif pengetahuan, yaitu: “jaraknya dihitung menggunakan apa?” dari hasil angket terbuka yang diberikan menunjukan bahwa 75% respon siswa menyatakan senang terhadap penyajian materi melalui pendekatan problem posing berbasis geogebra. berikut beberapa alasan mereka, yaitu: “saya menjadi paham dan melatih untuk berpikir kritis terhadap sesuatu” “saya bisa berperan aktif, baik dengan mengutarakan pertanyaan ataupun pendapat” “tidak membosankan” “saya dapat mengetahui sesuatu hal yang baru” “materinya beda dari yang biasanya” “saya bisa merefleksikan suatu soal terhadap keadaan nyata” “penyajian materi yang lebih interaktif” “komunikatif dan saling menjawab pertanyaan itu mengasyikan” “cara ini memancing saya bertanya” “suasananya tidak terlalu tegang” “setiap pendapat teman-teman diterima” infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 39 hasil dari item ke-2 angket menunjukkan 58% siswa tidak binggung terhadap materi yang disajikan, sisanya merasa sebaliknya. berikut beberapa alasan mereka, yaitu: “tidak mengerti tujuan sebenarnya dari materi yang disajikan” “untuk apa materi ini diberikan” “tidak mengerti dibagian harga tarif dan grafiknya” “tiba-tiba kami diberi arahan untuk membuat pertanyaan” “penjelasannya tidak rinci” hasil dari item ke-3 angket menunjukkan 67% respon siswa merasa tertantang untuk menjawab pertanyaan yang dibuatnya. berikut beberapa alasan mereka, yaitu: “supaya bisa memecahkan masalah sendiri” “saya yang membuat pertanyaan maka saya harus bisa menjawab” “rasa penasaran timbul setelah membuat pertanyaan” “saya suka pertanyaan yang dibuat” “saya mempunyai keingintahuan dan pendapat saya sendiri” “pertanyaannya sendiri menantang” “saya harus tahu jawabannya dengan bertanya/mencari” “saya dapat mengetahui sesuatu yang sebelumnya saya tidak tahu” “saya menjadi ingin berpendapat sesuai pikiran dan logika saya” hasil dari item ke-4 angket menunjukkan 78% respon siswa menunjukkan rasa tanggung jawab terhadap pertanyaan yang telah dibuat untuk diselesaikan. hasil dari item ke-5 angket menunjukkan 23% siswa menyatakan bahwa materi yang diberikan terkait dengan fungsi, 31% siswa menyatakan materi tersebut merupakan persamaan linier, sedangkan sisanya menyatakan materi lainnya. hasil dari item ke-6 angket menunjukkan 74% siswa berpendapat bahwa materi yang disajikan berbeda dari penyajian materi yang dilakukan oleh guru mereka sebelumnya. berikut beberapa alasan mereka, yaitu: “penyajian materi lebih menarik karena kita diajak untuk menganalisa terlebih dahulu” “pembelajaran kali ini lebih komunikatif dan hampir semua siswa aktif” “pada umumnya, guru-guru menyajikan presentasi/materi yang kurang interaktif” “bisanya hanya materi soal abstrak yang sulit terpikir oleh logika yang real dalam kehidupan” “yang disajikan di sini adalah masalah di kehidupan sehari-hari” dari 35% pertanyaan yang merupakan masalah matematika dan cukup informasi, pertanyaanpertanyaan tersebut belum mengarah pada pertanyaan yang generatif seperti yang dicontohkan dalam brown dan walter (2005), yaitu diantaranya: “gagasan apa yang terdapat didalam masalah ini?” “bagaimana keterhubungannya?’ “dimana saja kita dapat melihat permasalahan seperti ini?” “apakah kami cukup gagasan untuk menyelesaikan masalah tersebut?” kemudian, hasil analisis jenjang kognitif taksonomi bloom terhadap pertanyaan tersebut menunjukkan bahwa guru masih sering memberikan soal-soal pemahaman. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 40 pada penelitian ini sulit mengkategorikan problem posing yang diajukan siswa yakni membedakan apakah pertanyaan tersebut merupakan kategori 3 atau 4 terlihat dari persentase kategori 3 sebesar 2% dan kategori 4 sebesar 5%. hal ini seperti yang terjadi pada hasil penelitian yang dilakukan oleh leung (2012). hasil angket menunjukkan bahwa siswa senang dalam pembelajaran matematika yang diterapkan pendekatan problem posing dan merasa bertanggung-jawab terhadap pertanyaan yang telah dibuat untuk diselesaikan. hal ini sesuai dengan apa yang diungkapkan oleh brown and walter (1993) dan silver (1994) (dikutip dalam english, 1997), yaitu bahwa pendekatan ini memberi kesempatan seluas-luasnya bagi siswa untuk mengeksplorasi situasi masalah. suasana yang tercipta membuat pembelajaran matematika menyenangkan dan produktif. lebih khusus, problem posing dapat memberikan semangat dan menyebabkan timbulnya berpikir fleksible dan beragam, mendorong para siswa untuk bertanggung jawab atas apa yang mereka pelajari melalui pertanyaan yang mereka buat, mengantisipasi salah paham antara guru dan siswa, memperkaya konsep-konsep dasar, menghilangkan kesalahan sudut pandang dari sifat dasar matematika, dan mengurangi kecemasan dalam belajar matematika. hasil angket juga menunjukkan bahwa siswa merasa senang dikarenakan materi yang disajikan berupa situasi yang berhubungan dengan kehidupan sehari-hari dan ditampilkan pada geogebra sehingga materi yang disajikan menjadi lebih interaktif. v. kesimpulan dan saran dari hasil studi ini dapat disimpulkan bahwa 1) untuk menghasilkan materi problem posing berbasis komputer, pertama, guru harus memilih konsep yang ingin disampaikan. kemudian, guru mencari konteks yang sesuai dengan konsep tersebut. setelah konteks terpilih dan bersesuaian maka dipilihlah software yang tepat untuk merepresentasikan gagasan ke dalam bentuk lain yang dinamis, 2) dari 240 pertanyaan yang diajukan siswa, hanya 35% yang merupakan masalah matematika yang masuk akal dan cukup informasi untuk diselesaikan. dari 35% pertanyaan tersebut, menunjukkan bahwa 75% siswa masih dalam tingkat pemahaman berdasarkan taksonomi bloom. kemudian 75% respon siswa menyatakan senang terhadap penyajian materi melalui pendekatan problem posing berbasis geogebra. oleh karena itu, disain materi perlu dibatasi ruang lingkup problem atau pernyataan yang akan disajikan pada mahasiswa. bagi guru yang ingin mendesain mpp berbasis komputer, maka guru harus menguasai software yang akan digunakan dan memahami hubungan antara konteks dan konsep yang akan di disajikan. kemudian, guru dalam membuat soal sebaiknya bersifat penerapan, analisis, sintesis, dan evaluasi untuk membentuk kebiasaan berpikir siswa. daftar pustaka beal, c. r. dan cohen, p. r. 2012. teach ourselves: technology to support problem posing in the stem classroom. published online in scires. vol. 3, no. 4, 513 – 519. diakses pada tanggal 13 november 2015, dari http://www.scirp.org/journal/ce http://www.scirp.org/journal/ce infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 41 chang, et al. 2011. embedding game-based problem-solving phase into problem-posing system for mathematics learning. journal computers & education. diakses pada tanggal 13 november 2015, dari www.elsevier.com/locate/compedu bonotto, cinzia. 2012. artifacts as sources for problem-posing activities. springer science+business media dordrecht. brown, s. i. dan walter, m. i., 2005. the art of problem posing, 3 rd ed. new jersey: lawrence erlbaum associates, inc. english, l. d. 1997. promoting a problem-posing classroom. proquest education journals. united states: nctm. ferrigo, m. problem posing in mathematical education. department of mathematics. pisa university. diakses pada tanggal 13 november 2015, dari http://www.dm.unipi.it/~georgiev/club/progects/dynamat/public/d10_disseminat ions/pisa/13_it_nitra_ferrigo.pdf fukuda, c., & kakihana, k. (2009). problem posing and its environment with technology. in proceeding of 33rd conference of japan society for science education. haji, s. 2011. pendekatan problem posing dalam pembelajaran matematika di sekolah dasar. triadik, 14(1), 55-63. herawati, o. d. p., siroj, r. a., & basir, m. d. 2010. pengaruh pembelajaran problem posing terhadap kemampuan pemahaman konsep matematika siswa kelas xi ipa sma negeri 6 palembang. jurnal pendidikan matematika, 4(1), 70-80. leung, s-k. s. 2012. teachers implementing mathematical problem posing in the classroom: challenges and strategies. springer science+business media b. v. robiyana, a. 2013. eksperimentasi pembelajaran matematika menggunakan model problem posing materi segitiga kelas vii semester genap smp negeri 16 purworejo tahun pelajaran 2011/2012. ekuivalen-pendidikan matematika, 3(1). stoyanova & ellerton, 1996. a framework for research into student’s problem posing in school mathematics. diakses pada tanggal 13 november 2015, dari http:// www.merga.net.au/documents/rp_stoyanova _ellerton _1996.pdf susanti, e. l., sukestiyarno, y. l., & sugiharti, e. 2012. efektivitas pembelajaran matematika dengan metode problem posing berbasis pendidikan karakter. unnes journal of mathematics education, 1(1). sumarmo, u. 2015. mathematical problem posing: rasional, pengertian, pembelajaran, dan pengukurannya. pascasarjana stkip siliwangi bandung dan pascasarjana upi. diakses pada tanggal 13 november 2015, dari http://utarisumarmo.dosen.stkipsiliwangi.ac.id/files/2015/09/problem-posing-matematikpengertian-dan-rasional-2015.pdf zakaria, et. al. 2012. teachers’ creativity in posing statistical problems from discrete data. proquest education journal, vol. 3, issue. 8. issn: 21514755 widodo, a., & pujiastuti, s. 2006. profil pertanyaan guru dan siswa dalam pembelajaran sains. jurnal pendidikan dan pembelajaran, 4(2), 139-148. http://www.elsevier.com/locate/compedu http://www.dm.unipi.it/~georgiev/club/progects/dynamat/public/d10_disseminations/pisa/13_it_nitra_ferrigo.pdf http://www.dm.unipi.it/~georgiev/club/progects/dynamat/public/d10_disseminations/pisa/13_it_nitra_ferrigo.pdf http://?/?www.merga.net.au/?documents/rp_stoyanova%20_ellerton%20_1996.pdf http://?/?www.merga.net.au/?documents/rp_stoyanova%20_ellerton%20_1996.pdf http://utari-sumarmo.dosen.stkipsiliwangi.ac.id/files/2015/09/problem-posing-matematik-pengertian-dan-rasional-2015.pdf http://utari-sumarmo.dosen.stkipsiliwangi.ac.id/files/2015/09/problem-posing-matematik-pengertian-dan-rasional-2015.pdf http://utari-sumarmo.dosen.stkipsiliwangi.ac.id/files/2015/09/problem-posing-matematik-pengertian-dan-rasional-2015.pdf sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p109-116 109 problem solving in the context of computational thinking *1 swasti maharani, 2 muhammad noor kholid, 3 lingga nico pradana, 4 toto nusantara 1,4 universitas negeri malang 2 universitas muhammadiyah surakarta 1,3 universitas pgri madiun article info abstract article history: received dec 16, 2018 revised may 15, 2019 accepted sept 2, 2019 computational thinking is needed in the 21st century, where we live in an era of digitalization. also, there is a global movement to incorporate computational thinking into the education curriculum, especially mathematics education. the different of this research with others is this research compares the polya problem solving and computational thinking. this research was conducted to find out how the relationship/relationship of the polya problem-solving with the steps of computational thinking. the method used in this research is descriptive qualitative. the subject of this study was mathematics education students. the results showed that the relationship between problem-solving and computational thinking of respondent when solving the problem is when defining the problem in the context of problem-solving, the respondent performs the stage of decomposition and abstraction in the context of computational thinking. during the planning process of the solution process, respondents carried out the generalization stage. when the scene is carrying out the plan and the problem solver to look back to evaluate the solution, the respondent performs the debugging and algorithmic steps. keywords: computational thinking, graph, problem solving, polya, mathematics education copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: swasti maharani, department of mathematics education, universitas pgri madiun, jl. setiabudi no.85, kanigoro, kota madiun, jawa timur 63118, indonesia email: swasti.mathedu@unipma.ac.id how to cite: maharani, s., kholid, m. n., pradana, l. n., & nusantara, t. (2019). problem solving in the context of computational thinking. infinity, 8(2), 109-116. 1. introduction in the current era of globalization in the 21st century, digital technology plays an important role in everyday life. in response to the increasing demand to compete in the global economy, countries need to prepare students with appropriate technical knowledge and communication skills to compete (tsai & tsai, 2017). combining knowledge and technology is a solution to a problem that will become a trend (voskoglou & buckley, 2012). one step in dealing with this is to include computational thinking into the curriculum (bower, wood, howe, & lister, 2017; weintrop et al., 2016; voogt, fisser, good, mishra, & yadav, 2015; geary, saults, liu, & hoard, 2000). however, this has not been done in indonesia. maharani, kholid, pradana, & nusantara, problem solving in the context … 110 computational thinking is a basic ability for students in education, which is the same basis as the ability to read, write and arithmetic calculations (zhong, wang, chen, & li, 2016; hu, 2011). learning by using computational thinking, as a basic skill throughout the school curriculum, will enable students to learn abstract thinking, algorithmic and logical, as well as ready to solve complex and open problems. supported by adler & kim (2017) who said that honing computational thinking would be beneficial in education, and beneficial for their future. computational thinking is everyone's basic ability to learn which is an important preparation for the future to educate young people with computational thinking. activity-based learning strategies are strategies to help young people's cognitive growth, and can guide their learning effectively through manipulation and real expressions. (cho & lee, 2017). computational thinking is considered an important competency because students currently not only work in fields affected by computing, but also need to face computing in their daily lives and in today's global economy (bower et al., 2017; grover & pea, 2013). one of the subjects in the school curriculum is mathematics, so it does not rule out the possibility that applying computational thinking in mathematics can improve students' conceptual mathematics. mathematics requires learning activities that provide direct experience to encourage problem solving skills (sung, ahn, & black, 2017). computational thinking and learning mathematics have reciprocal relationships, using computing to enrich mathematics and science learning, and apply the context of mathematics and science to enrich computational learning (weintrop et al., 2016). the main motivation for introducing the practice of ct (computational thinking) into the mathematics classroom is in response to increasingly computerized disciplines because they are practiced in the professional world (acharya, 2016). mathematical ability is considered a core factor that predicts students' ability to learn (grover & pea, 2013). some researchers put forward convincing arguments that mathematical thinking plays an important role in ct (gadanidis, 2017; rambally, 2017; son & lee, 2016) because solving math problems is a construction process (benakli, kostadinov, satyanarayana, & singh, 2017; lockwood, dejarnette, asay, & thomas, 2016; merle, 2016). the construction process to complete this solution requires an analytical perspective to solve unique and fundamental problems for students. based on the results of previous studies, computational thinking can improve the mastery of material number sense and arithmetic abilities (hartnett, 2015) which is influenced by thinking style, academic success and attitude towards mathematics (durak & saritepeci, 2017). in addition, computational thinking can also be influenced by the level of class and the duration of ownership of mobile technology (korucu, gencturk, & gundogdu, 2017). cognitive habits that can assist in the development of computational thinking are spatial reasoning and intelligence (ambrosio, almeida, macedo, & franco, 2014; yasar, maliekal, veronesi, & little, 2017). problems have an important role in mathematics. most of the learning in school is designed in such a way based on mathematical problems (reiss & törner, 2007). during this time in solving students' math problems more on solving the problem. solving the problems that are often reviewed are steps from polya including the problem identification stage, planning problems, implementing the plan, and checking the answers (reiss & törner, 2007). in addition, computational thinking also has a role in solving mathematics, so it needs to be revealed how to solve mathematical problems in the context of computational thinking. volume 8, no 2, september 2019, pp. 109-116 111 2. method this research is a qualitative descriptive research with the respondent is 30 of mathematics education students at universitas negeri malang. the characteristics of the subject is mathematics education students who have been finished graph subject. the instrument used is one math problem consisting of problem solving question. the technique used in the determination of the respondent is the method of random sampling, because this research want to know the relationship of polya problem solving and computational thinking. all of students do the polya problem solving on solve the mathematics problem. there are five stages in this study. first, giving problem solving question to respondent and asking the respondent to do it. the question is “map can be easily represented by graph. a country symbolized by a vertex and edge (line between two vertexes) describes two neighboring countries on graph. the picture below represents a map into the graph. specify an appropriate map for the given graph!” (figure 1) figure 1. question the second stage, observing. researchers recorded directly respondent and also by recording directly any activity of research respondents when solving problem solving question based on the observation sheet to classify the tendency of computational thinking. observations focused on behavioral trends in performing computational thinking during problem solving task. the third stage, analyzing the components of computational thinking that appear on respondent of research based on the results of direct observation. the results of the analysis in the form of conclusions about the behavior of research respondents whether the responden to do computational thinking or not. fourth, perform triangulation of data to confirm the results of the analysis is the conclusion whether the responden to do computational thinking or not by conducting an in-depth interview (in-deep interview). interview guidelines used are with a structured and open format. in addition to interviews, there is also a data reduction stage that is not required after in-depth interviews. finally, summarizes the results of the analysis of the components of computational thinking of prospective mathematics teachers based on the results of observation and interviews so that data can be obtained by a computational thinking of the students of mathematics education in solving the problem. the results obtained at the last stage is the classification of computational thinking of prospective mathematics teachers when solving the problem. the indicator of computational thinking when solving the problem can be viewed on the table 1. maharani, kholid, pradana, & nusantara, problem solving in the context … 112 table 1. indicator of computational thinking when solving the problem the component of computational thinking students activity abstraction students can decide on an object to use or reject, can be interpreted to separate important information from information that is not used generalization the ability to formulated a solution into general form so that can be applied to different problems, can be interpreted as the use of variables in resolving solutions decomposition the ability to break complex problems into simpler ones that are easier to understand and solve algorithmic the ability to design step by step an operation/action how the problems are solved debugging the ability to identify, dispose of, and correct errors 3. results and discussion 3.1. results the results show that the students can solve the problem with computational thinking components. the responden need about five minutes to read the problem. responden known that map 1 and map 2 didn’t correct answer. this step responden did decomposition step, because responden break the map from 4 into 2 maps. if in this stage of problem solving, enter the stage define the problem. the respondent draws vertices and edges according to the problem, gives a symbol on each vertex, separating any letters connected to two countries, three countries and so on. this process/stage can be called the abstraction stage because the respondent can separate important information that can be used. the way to do that is by identifying each map (map 3 and 4) which is in accordance with the graph drawing that he made earlier. this stage can be said to be the generalization stage, because the respondent can make a general form which in this context is a graph on a question that has been given a symbol. this process can be called planning the use of strategies in problem solving. when working on, the respondent realizes that there is an error he made that there is a writing error that is "connected" is replaced with "neighbor". this stage can be called the debugging stage, because the respondent corrects the error. in working on map 3, the respondent draws a map and gives a symbol to each country, the same as repeat as in the initial example. after finding the answer, namely map 3. then the respondent checks the map 4. the respondent draws map 4 and gives the symbol the same as the previous way. fear of map 4 is also true because questions are not multiple choice questions. at this stage, it can be said that respondents carried out an algorithmic stage. when viewed from the side of problem solving this stage enters the implementation phase of the plan / problem solving strategy while checking the answers. the answer of responden can be viewed on the figure 2. volume 8, no 2, september 2019, pp. 109-116 113 figure 2. answer one of respondent figure 2 show that computational thinking student on solving mathematics problem especially graph. first, the student did the abstraction, then decomposition, debugging, generalization, and the last algorithmic. 3.2. discussion previous studies discussed about what it might mean and what we might do about computational thinking (hu, 2011), problem solving in the mathematics classroom in germany (voskoglou & buckley, 2012; voskoglou, 2013) implications for teacher knowledge in k-6 computational thinking curriculum framework (angeli et al., 2016), the possibility of improving computational thinking through activity based learning (cho & lee, 2017), a framework of curriculum design for computational thinking development in k-12 education (kong, 2016). the level of participants' computational thinking skills differed significantly in terms of their grade level, not significantly different in terms of their gender (korucu et al., 2017). the steps taken by respondents are first decomposition, abstraction, generalization, debugging and algorithmic. these steps do not match the order of computational thinking indicators. this is in line with the results of research conducted by voskoglou & buckley (2012) which states that the sequence of problem solving steps seen from computational thinking does not have to be in order. when performing the decomposition and abstraction stage, the respondent understands the problem by reading the questions carefully for five minutes, and determining that maps 1 and 2 do not fulfill the reasons. this means that respondents have understood what was asked about the problem and identified the reasonable parts (reiss & törner, 2007). next step, the generalization stage of the respondent can make a general form which in this context is a graph on the question that has been given a symbol. at this stage if viewed in terms of problem solving can enter the planning process of the solution stage because the respondent tries to make his own formula to complete, it is a strategy to solve. the respondent identifying auxiliary problems, changing the formulation, or checking the relevance of the data (reiss & törner, 2007). respondents did debugging while the work takes place, before doing algorithmic respondents have debugged. then the respondent performs an algorithmic process that is completing map 3 and map 4 according to the general form that was made earlier. in this process the respondent also checked the map 4 even though he had found an answer maharani, kholid, pradana, & nusantara, problem solving in the context … 114 namely map 3. in this case the respondent did the stage carrying out the plan and the problem solver to look back and to evaluate the solution. this means to check every single part of the solution and to make sure (or, preferably, to prove) that it is correct and to show that it is correct and all arguments are valid (reiss & törner, 2007). in general, computational thinking is a problem solving who not only on information technology but on mathematics education too. students who use the computational thinking on solving the mathematics problem would be easy to solve other mathematics problems. 4. conclusion the relationship between problem-solving and computational thinking of respondent when solving the problem is when defining the problem in the context of problem-solving, the respondent performs the stage of decomposition and abstraction in the context of computational thinking. during the planning process of the solution process, respondents carried out the generalization stage. when the scene is carrying out the plan and the problem solver to look back to evaluate the solution, the respondent performs the debugging and algorithmic steps. computational thinking supported students to solve the mathematics problem. the development of computational thinking was needed for future research that will be affected to learning especially mathematics learning. for example, the assessment of computational thinking, the characteristic of computational thinking, the expansive of each component of computational thinking and others. acknowledgements we want to thank the anonymous referees for their constructive criticism and many helpful comments and suggestions, which undoubtedly improved the quality of the paper. thank you to the ministry of research, technology, and the higher education republic of indonesia. 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(2016). an exploration of three-dimensional integrated assessment for computational thinking. journal of educational computing research, 53(4), 562–590. https://doi.org/10.1007/s10758-017-9328-x https://doi.org/10.1007/s10758-017-9328-x https://doi.org/10.1007/s10758-017-9328-x https://doi.org/10.1007/s10209-017-0542-z https://doi.org/10.1007/s10209-017-0542-z https://doi.org/10.1007/s10209-017-0542-z https://doi.org/10.1007/s10639-015-9412-6 https://doi.org/10.1007/s10639-015-9412-6 https://doi.org/10.1007/s10639-015-9412-6 https://arxiv.org/abs/1212.0750 https://arxiv.org/abs/1212.0750 http://www.ecsjournal.org/archive/volume37/issue1/8.pdf http://www.ecsjournal.org/archive/volume37/issue1/8.pdf https://doi.org/10.1007/s10956-015-9581-5 https://doi.org/10.1007/s10956-015-9581-5 https://doi.org/10.1007/s10956-015-9581-5 https://www.asee.org/public/conferences/78/papers/17618/view https://www.asee.org/public/conferences/78/papers/17618/view https://doi.org/10.1177/0735633115608444 https://doi.org/10.1177/0735633115608444 https://doi.org/10.1177/0735633115608444 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p103-110 103 mathematical representation of grade 7 students in set theory topics through problem-based learning ida lestari* 1 , nila kesumawati 2 , yunika lestaria ningsih 3 1,2,3 universitas pgri palembang article info abstract article history: received nov 25, 2019 revised jan 29, 2020 accepted feb 6, 2020 set theory has a wide role in mathematical concepts. students have to understand the set theory before learning other concepts such as algebra and probability. this study aims to determine the effect of the problem-based learning (pbl) model on the students’ mathematical representation in set theory topics. the method used in this study is a quasi-experiment design. the populations in this study were 289 students of 7th grade at secondary school in palembang. the sample of this study were students of class 7.8 (control group) and 7.10 (experimental group). data were collected through tests, interviews, and documentation. based on data analysis, known that pbl affects the students’ mathematical representation. students who had the pbl model get the better score of mathematical representation. they could use the symbol of set correctly, represent the set into venn diagram correctly and they also could explain their answer. furthermore, the implementation of the pbl model is offered. keywords: mathematical representation, problem-based learning, set theory copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: ida lestari, departement of mathematics education, universitas pgri palembang jl. jenderal a. yani lrg. gotong royong 9-10 ulu, palembang 30116, indonesia. email: idalestariida21@gmail.com how to cite: lestari, i., kesumawati, n., & ningsih, y. l. (2020). mathematical representation of grade 7 students in set theory topics through problem-based learning. infinity, 9(1), 103-110. 1. introduction set theory is an elementary concept that is crucial in mathematics. learning mathematical concepts need knowing the set theory (razmjooei, 2013). additionally, dogan-dunlap (2006) said that it’s the prerequisite concepts for algebra topics. “a set is defined in terms of certain properties shared by its elements. these properties must be well described, with no ambiguities, so that it is always clear whether a given element belongs to a given set or not” (andre, 2014). but in fact, many students have to struggle in understanding this concept. pinker (zazkis & gunn, 1997) mentioned that they felt difficult in learning the concept of set. signs and symbols have a wide role in the concept of set. bagni (2006) explained the representation of set, as follow : (1) verbal: the definitions of set, element, subset, union, etc, (2) symbolic: the set symbol (capital letter), the set operation symbols, the mailto:idalestariida21@gmail.com lestari, kesumawati, & ningsih, mathematical representation of grade 7 students. … 104 brackets, etc, and (3) visual: the presenting of set in venn diagram. these set representations are known as mathematical representations. the mathematical representation is a capability that has to master by students in the learning of mathematics. it is based on one of the goals of mathematics learning by the national council of teacher mathematics (nctm). according to nctm mathematical representation can be defined as the ability to restate the notation, symbols, tables, figures, charts, diagrams, equations or another mathematical expression into another form (johar & lubis, 2018). it is associated with the student's understanding of mathematical concepts (bolden, barmby, raine, & gardner, 2015). students who have good mathematical representation will easier in understanding the mathematical concepts. however, despite the crucial mathematical representation as to the goal learning of mathematics, students still lack this capability. previous studies said that many students still have a low level of mathematical representation (hernawati, 2016; hutagoal, 2013; noto, hartono, & sundawan, 2016). to improve the students’ mathematical representation, researchers chosen the problem based learning (pbl) model. according to delisle (happy & widjajanti, 2014), pbl has several advantages such as students are encouraged to have the ability to solve problems in real life, students can build their knowledge through learning activities, learning focuses on problems, scientific activities occur students at workgroups, students are accustomed to using various sources of knowledge, and student difficulties can be overcome through discussion activities. the pbl model presents contextual problems so that students need analytical skills to solve these problems (o'brien, wallach, & mash-duncan, 2011). pbl is a bridge that connects theory and real-world application in a more environmentally friendly and familiar that enables students to acquire practical skills (roh, 2003). according to previous research, it is known that pbl is effective in improving students’ mathematical performance (abdullah, tarmizi, & abu, 2010). it also has a positive impact on students’ in learning mathematics (ahamad, li, shahrill, & prahmana, 2017; botty, shahrill, jaidin, li, & chong, 2016). due to the importance of mathematical representation in learning mathematics (especially in learning set theory), this study will analyze the effect of the pbl model on the students’ mathematical representation. the purpose of this study is to determine the effect of the model pbl on the mathematical representation of secondary school students in learning elementary set theory. 2. method this study is used posttest-only control group design. the design of the study can be drawn as follow: r x o1 r x o2 r : randomly selected groups. o1 : posttest in the group that was given treatment. o2 : posttest in the group that was not given treatment. x : the treatment given to the experimental class using the pbl model the populations in this study were students of 7th-grade students at secondary school in palembang, south sumatera, indonesia. the samples were 27 students of 7.10 class as an experimental group (eg) and 27 students of 7.8 class as the control group (cg). volume 9, no 1, february 2020, pp. 103-110 105 both classes are equal in mathematics performance. data collected through tests, interviews, and documentation. the test is used to measure the students’ mathematical representation that consists of 5 questions. the example of the problem can be seen in figure 1. figure 1. the example of problem in postest the result of the test is analyzed based on mathematical representation scoring (table 1) developed by cai, s.jakabcsin, & lane (1996). table 1. the rubric scoring for mathematical representations score visual symbolic verbal 0 there is no answer, even if there is only a lack of understanding of the concept so that the information provided does not mean anything. 1 only a few of the drawings, diagrams are correct. only a few of the mathematical models are correct. only a few of the explanations are correct. 2 draw diagrams, pictures, but not complete and correct. find the mathematical model correctly, but wrong in getting a solution. the mathematical explanation makes sense but only partially complete and correct. 3 paint, diagram, picture, in full, but there are still a few mistakes. find the model correctly, then do the calculation or get the right solution but there are a few mistakes writing symbols. the mathematical explanation makes sense and is correct, even though it is not arranged logically or there are few language errors. 4 paint, diagram, picture, completely and correctly. find the mathematical model correctly, then do a calculation or get a solution correctly and completely. the mathematical explanation makes sense and is clear and logically arranged. data on students’ mathematical representation then analyze with statistical inference. the prerequisite statistics testing then conducted. the interviews are held after the posttest. the interview results then analyzed descriptively. 3. results and discussion 3.1. results this research was conducted in august 2019. the pbl model implemented in the experiment group (eg), consists of 27 students. meanwhile, the control group (cg) got lestari, kesumawati, & ningsih, mathematical representation of grade 7 students. … 106 expository learning. the pbl model was design based on lee & bae (ahamad et al., 2017) as follow: step 1: introduction, understand the problem and searching for information students are introduced to understand and analyze the set problem in the form of students’ worksheets (see figure 2). the worksheet also containing the use of signs and symbols in set theory. step 2: construct and gather solution students are work collaboratively to find out the solution. teachers controlling the students' work, and guiding the students who have troubles (see figure 3a). step 3: presentation and reflection students presented their findings in front of the class (see figure 3b), the teacher checked their work and ask students to do some exercise. figure 2. the example of students’ pbl worksheet (a) (b) figure 3. students work collaboratively and presented their findings. volume 9, no 1, february 2020, pp. 103-110 107 the learning process was lasting in three meetings, and the posttest conducted in the fourth meeting. the statistical description of students’ test results in mathematical representation (eg and cg) can be seen in table 2. table 2. descriptive statistics of students posttest eg cg ̅ 80.00 71.85 s 7.34 7.23 max 95 90 min 65 60 the result of the prerequisite test for hypothesis testing can be seen in table 3 (normality test) and table 4 (homogeneity test). table 3. the result of normality test groups n k-s sig conclusion eg 27 0.710 0.694 normal cg 27 0.814 0.521 normal table 4. the result of homogeneity test groups n f sig conclusion eg and cg 54 0.029 0.866 homogeneous the hypothesis used in this research is the pbl model affects on students’ mathematical representation. it means that there are differences mean between the two groups. hypothesis testing is using the parametric statistics-t. the result of the t-test can be seen in table 5. table 5. the result of the t-test groups n t sig conclusion eg 27 4.111 0.000 different cg 27 there are three aspects of mathematical representation in this study. the posttest is consists of 5 questions. questions number 1 is examined verbal representation, symbol representation in question number 2 and 3, and the visual representation in question number 4 and 5. furthermore, the score of every aspect of mathematical representation can be seen in table 6. table 6. students’ mathematical representation of eg and cg aspects of mathematical representation eg cg verbal representation; the ability to describe the problem in the form of written words. 88.89 81.48 symbol representation; the ability to describe the problem in the form of model / mathematical notation. 70.37 57.40 lestari, kesumawati, & ningsih, mathematical representation of grade 7 students. … 108 aspects of mathematical representation eg cg visual representation; the ability to describe the problem in the form of pictures / graphics. 85.18 81.48 based on table 6, it is known that the lowest score both in eg and cg was in the aspect of symbol representation. interviews conducted to the chosen students to explore this finding. some students stated that there are so many symbols in learning set theory, they difficult to remember and explain. they did not realize that the number of elements of set a is n(a) even though they know the number. they usually wrote the number and calculated it. 3.2. discussion based on the result, it is known that pbl affects on students’ mathematical representation in set theory topics. it is mean that pbl gives impact to students in learning mathematics. this finding is in a line with the previous studies (abdullah et al., 2010; ahamad et al., 2017; botty et al., 2016;). the first step of pbl is introducing students to the problem. then students begin to understand the problem, searching the information to solve the problem. this step is to encourage students to discuss and sharing their idea to get the information (ahamad et al., 2017). the proposed problem in pbl helps students to think critically (abdullah et al., 2010; padmavathy & mareesh, 2013). after getting the information to solve the problem, students continue the activity to solve it. they working together, discussing, and struggling to complete their worksheet. this step is helping students in understanding the meaning of signs and symbols of set theory, and they also learn how to represent the set problem into a correct diagram venn. this step is useful to improve their mathematical representation. based on tabel 6, known that the average score of symbol representation students in eg is higher than cg one. the example of students’ posttest results can be seen in figure 4. (a) students’ work at cg (b) students’ work at eg figure 4. students’ result for test of symbol representation based on figure 4, both students can solve the given problem. however, students in eg show better performance in using signs and symbols. they can define the set from the problem. they make a correct notation for the element of the set, and the calculation is also correct. volume 9, no 1, february 2020, pp. 103-110 109 4. conclusion this study has shown that the pbl model affects on students’ mathematical representation in learning the set theory topics. the t-test result reported that there is a difference mean between pbl and the expository model. after the pbl conducted, students in eg could write the problem into the notation of set. they make correct signs and symbols in set operation to solve the problem, they also have a good visual representation by forming the set into a correct diagram venn. it can conclude that students who have a pbl model can develop their mathematical representation. this study is limited to elementary set theory topics, further study can expand for other topics. the symbol representation is still the hardest thing for students to learn. suggestions for others study to analyze and improve this ability. references abdullah, n. i., tarmizi, r. a., & abu, r. 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(2020). watson’s categories analysis of sequences and series question. infinity, 9(1), 31-40. 1. introduction children are provided with mathematics as early as possible from kindergarten to high school and even college. mathematics are able to solve problems (ratnasari, rosita, & pramuditya, 2017). through line and line material learning, students are expected to be able to think at a high level to solve a problem. the results of observations showed that in the row and series material there were still many students who made mistakes solving the problems. in general, students feel confused if the test is given a slightly different problem when studying in class. stude nts feel difficulties in the use of formulas between arithmetic and geometry. one of the problems is determining the middle term of arithmetic lines 6, 9, ..., 36. one student answer can be seen in figure 1. mailto:anggi3007@yahoo.co.id maharani, wahyuni, & oktavianingsih, watson’s categories analysis of sequences and series … 32 figure 1. work student result the problem in figure 1 students are asked to find the number of middle tribes, students must first look for ut. one student's solution to the problem shows that students are still wrong in answering it. students have a problem distinguishing what is the n-th term and middle term, so the formula or procedure used by students in solving the problem is wrong. errors in solving problems are interpreted as deviations from the answers that have been determined truth (farida, 2015; putri, 2014). hudjono said that to solve or obtain, certain rules or laws are needed (irfan, 2015). the problems are situation when the student has obstacles in their learning process (widodo, hidayanti, & gunawan, 2019) while an important part of learning mathematics is solving problems (astutik & nuriyatin, 2016). many problems will slightly affect the problems faced by students. test questions in the form of description (essay) provide opportunities for students to decipher the answers in accordance with the knowledge they have. students cannot guess the answers so that problems often arise in the form of essays. through students' answers in the form of a description (essay), the teacher will be easier to analyze where the location of student errors in solving a given problem. the results of the analysis can be considered improvements in subsequent learning, for example in the use of teaching media. the problems that arise in the education system in indonesia are very complex and a common mistake made by students in solving mathematical problems is due to a lack of understanding of concepts and inaccuracy in counting (perbowo & anjarwati, 2017; rahayu, 2019). watson and asikin revealed these errors and categorized based on watson's category, namely: a) inappropriate data (id); b) inappropriate procedure (ip); c) omitted data (od); d) omitted conclusion (oc); e) response level conflict (rlc); f) undirected manipulation (um); g) skill hierarchy problem (shp); h) above other (asikin, 2003; kasana & khotimah, 2019). watson is a pure behaviorist. he used pavlov's discovery as a basis for his learning theory. his study of learning is aligned with other sciences that are oriented solely on empirical experience, that is, as far as it can be observed and measured (nahar, 2016). first categories is inappropriate data (id). in this case, students use data that is not quite right in other words incorrectly entering variable values; second categories is inappropriate procedure (ip). students use procedures or methods that are not appropriate, for example using formulas in an inappropriate way; third categories is oomitted data (od). missing one or more data from students' responses, thus the solution becomes incorrect; fourth categories is omitted conclusion (oc). students show reasons at the right level and then fail to conclude. in solving problems students have not yet reached the final stage of what was asked in the problem; fifth ctegories is response level conflict (rlc). students do not understand the form of questions, so what is done is to do simple operations with existing data and then made the final result in a way that is not in accordance with the volume 9, no 1, february 2020, pp. 31-40 33 actual concept, or students just simply write the answer without any reason or logical way; sixth ctegories is undirected manipulation (um). there is a completion of the process of changing from one stage to the next there is something illogical; seventh ctegories is the skill hierarchy problem (shp). students cannot solve problems because of a lack of or not visible skills. for example, students can change the basic formula into the requested formula; and last categories is above other (ao). apart from the seven categories of errors that have been explained above, there is an eighth category where students do not respond to questions. for example, students do not answer the problem at all, or students only write questions back. 2. method this research is a descriptive study with a qualitative approach involving 18 students high schools of a social class in the city of cirebon, indonesia. data collection methods include tests and interviews. qualitative data analysis was carried out in three stages namely data reduction, data presentation, and conclusion drawing. the test questions consist of four problem questions that had been validated in advance by researchers using anates software. the indicators used in the problem are counting the number of terms of the middle term of an arithmetic sequence, calculating the n-th term of an arithmetic sequence, counting the number of n first terms of an arithmetic series, and counting the number of the first five terms of a geometric sequence. 3. results and discussion some research on the watson’s category analysis has been carried out (kristayulita & nurhardiani, 2011; munawaroh, rohaeti, & aripin, 2018) but they did not discuss the watson category in the row and series material for social students. here is the result analysis of student answers accompanied by an interview. 3.1. student answer the following are examples of students' answers to each question problem number 1 if arithmetic ranks 6.9, .... 36. determine how many middle terms! here is one example of student answers: figure 2. work student result for number 1 in this problem, students are asked to find the number of middle syllables. figure 2 appears that students are still using the formula incorrectly. students have not been able to distinguish what is an n-th term (un) and what is the middle term (ut), it also appears that students are still wrong in entering data n, which should be n itself is t. student mistakes in maharani, wahyuni, & oktavianingsih, watson’s categories analysis of sequences and series … 34 answering question number 1 based on the watson category are then analyzed and poured into the following figure 3. figure 3. error results of problem number 1 figure 3 shows that the categories of errors made by students varied. there are 18 students who fall into the incorrect procedure category and 13 students who make mistakes in the inference category are missing, for the incorrect procedure category students are still wrong in distinguishing between ut and un. from the answers, there are some students who are still wrong in using the formula because they are still using the un formula in solving problems even though the question asked in the problem is how many middle terms are not the third term. students are still wrong in using the formula, there are also some students who make mistakes in the category of missing conclusions, namely in solving problems students have not reached the final stage requested in the problem because when determining the student's formula was wrong. the next category is the indirect manipulation category and the problem of skill hierarchy consists of 10 students. in the category of indirect manipulation, students use illogical reasons in solving problems, whereas in the hierarchy category students' skills are still wrong in operating addition and multiplication, when summing and multiplication operations are found students still don't understand which one should be operated first, namely in operation 6 +3 (36-1) some students operate first adding up between 6 + 3 = 9 then continuing to operate (36-1) = 35 and then the results of both times multiplied by 9 (35) = 315 should students first operate contained in brackets and then prioritizing multiplication and finally the sum is 6 + 3 (361) = 6 + 3 (35) = 6 + 105 = 165. problem number 2 an arithmetic sequence with u_6 = 11 and u_10 = 23. the 11th term is ..... here is one example of student answers: figure 4. work student result for number 2 0 10 20 id ip od oc rlc um shp ao problem number 1 volume 9, no 1, february 2020, pp. 31-40 35 in problem number 2 students are asked to find the value of u_11, seen in figure 4. the answer of one of the students obtained that the student is right in using the formula, the steps taken by the student are correct, that is finding the value of b first then looking for the value of a and new students can look for the value of u_11. the students' answers become wrong because students make changes in the illogical stage when operating the value of a, so that the conclusion becomes wrong. student mistakes in answering question number 2 based on the watson category are then analyzed and presented in the following figure 5. figure 5. error results of problem number 2 figure 5 it seems very prominent that the ip category errors were made by all students. the mistakes were made by students because students were not quite right in carrying out the completion steps. some students are right in using the formula that must be used but the procedure in solving problems is not right for students. students make a mistake at stage a + 15 = 11 they write the results, namely a = 15-11 which produces a value a = 4, there are also those who answer a = 11-15 the result is a = -4, students should answer in the way a + 15 = 11 by means of the two segments reduced by 15 to eliminate the value of 15 in the left section, namely a + 15-15 = 11-15, producing a = 11-15 produces the value a = -4. problem number 3 a bookstore in the first month sold 50 books, in the second month there were 65 and added 10 more each month. how many books did the store sell for six months? here is one example of student answers: figure 6. work student result for number 3 in question number 3 students are asked to look for the number of books sold for 6 months, namely s_6. figure 6 it appears that students' answers are not up to looking for 0 10 20 id ip od oc rlc um shp ao problem number 2 maharani, wahyuni, & oktavianingsih, watson’s categories analysis of sequences and series … 36 s_6, but only to the stage of looking for u_6. student mistakes in answering question number 3 based on the watson category are then analyzed and poured into the following figure 7. figure 7. error results of problem number 3 figure 7 it can be seen the error category is oc or the conclusion is missing where some students are still wrong in using the formula. the category error in the conclusion is missing frequently occurs in problem number 3 because only work up to the stage of finding the value of un. so that students make a mistake in the category oc that is not reaching the final conclusion requested for the problem because in that problem asking students to find the value of sn not the value of un. problem number 4 the third and fifth rows of the geometry are 64 and 4. if the ratio of the sequences is positive, specify s_5! here is one example of student answers: figure 8. work student result for number 4 based on problem number 4, students are asked to look for s_5 in a geometric sequence. figure 8 shows that students do not understand the concept of geometry because in these answers’ students have not written the formula to be used. students arrive at the stage of finding the value of r. student mistakes in answering question number 4 based on the watson category are then analyzed and poured into the following figure 9. 0 5 10 id ip od oc rlc um shp ao problem number 3 volume 9, no 1, february 2020, pp. 31-40 37 figure 9. error results of problem number 4 figure 9 show the students who make mistakes. the most prominent is the type of procedure is not right and the type of conclusion is lost. most students have found the value of r but stopped because they did not know the next procedure to do in answering the questions given. so, students still don't seem to understand the material in this problem. students cannot perform procedures correctly. students also make mistakes of type oc, ie conclusions are lost, students have not reached the final stage requested for the problem because students also do not perform procedures correctly ie do not know the stage to be done next, the last there 2 students who don't do the questions. 3.2. interview result the following are the results of interviews based on the results of students' answers that have been identified in accordance with the watson categories. inappropriate data/id based on the results of interviews conducted with s11 students, the researchers found that in the incorrect data category, they were still confused and did not understand the sequence and series question. the student could not distinguish between term n and term the middle. when entering data into student variables is still wrong, because the students are wrong in determining the formula. so, students are confused to enter a value in the variable. inappropriate procedure/ ip based on the results of interviews conducted with s12 students, researchers found that in the category of procedures that were not right. students working on material and row problems were still wrong in using the formula. students were still confused about what formula to use. another factor was because students did not like math lessons because students still consider math lessons difficult. even students do not pay attention to the teacher when mathematics begins. omitted data/od based on the results of interviews conducted with s13 students, researchers found that in the category of missing data errors. students lacked focus on working on math problems. students also had negative thoughts in the first place which considered mathematics difficult and could make a headache. 0 5 10 15 20 id ip od oc rlc um shp ao problem number 4 maharani, wahyuni, & oktavianingsih, watson’s categories analysis of sequences and series … 38 omitted conclusion (oc) based on the results of interviews conducted with s14 students, the researchers found that in the category of conclusion was lost. students in working on the row and series questions were still confused about what steps to do next. another factor was that students lacked training in doing math problems because they were lazy in counting. response level conflict (rlc) based on the results of interviews conducted with s8 students. researchers found that in the category of conflict the level of response made by students was caused because students were still having difficulty in operating complicated forms. students immediately guessed the answers because they considered it complicated to continue counting. undirected manipulation error (um) the results of interviews conducted with s5 students, researchers found that the category of manipulation was not directly carried out by students because students were not careful in solving problems, and students were still confused when faced with things like moving positions, confused whether the sign of the operation changed or not. skill hierarchy problem (shp) the results of the interviews conducted with s6 students. researchers found that in the category of skills the hierarchy was carried out by students. students rushed in working on the questions and because at the time of the interview students were aware of these mistakes. other category error the results of the interviews conducted with s4 students. researchers found that in other categories students did not answer the questions or did not respond to problems at all. students were hesitant in taking steps to work on the problems and did not like math lessons so that when finding difficult questions, it would be lazy to think. percentage of error type percentage of types of student errors based on the number of questions and the watson category, appear in the following table 1. table 1. percentage of error types question id ip od oc rlc um shp ao 1 16,67% 30% 1,67% 21,67% 5% 16,67% 6,67% 1,67% 2 2,38% 42,86% 0% 4,76% 2,38% 33,3% 14,28% 0% 3 7,41% 29,63% 3,7% 33,3% 14,8% 3,7% 7,4% 0% 4 1,89% 33,96% 5,66% 24,52% 26,4% 9,43% 9,43% 3,77% whole 7,69% 34,06% 2,75% 20,33% 7,69% 16,48% 9,34% 1,65% volume 9, no 1, february 2020, pp. 31-40 39 table 1 for question number 1 it appears that ip and oc errors are the most mistakes made by students by 30% and 21.67%, respectively. students are still mistaking in determining the formula and there are still many students who do not understand. so, the oc error also occurs because at first. the students do not know what formula to use. the students also do not reach the conclusion stage. for the second number, the most mistakes made by students are the um type, which is indirect manipulation. students are still wrong in changing from one stage to the next stage which is 33.3%. the error is because students are still confused about manipulating because of a lack of practice in working on math problems. the third number, about the number of nth-term students. there are still many errors of ip type, which is an incorrect procedure of 29.63%. students are still wrong in making the right completion steps. 33, 3% of students have not reached the final stage that was asked for the problem even though the initial steps used were correct but there were students stopping when completing the questions including. students forgot the formula, forgot about the steps to be taken next and lacked time in answering. the last for number four, there are also the same errors as other questions. namely errors in the ip category as much as 33.96%. this shows that students cannot understand the material in the problem. students do not understand the material. students are wrong in the procedure which is used which results in an error type oc that is a missing conclusion also occurs the same error made by the students is 26.4%. students are not able to complete the answers requested on the problem. the percentage of all errors made by students are found in the category of improper procedures and missing conclusions. students do not understand the material being taught. if students have made mistakes in the category of incorrect procedures, some students will also make mistakes in the category of conclusions lost. 4. conclusion judging from the overall type of error based on the watson category there are two categories of errors that are predominantly committed by students, namely the category of incorrect procedures and the category of inferences lost. based on the results of the study, some solutions that can be done by the teacher to minimize student errors in answering sequence and series questions are (1) develop teaching materials that are followed by guided answers; (2) the teacher should provide more opportunities for students to ask and answer questions on the board; (3) apperception that is less will adversely affect the success of students in the next material. acknowledgements the authors would like to thank all participants involved in this research. we would also like to thank the head master of sman 7 cirebon for supported, giving the opportunity and facilities for this research. references asikin, m. 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(2019). designing camtasia software assisted learning media toward students’ mathematical comprehension in numeral. infinity, 8(2), 143-156. 1. introduction mathematical comprehension is one ability that need to have to learn mathematic. that ability is an important foundation for thinking in solving mathematical and real-life problems. besides that mathematical understanding ability is very supportive for the development of other mathematical abilities such as communication skills, problem solving, reasoning, connection and other mathematical abilities. the problems that require understanding of mathematical concepts have not been optimally mastered by students, such as the weak understanding of one concept with other concepts needed to solve of mathematical problems (kariadinata, yaniawati, sugilar, & riyandani, 2019). students mailto:setiyani_0401509081@yahoo.com setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 144 must have ability mathematical comprehension as mathematical understanding is a musthave ability and should be developed in each students self. the ability of students' mathematical understanding can be measured based on the indicators. while its level can be seen based on completeness on indicators of mathematical understanding. but based on the results of observations conducted by researchers there are still indications that students do not have a level of understanding in accordance with the current level of thinking, as experienced by students in mtsn 2 cirebon in the subject matter numbers. the source states that students are still not maximal in understanding the concept of addition to fractions as shown in tab 1. figure 1. student exercise result from the operation steps to add the fraction number is correct, that is to look for the smallest guild multiplication (sgm) to equate the denominator, but when determining the numerator 34% of students are still confused and caused the students’ answer to be incorrect. students still do not master the concept of multiplication in fractions in operating the multiplication of fractions by simply multiplying the numerator with the numerator and denominator with the denominator. but from 25% of students’ answer above use the steps of the operation of addition or subtraction is by looking for the sgm and equating the denominator. that is what caused student answers less precise as shown in figure 2. figure 2. students who have not mastered the concept of multiplication in fractions. students' lack of understanding of the daily problems given related to the subject matter number with the question "andi is an amateur diver, first practiced diving at a depth of 3 meters uder sea level. after feeling swift diving at a depth of 3 meters, then andi decided to dive again 7 meters under the sea level. what is the difference in depth in the two conditions, present using a number line! ". student answers as shown in figure 3. volume 8, no 2, september 2019, pp. 143-156 145 figure 3. student who do not understand the mathematical concepts related to indicators change the form of representation to other form. figure 3 shows students trying to answer using a number line. but what can be seen in the answers of 47% of students shows that they lack understanding in using number lines and on the operation, hence students' answers are less precise. students also have not been able to identify the properties of a concept related to material numbers with the question "are commutative and associative properties also valid for integer reduction operations. if yes, show it. if not, explain with an example of a denominator? " student answers are shown in figure 4. figure 4. students who have not been able to identify the properties of a concept figure 4 shows the answers of students in proving the properties of a concept in material numbers. in proving the nature of a concept, it must be understood first what communicative properties are and how to prove them. the communicative nature is where a + b = b + a, but what is requested is in the reduction, then the fixed answer "communicative nature does not apply to subtraction, because the result is definitely not the same" so that the answers of 22% of students above prove a lack of communicative right. based on the answers above, it shows that there is still minimal of students’ mathematical comprehension ability in mtsn 2 cirebon. if this fact is ignored, as time goes by this issue can decrease students’ interest in learning if, because they feel that mathematics is difficult. mathematical comprehension ability is the initial ability that must be given as early as possible to students. based on the results of interviews with several students in vii grade at mts 2 cirebon, the researcher found some difficulties that students faced when studying material numbers. 20% of students stated that they found it difficult when they encountered questions about proving the properties of numbers, as well as the operation of negative summation. in the learning process students express that they are more interest of learning with media and some exercises. based on the results of the interview above, the learning process requires learning media as teaching aids in delivering material. learning materials is very important for teachers and students. it have a major contribution to the success of the learning process is implemented (noto, pramuditya, & fiqri, 2018). criticos state that media is one component of communication, namely as a messenger from the communicator to the communicant (daryanto, 2013). if communication does not go well, the message conveyed by the teacher is difficult for students to understand. conversely, if communication is setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 146 effective and efficient, more and more learning objectives are achieved. the use of learning media in the teaching process can make students better understand and learn according to their interests and abilities (kintoko & sujadi, 2015). the word media comes from latin is the plural form of the word medium boundary about the notion of media is very broad, but we limit it to the world of education only, namely the media used as tools and materials for learning activities. learning media is very important to be used by teachers in the teaching and learning process. learning media provide motivation to students in mathematics. with learning media students are more enthusiastic in learning in the classroom. of course the learning media needs to be created interestingly and designed practically, these aims to provide students with interest and interest in following the teaching and learning process in the classroom. media learning cd can be used as an alternative to convey messages by the message giver (teacher) to the recipient (student). the use of the learning cd media allows students not only to study at school, but also to learn by themselves at home. learning cds have characteristics such as attractive shapes and colours that can increase the activity of students to learn and clarify a concept. so with the use of this media, the learning process will feel more interesting and not boring and will stimulate students to learn mathematics. by using the learning cd, the contents of the subject matter can be modified using various applications such as camtasia studio and microsoft power point which can be combined into a more interesting unified learning media. so that it can help teachers in delivering subject matter and can help students understand mathematical concepts. the development of interactive learning cd assisted by camtasia studio software on integer material has been studied before. the feasibility test results are very good and the results of student trials using interactive learning cd using the problem based learning model assisted by camtasia studio software on integer material are better than conventional learning in class vii of the 2nd middle school in the 2015/2016 (prasetyawan, 2017). based on the description of the background above, the author wants to find out how camtasia software design helps learning media in students' mathematical understanding ability in numeral material? is the design of learning media assisted by camtasia software on the ability of students' mathematical understanding of material numbers that have been developed practice? 2. method this study uses the r & d (reserch and development) method with the addie model. to compile various systems, both formal and non-formal systems of instructional systems that are often used, namely addie. the addie model is one of the most commonly used models in the field of instructional design to produce effective designs (aldoobie, 2015). the addie model consists of five stages, namely: analysis, design, development, implementation, and evaluation. the complete stages are shown in figure 5 (branch, 2009). volume 8, no 2, september 2019, pp. 143-156 147 figure 5. model addie the explanation of the addie model which consists of five stages is as follows: the analysis stage consists of two stages, namely needs analysis and identification of needs. at this stage, analysis is carried out to determine the learning needs of students, by conducting a needs analysis, identifying problems, and performing task analysis. analysis must include learning characteristics, motivation, technological abilities, and learning objectives (wang & hsu, 2009). the design stage is the stage after learning needs are identified then selecting media and then designing learning media. this stage teaching and learning activities are designed. the activities carried out are formulating competencies determining learning materials, strategies, evaluations, sources and media makers. the development stage is in the form of making or producing or realizing a product specification learning that has been determined by the design resistance. in this phase the researcher builds and develops with the help of media software and supporting documentation (muruganantham, 2015). besides that at this phase also began to make examples of real products and good designs (aldoobie, 2015). after the product is produced the next step is validating the product by experts. the implementation stage is the stage of utilizing or using learning products in learning activities. activities that need to be prepared include preparing classrooms, schedules, tools and media, preparing students physically and mentally. the purpose of this phase is to convey or promote product results to students on the material and learning objectives (muruganantham, 2015). the evaluation stage is a process to see whether the learning system that is being built is successful, in accordance with initial expectations or not. to find out this, an evaluation of the product has been made with the aim to find out whether students can immediately understand the material or competencies taught in class. the results of this evaluation are used to provide responsiveness or opinion to the product maker. this addie model provides an opportunity to evaluate activities at each stage. the modified research flow that the author did can be seen in figure 6. analiysis impementation development design evaluation revision revision revision revision setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 148 figure 6. addie research flow 3. results and discussion learning media design carried out by researchers has some stages, namely analysis, design, development, implementation and evaluation which will be presented by the researcher as follows. 3.1. analysis phase analysis is the initial stage carried out by researchers in the process of making learning cds. this stage is used to find out the learning needs and difficulties experienced by students during the learning process in material numbers. this stage consists of two activities namely needs analysis and identification of needs. the two activities will be presented as follows: 3.1.1. needs analysis needs analysis is a way of knowing student learning difficulties and what are needed by students in the teaching and learning process. at this stage the researcher analyzes the needs of students by conducting interviews and giving questions about the test of mathematical abilities in the material numbers. interviews were conducted to the mathematics teachers and some seventh grade students at mts negeri 2 cirebon to find out the opinions of teachers about the difficulties experienced by students and how teachers overcome learning difficulties and know the opinions of students regarding the teaching and learning process in popular classes and difficulties experienced by students. student test results on number material show that students are still not maximal in understanding the concept of addition of fractions, the concept of multiplying fractions, solving problems in daily life, and not being able to identify the properties of a concept in numbers material. from the students’ interview found some difficulties that students faced analysis phase needs analysis give questions about testing to students conduct interviews with students and teachers identification of needs design phase media selection learning media design development phase learning media development learning media validation valid ? instructional media implementation of learning media implementation phase evaluation phase evaluation of learning media information : : flow of activities : cycle line revision volume 8, no 2, september 2019, pp. 143-156 149 when learning material numbers. 20% of students stated that they found it difficult when they encountered questions about proving the properties of numbers, as well as the operation of negative additions. in the learning process students reveal that students are more interest to learn learning media and lots of practice questions. while the results of the interview with teachers found some difficulties experienced by students when studying material numbers. mathematics teachers at mts negeri 2 cirebon stated that students were still having difficulty in learning number material, it can be seen from the learning evaluation carried out by the teacher there was still many students who got scores under minimum completion criteria. the way to overcome this is by giving practice questions that contain comprehension skills, where comprehension ability is the initial ability that must be possessed by students. the learning process still uses lectures and does not use media in learning related to material numbers. 3.1.2. identification of needs at this stage, it is identifying the needs that have been obtained from the results of the trial questions and the results of interviews with students and mathematics teachers. from the needs analysis, several important points are obtained: students are still not maximal in understanding material numbers, the level of students' mathematical understanding ability is still not maximal related to material numbers, learning media have not use in the learning process, and students are more excited about learning with media and lots of practice questions. the results of identification of these needs are used by researchers as a reference in designing learning media for the learning process of students in the classroom. 3.2. design phase design is the second stage carried out by researchers. based on the identification of needs researchers will design or create learning media that appropriate to the needs and overcome the difficulties experienced by students. at this stage includes activities: collect material references in this activity the researcher makes material by reference to various sources or reference books. beside compeling material, the researcher also determines the sub-chapter of the material that will be discussed in the learning media created. based on the results of the analysis phase, the researcher took the sub chapter of integer operating material and the operation of fraction numbers. the material included in the learning media has been adjusted in terms of language and structure to make it easier for students in the learning process. the language used in the preparation of the material is to use polite language according to eyd (enhanced spelling), but still easy to understand by students. as explained in the analysis phase, that in the learning process at mtsn 2 cirebon class vii related to material numbers has not used learning media and only with lectures. this makes students bored in learning, so researchers are interested in creating learning media that aim to attract interest and provide learning enthusiasm for students in the form of learning videos assisted by camtasia software and produce products in the form of learning cd. in this activity the researcher designed learning media with the help of camtasia software. the integer material is designed in this stage assisted by camtasia software makes it easier for students to understand each step of the problem solving. besides that this application can be installed via an android phone, so students can study anywhere and anytime. next is a presentation on the design of learning media design. making a camtasia video in the initial stages starts with entering the title of the video. select the menu to enter the text as shown in figure 7. setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 150 figure 7. to write text from this step, the researcher will start assembling and designing operating material with a round and fraction operation. the following page is for entering material text in the camtasia video setup (figure 8). figure 8. includes material text in compiling the flow of activities in the camtasia video researchers also pay attention to the material, examples of questions and practice questions must be in accordance with the initial objectives of making learning media assisted by camtasia software towards mathematical understanding skills. the following are the design results that the researchers made, namely the video camtasia with integer operating material and fractional number operations on mathematical comprehension abilities for class vii junior / mts students. the following is the display made by the researcher: 3.2.1. making slides in making the initial slide, the researchers made several slides. the slide is the opening, an explanation of ki (core competence) and kd (basic competence), an explanation of mathematical indicators, indicators of mathematical understanding and learning objectives. the purpose of the slides is to make the teacher or students loyal to the media know the goals that must be achieved by the students and the contents of the video. 3.2.2. making material slides in making the material slide the researcher is include integer operating material and fractional number operations in the camtasia video. the content of the material is in the form of an explanation of each sub chapter of the material number. the material that will be volume 8, no 2, september 2019, pp. 143-156 151 presented are integer addition operations, integer reduction operations, integer multiplication operations, integer division operations, fraction operations with fractions, fractional reduction operations, fractional multiplication operations, and fraction division operations. 3.2.3. making sample slides of questions along with discussion and understanding ability test exercises in making examples of questions, the researcher encloses discussion and problem training. it is intended that students understand from each example of the problem given. the placement of each sample slide is given directly after an explanation of each material. 3.2.4. making the final slide this slide contains the closing of the camtasia video by providing motivation for students, and the biodata of the maker 3.3. development phase this stage is the third stage carried out by researchers. at this stage the researcher embodies the design into a product in the form of a cd learning. to create the product there is a development where the results of the video learning design in the camtasia software are converted into mp4 by clicking produce and share then select hd so that the quality is good then next as shown in figure 9. figure 9. changing to mp4 after mp4 learning videos are created, the next step is burning the mp4 video into a cd / dvd that has been prepared to become a learning cd. when it has turned into a learning cd, it is then continued to the learning cd validation stage that has been made. the goal is to find out whether the product is valid or not. in the validation stage this learning cd was validated by three mathematics lecturers and one mathematics teacher. the last step in making a valid learning cd is to test the validity level of the learning cd. this stage aim to find out whether the learning cd is worthy of use or not. validation used is an assessment in the form of a questionnaire. the validator consists of three mathematics lecturers at swadaya gunung jati university and one mathematics teacher at mtsn 2 cirebon. setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 152 this validation is done to find out whether the product created by the researcher is valid or not. based on the results of validation that each validator provides a good assessment of the learning cd produced. this is the result of the overall assessment which is quite high with the acquisition of results about 62 to 72 from a maximum score of 80. the next step that must be done is to analyze the data. data analysis will be divided into two stages, namely the analysis of validation data for each validator and overall data analysis. recapitulation of analysis calculations from each validator as shown in table 1. table 1. recapitulation of each validator validator score achieved expected maximum score overall validation criteria (%) v-1 68 80 85 v-2 62 77.5 v-3 72 90 v-4 70 87.5 average 85 based on the calculation of the analysis results from each validator, it has been obtained for v-1 = 85% (very valid), v-2 = 77.5% (quite valid), v-3 = 90% (very valid), and v-4 = 87.5% (very valid). this shows that there is one validator who gives an evaluation with sufficiently valid criteria and there are three more validators who provide very valid criteria. so based on these assessments the resulting learning cd can be used. overall validation then the overall analysis calculation is done to find out that this learning cd is valid or not. the following formula will be used. based on the results of these calculations, obtained for v_combined = 85%. furthermore, the calculation results are interpreted with the following criteria (table 2). table 2. validation criteria validation criteria level of validation 85% ≤ v ≤ 100% very valid or can be used without revisions 70% ≤ v < 85% it is quite valid or can be used but needs to be small revisions 50% ≤ v < 70% less valid, it is recommended not to be used because it needs major revisions v < 50% invalid or may not be used based on the validation criteria, the assessment of the validator is v_combined = 85% included in very valid criteria. 3.4. implementation phase the implementation phase is the fourth stage. at this stage the researcher carried out the implementation of the product created in the form of a mathematics learning cd at volume 8, no 2, september 2019, pp. 143-156 153 mts 2 cirebon. the implementation of a mathematics learning cd is using the talking stick model with a scientific approach. in the core learning process, the researcher explained the material about summing operations and reduction by round using learning media in the form of videos. after explaining the researcher also gave an example of a question along with explaining how to solve the sample problem and the student observing then giving the question what the students did not understand. when the question and answer process between researchers and students is completed and students understand what is conveyed by the researcher, then the researcher gives questions to students and prepares music and sticks. students do exercises on the questions given by the researcher individually. the researcher does a talking stick game to test students' mathematical abilities. for students who get a section to discuss the questions the researcher cooperates with the answers that have been obtained. after the game is finished the students record all the answers that have been answered with the talking stick game. at the end of the implementation the researcher gave a brief explanation and gave conclusions and offered to students if they wanted to have a learning video, they could copy paste or share it to pc / handphone. before implementation, the researcher gave a practical questionnaire to students which aimed to find out the practicality of the learning cd. 3.5. evaluation phase the evaluation phase is the fifth stage. at this stage the researcher asks for input from the teacher and the students that aimd to find out whether the learning media in the form of video learning cd, can overcome student learning difficulties in material numbers and realize the desired needs of students in the learning process. judging from the students' answers and the input of teachers and students that with the use of learning cds in the learning process in the classroom can overcome students' learning difficulties in material numbers and give enthusiasm to students in the learning process. this shows that by making learning media in the form of learning vide in cds can overcome the difficulties of students in understanding material numbers and realizing the needs of students in the learning process. after five stages, the researcher then processes the practical data that is filled in by the students at the implementation stage. the data is in the form of quantitative data. practicality is used to find out the products produced, namely learning media in the form of mathematical learning cds in the form of videos that are practical or not used in the learning process in mathematics subjects with sub chapter numbers in addition, subtraction, multiplication, integer division and fraction numbers. based on table 3, it appears that each assessor gives a good assessment of the learning cd produced. this is the result of a high overall assessment with yields around 26 to 31 from a maximum score of 32. the next step that must be done is to analyze practical data. it aims to find out the scores obtained from each assessor so that they can formulate practical criteria or not. recapitulation of analytical calculations from each assessor as shown in table 3. table 3. recapitulation of analysis calculations from each assessor assesor (p) score achieved overall assessment criteria (%) p-1 29 90 p-2 31 96 p-3 30 93 p-4 28 87 setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 154 assesor (p) score achieved overall assessment criteria (%) p-5 28 87 p-6 29 90 p-7 27 84 p-8 26 81 p-9 27 84 average 97.66 expected maximum score = 32 tabel 3 shows that the assessment of all students with criteria is very practical. as the result based on the assessment the learning cd produced is practical. mathematics learning is done very much. one of the learning media development approaches with rectangle-based geographic post problems. after learning is done using this media, this media can facilitate students in asking about the characteristics of getting a quadrilateral, facilitating students to learn the relationship between the types of building rectangles that have the same nature, and provide opportunities for teachers to conduct discussions about mathematical communication students when asking and writing (saputro, 2016). the results of the research carried out have the advantages of the research conducted by latif, darmawijoyo, & putri (2013) which produced a product from the camtasia software in the form of videos that were learned through edmodo and the learning process needed the internet, different from what the researchers produced. learning process does not require an internet connection.the disadvantage of learning media that uses camtasia software is not the availability menu equation. thus it was developed on mathematical subjects that use formulas difficult to use. 4. conclusion making learning media with software aided by camtasia on the ability of students' mathematical understanding of material numbers that have been completed, this is proved by each example and practice about the problem of the ability to access understanding. and in the learning process using learning media consisting of learning cds that make students more enthusiastic in learning and more practical for students because the created learning cd contains videos that can be used anywhere both on pc / hanphone, so students can learn everytime students want. assessments of products produced by researchers by experts indicate that learning is made very valid, with a presentation of 85%. it can be deny that learning cds can be used by students. practicality of learning cd media conducted by nine students consisting of three high abilities, three moderate abilities, and three low abilities. based on the results of nine seventh grade students at mts negeri 2 cirebon who are capable of high, medium and low shows that the learning cd produced is very practical, with presentations of 97.66%. it is important to state that the learning cd is very practical for learning class vii students in material numbers. suggestions that researchers can convey based on the results of research in the framework of making mathematics learning cds are firstly in the process of implementing media learning cds in the form of videos do not use a projector (infocus), so each student must have the video. secondly, in evaluating the questions there needs to be a more diverse question additionhe conclusion should contain the confirmation of the problem that has volume 8, no 2, september 2019, pp. 143-156 155 been analyzed in result and discussion section. the conclusion should contain the confirmation of the problem that has been analyzed in result and discussion section. references aldoobie, n. (2015). addie model. american international journal of contemporary research, 5(6), 68-72. branch, r. m. (2009). instructional design: the addie approach (vol. 722). springer science & business media. daryanto, d. (2013). media pembelajaran: perannya sangat penting dalam mencapai tujuan pembelajaran. yogyakarta: penerbit gava media. kariadinata, r., yaniawati, r. p., sugilar, h., & riyandani, d. (2019). learning motivation and mathematical understanding of students of islamic junior high school through active knowledge sharing strategy. infinity journal, 8(1), 31-42. kintoko, k., & sujadi, i. (2015). pengembangan media pembelajaran matematika berbantuan komputer dengan lectora authoring tools pada materi bangun ruang sisi datar kelas viii smp/mts. jurnal pembelajaran matematika, 3(2). latif, y., darmawijoyo, d., & putri, r. i. i. (2013). pengembangan bahan ajar berbantuan camtasia pada pokok bahasan lingkaran melalui edmodo untuk siswa mts. kreano, jurnal matematika kreatif-inovatif, 4(2), 105-114. muruganantham, g. (2015). developing of e-content package by using addie model. international journal of applied research, 1(3), 52-54. noto, m. s., pramuditya, s. a., & fiqri, y. m. (2018). design of learning materials on limit function based mathematical understanding. infinity journal, 7(1), 61-68. prasetyawan, a. (2017). pengembangan cd pembelajaran interaktif dengan menggunakan model problem based learning berbantuan software camtasia studio pada materi bilangan bulat. aksioma: jurnal matematika dan pendidikan matematika, 7(1), 26-35. saputro, b. a. (2016). learning media development approach with a rectangle problem posing based geogebra. infinity journal, 5(2), 121-130. wang, s. k., & hsu, h. y. (2008). using addie model to design second life activities for online learners. in e-learn: world conference on e-learning in corporate, government, healthcare, and higher education, 2045-2050. https://books.google.co.id/books?hl=id&lr=&id=mhswjpe099ec&oi=fnd&pg=pr3&dq=branch,+r.+m.+(2009)+internasional+desain+the+addie+approach.+london+:+springer+is+part+of+springer+science+and+business+media.&ots=jo-pbazx4z&sig=xqggeg41igcyiqph68pay1busgk&redir_esc=y#v=onepage&q&f=false https://books.google.co.id/books?hl=id&lr=&id=mhswjpe099ec&oi=fnd&pg=pr3&dq=branch,+r.+m.+(2009)+internasional+desain+the+addie+approach.+london+:+springer+is+part+of+springer+science+and+business+media.&ots=jo-pbazx4z&sig=xqggeg41igcyiqph68pay1busgk&redir_esc=y#v=onepage&q&f=false http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/925 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/925 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/925 http://www.jurnal.fkip.uns.ac.id/index.php/s2math/article/view/5728 http://www.jurnal.fkip.uns.ac.id/index.php/s2math/article/view/5728 http://www.jurnal.fkip.uns.ac.id/index.php/s2math/article/view/5728 https://journal.unnes.ac.id/nju/index.php/kreano/article/view/2937 https://journal.unnes.ac.id/nju/index.php/kreano/article/view/2937 https://journal.unnes.ac.id/nju/index.php/kreano/article/view/2937 http://www.allresearchjournal.com/vol1issue3/partb/pdf/67.1.pdf http://www.allresearchjournal.com/vol1issue3/partb/pdf/67.1.pdf http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/530 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/530 http://journal.upgris.ac.id/index.php/aksioma/article/view/1407 http://journal.upgris.ac.id/index.php/aksioma/article/view/1407 http://journal.upgris.ac.id/index.php/aksioma/article/view/1407 http://journal.upgris.ac.id/index.php/aksioma/article/view/1407 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/218 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/218 https://www.learntechlib.org/p/29946/ https://www.learntechlib.org/p/29946/ https://www.learntechlib.org/p/29946/ setiyani, putri, & prakarsa, designing camtasia software assisted learning media … 156 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p93-102 93 advocacy approach with open-ended problems to mathematical creative thinking ability ibrahim *1 , sri adi widodo 3 1 universitas islam negeri sunan kalijaga yogyakarta 2 universitas sarjanawiyata tamansiswa yogyakarta article info abstract article history: received oct 9, 2019 revised jan 31, 2020 accepted feb 5, 2020 the purpose of this study is to find out the increase in students' ability to think creatively in advocacy learning by using open-ended problems. this type of research is an experiment with nonequivalent control group design. the sample in this study were 72 students taken using random sampling techniques. the variables in this study are learning models, mathematical creative thinking abilities, and general mathematics abilities. the instruments used in this study were creative thinking tests and general mathematics tests. data analysis techniques used in this study are statistical inference using the mann-whitney test and one-way anova. the results showed that students who were treated with an advocacy approach by presenting open-ended problems improved their mathematical creative thinking abilities better when compared with conventional learning. keywords: advocacy approach, creative thinking ability, general mathematics ability, open-ended copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: ibrahim, departement of mathematics education, universitas islam negeri sunan kalijaga, jl. laksda adisucipto, caturtunggal, sleman, yogyakarta 55281, indonesia. email: ibrahim@uin-suka.ac.id how to cite: ibrahim, i., & widodo, s. a. (2020). advocacy approach with open-ended problems to mathematical creative thinking ability. infinity, 9(1), 93-102. 1. introduction creative thinking processes are rarely trained in the general education process (ariandari, 2015; susilo, 2012). the ability to think creatively is needed by students in solving mathematical problems (asriningsih, 2014; noer, 2011; saefudin, 2012; simbolon, surya, & syahputra, 2017; siswono, 2005; sumarmo, hidayat, zukarnaen, hamidah, & sariningsih, 2012). the results of previous studies indicate that the ability of students to understand problems and plan a solution is still very low (pardimin, widodo, & purwaningsih, 2018; siswono, 2005, 2010; widodo, darhim, & ikhwanudin, 2018; widodo, turmudi, & dahlan, 2019). understanding a problem is shown by knowing what is known and what is asked (pardimin & widodo, 2017; siswono, 2005; widodo et al., 2018), planning for solving a problem is shown by organizing information or data creatively by using certain strategies to find possible solutions (siswono, 2010). understanding and planning problem solving requires an adequate creative thinking ability mailto:ibrahim@uin ibrahim & widodo, advocacy approach with open-ended problems to mathematical … 94 of students because these abilities are high-level and critical thinking skills (krulik & reys, 1980; krulik & rudnick, 1999; posamentier & krulik, 2009; siswono, 2011). to measure the ability to think creatively, in fact, the government, through the ministry of education and culture, has used it on standardized questions i.e national exam questions (istianah, 2013; widana et al., 2019; wulandari, 2014). related to this, teachers need to train students to have adequate creative thinking skills, especially in learning mathematics. one that can be used by teachers to improve the ability to think creatively by using learning models that can support students' thinking power to be creative (noer, 2011). this is because learning provides opportunities for student activity to construct their own knowledge, in general can increase the flexibility of mathematical skills and improving creative skills. thus, it is thought to lead to learning that can develop mathematical creative thinking skills that must depart from learning that makes students active. in learning that prepares students active, students are given the freedom to think and question again what they receive from their teacher. therefore, there needs to be an effort to search for and seriously apply the results of research on mathematics learning approaches, which can actively involve students in the classroom and be able to improve students' creative thinking abilities in learning mathematics. the ability to think creatively is possible to be enhanced by learning (hasratuddin, 2010; ismaimuza, 2013; noer, 2011; siswono, 2005; soviawati, 2011; sumarmo et al., 2012). hasratudin (2010) and soviawati (2011) research results show that in general realistic mathematics learning can improve students' creative thinking abilities. the results of research conducted by ismaimuza (2013) on junior high school students in palu obtained that students' creative thinking abilities by using problem-based learning with cognitive conflict are better when compared to using conventional learning. the same thing was expressed by siswono (2005), which stated that with learning in the form of problem submission, students' creative thinking skills could increase. one learning that can be used to improve the ability to think creatively including learning with an advocacy approach (nurhasanah & julyanti, 2019; suhartono & patma, 2018; tandililing, 2013). an advocacy approach is an approach that seeks to enable students to be actively involved in the process of learning mathematics in class. student activity is manifested in proposing the resolution of a mathematical problem given by the teacher through a process of debate. by actively involving students in the debate process, it is hoped that students' mathematical creative thinking skills will continue to be well trained through debate learning that debate engages students. the advocacy approach is commonly used in social science learning and can provide opportunities for students to discuss social problems or issues and personal problems through direct involvement and personal participation in the debate process (suhartono & patma, 2018). however, judging from its characteristics, an advocacy approach does not rule out the possibility to be used in mathematics learning (nurhasanah & julyanti, 2019). this is because this learning can invite students to think creatively in the learning process, so it has an opportunity to improve students' mathematical creative thinking itself (mustikasari, zulkardi, & nyimas, 2010). based on this, the purpose of this study was to determine the increase in students' ability to think creatively in advocacy learning by using open-ended problems. 2. method this study uses an experimental method with a nonequivalent control group design (creswell, 2012a, 2012b). this is because the researchers tested the application of an volume 9, no 1, february 2020, pp. 93-102 95 advocacy approach by presenting open-ended problems in learning mathematics in schools. this study involved two classes that were made in two categories of sample groups, namely the experimental group that was given the treatment of an advocacy approach by presenting open-ended problems in mathematics learning, and the control group that was given the treatment of conventional mathematics learning, namely learning mathematics that is often done by teachers. the subjects in this study were 72 students who were divided into two classes. where the two classes used as research subjects were taken using random cluster sampling. this sampling technique is done randomly based on classes that have been formed not based on students. the instrument used in this study was a test of mathematical creative thinking ability. this test is given before and after treatment to students who are subjected to research. mathematical creative thinking ability test consists of 7 items in the form of description. indicators of mathematical creative thinking ability tests include fluency, flexibility, authenticity, and elaboration in dealing with the mathematical situation it faces (siswono, 2004, 2005, 2010, 2011). the data obtained in this study are data about learning models, students' creative thinking abilities, and general mathematical abilities. for general mathematical abilities are divided into 3 groups: upper, middle, and lower groups. the data processed in this study is normalized gain (n-gain) data in the form of a percentage. first, a descriptive statistical analysis is performed, calculating the mean, variance, and standard deviation of each group of data, so that a general picture can be obtained. second, statistical inference analysis is performed by applying one-way anova statistics by considering assumptions that must be met, such as normality assumptions and homogeneity assumptions of variance (stahle & wold, 1989). if these assumptions cannot be fulfilled then statistical inference analysis uses the mann-whitney test (glass, peckham, & sanders, 1972; liliefors, 1967; martin & games, 1977). 3. results and discussion 3.1. results table 1 is the average results and variants of the mathematical creative thinking ability test scores from the experimental and control groups. table 1. mathematical creative thinking ability test data group pretes postes n-gain (%) mean variance mean variance mean variance experiment 1.15 5.15 13.47 70.39 37.07 527.62 control 1.43 4.33 9.71 34.81 24.94 252.81 based on table 1, it is found that the average for the experimental group is better when compared to the experimental group. as in the pre-test mathematical creative thinking ability, the experimental group obtained an average of 1.15, while for the control group of 1.43. in the post-test, the experimental group obtained an average of 13.47, while that of the control group was 9.71. in n-gain, the experimental group obtained an average of 37.07, while that of the control group was 24.94. furthermore, to find out better learning between the two lessons used, the n-gain average difference test on the creative thinking abilities of students is carried out. the average for n-gain in the experimental group was 37.07, with a variance of 527.62, while ibrahim & widodo, advocacy approach with open-ended problems to mathematical … 96 the n-gain for the control group was 24.94, with a variance of 252.81. because one of the two n-gain data to be tested does not meet the normality assumption, the test used in this case is the mann-whitney upper-tailed test. the summary of the statistical tests is presented in table 2. table 2. summary of mann-whitney n-gain test results comparison n-gain w p decision experiment vs control 1533.5 0.0068 there is different from table 2, it shows that the significance coefficient is 0.0068 so that the increase in mathematical creative thinking abilities in students using an advocacy learning approach with the presentation of open-ended problems is significantly better than students who have conventional mathematical learning. furthermore, because the increase in students' mathematical creative thinking abilities in the experimental group is better when compared to the control group, the n-gain is analyzed based on general mathematical ability based on the upper, middle, and lower groups in the experimental group (groups of students who use an advocacy approach with problem presentation open-ended). table 3. the mean and deviation n-gain based on general mathematical ability general mathematics ability group n n-gain mathematical creative thinking abilities (%) mean standard deviation upper 8 59.94 24.52 middle 22 32.76 18.87 lower 6 22.38 13.42 table 3 is the mean and standard deviation of n-gain data on mathematical creative thinking abilities of students in the experimental group based on general mathematical abilities. based on table 3, it was found that the average mathematical creative thinking ability in the experimental group students based on general mathematical abilities was relatively different. this is because the mean score of students in the general mathematics ability group in the upper category is 59.94, the mean student in the general mathematics ability group is 32.76 in the middle category, and the mean student in the general mathematics ability group in the lower category is 22.38. these results indicate that the upper group has an average n-gain, which is relatively higher than the other two groups. table 4. one-way anova of n-gain based on general mathematics ability in the experiment group n-gain comparison of general mathematics ability group f p mathematical creative thinking abilities upper, middle, and lower groups 5.75 0.007 the results of this average difference need to be tested further, this is done to find out whether the three general mathematics ability groups have significant differences. volume 9, no 1, february 2020, pp. 93-102 97 the calculation one-way anova of n-gain based on general mathematics ability in the experiment group, obtained thet f is 7.72 with a significance coefficient of 0.002 (table 4). based on these results, it can be concluded that the improvement of students' mathematical creative thinking skills between the upper, middle, and lower groups in the experimental class is in different conditions. to find out which general mathematics ability group has better creative thinking skills, followed by post-hoc anova test, in this study the post-hoc anova test used the turkey test. based on the tukey test, in the experimental group there were significant differences between the upper group and the two other n-gain groups. these results can generally be seen in table 3, that the mean of general mathematics ability in upper is higher than the mean of other general mathematics ability. 3.2. discussion the results of the n-gain analysis on students' mathematical creative thinking abilities have increased, and the experimental class students looked better improved than the control class. students can analyze the arguments of a statement or conclusion given if the student has a minimum of knowledge of mathematical concepts that are relevant and correct and can show the relationship with the statements and conclusions given. in this connection, learning through an advocacy approach by presenting open-ended problems has given students broad opportunities to think freely, raise opinions or ideas, ask questions, and criticize the opinions of friends, especially in the process of solving problems given by the teacher. this is what seems to make students in the experimental class accustomed to display their arguments that support the ideas, statements, or conclusions that they propose. also, with the learning process carried out in the experimental class, students seemed to be doing a lot of exploring mathematical concepts in depth. thus, students in the experimental class have more opportunities to gain knowledge of relevant and correct mathematical concepts, and show their relationship to the statements and conclusions they provide, than students in the control class. students can do and consider induction if students can predict, test a rule of observed patterns, and proceed with formulating it. in this study, the ability of students to do and consider induction is certainly influenced by their mastery of the concept of function. one of the material functions related to determining the formula of a function if known several values of the function. learning through an advocacy approach by presenting open-ended problems has provided opportunities for students to explore and try various ways of solving. this is what seems to make students in the experimental class accustomed to estimating, testing a rule of observed patterns, and then proceeding with formulating it. meanwhile, students in the control class have inadequate opportunities regarding things experienced by students in the experimental class; this is due to the characteristics of learning in the control class. students who can smooth or think fluently are students who can express many ideas, answers, and problem-solving. in this connection, learning through an advocacy approach by presenting open-ended problems has given students ample opportunity to think freely, submit opinions or ideas, and ask questions. this is what seems to make students in the experimental class accustomed to displaying their ideas in solving problems or problems given by the teacher. students who can authenticate or think original are students who can come up with ideas in original ways, not cliched, rarely given by most people. meanwhile, students who can be flexible or think flexible are students who can express various solutions or approaches to problems. in this connection, learning through an advocacy approach by presenting open-ended problems has conditioned students to have a different solution from ibrahim & widodo, advocacy approach with open-ended problems to mathematical … 98 the problem or problem is given, consider solving other students who are considered new, so that inviting other students to give opinions, comments, or criticism, in a debate. this is what seems to make students in the experimental class accustomed to finding a variety of answers that are relatively new to students. students who can elaborate or think in detail are students who can describe something in detail, enrich, and develop an idea. in this connection, learning through an advocacy approach by presenting open-ended problems has conditioned students to prepare or add details of an idea to ward off criticism of other students. this is what seems to make students in the experimental class accustomed to describing things in detail, building relationships, and enriching and developing ideas. the findings of this study indicate that the posttest score of students' creative thinking ability in the experimental class, which is 13.47, with an ideal score of 35. these data indicate that the mathematical creative thinking ability of students in the experimental class, the results are not optimal. this is most likely caused by students who are not accustomed to learning through an advocacy approach by presenting open-ended problems. also, it is suspected that the results are not optimal in the experimental class due to the portion of the difficult questions on the set of tests used, which is more than 70% of the total number of questions. this is consistent with students' recognition of the results of interviews and daily journal notes, which state that the questions given in the learning process or tests are more difficult than the questions that are normally given by the teacher in conventional learning or conventional learning. thus, the results of this study might be different if the number of questions is difficult to reduce portions. as for the differences in the improvement of students' mathematical creative thinking abilities between the upper, middle, and lower groups in the experimental class, based on the results of the one-way anova statistical test, there are significant differences. this means that learning through an advocacy approach by presenting openended problems has a different effect on increasing mathematical creative thinking abilities, for each group of students' general mathematical abilities. several reasons can be put forward to explain the difference in the increase in mathematical creative thinking abilities between the three groups of students in the experimental class. one possible reason is that students for the lower and middle groups are not yet accustomed to openly expressing ideas, questions, or answers, thus hampering the development of students' mathematical creative thinking abilities. this is consistent with the results of the interview, representatives of the lower classes, who stated that some of their friends did not like the difficult or challenging problems or problems contained in the teaching material, so they were lazy to solve them. also, there were friends who were shy to ask questions. thus, for the lower class students who still feel ashamed to express ideas, questions, and criticize the opinions of other students, it will inhibit the increase in mathematical creative thinking abilities. the difference in the increase of creative thinking skills between the three groups of students in the experimental class can be attributed to the concept of zone of proximal development from lev vygotsky which states that children's cognitive abilities are divided into two stages, namely the stage of actual development and potential development (bruner, 1984; herlina, 2013; kotliar, sokolova, & tarasova, 2009). in this connection, the intervention given by the teacher to students in the learning process is adjusted according to differences in the level of the actual development of students to reach their potential level of development (permatasari, wayhuno, & adi, 2017; smagorinsky, 2018). this difference in intervention seems to be the reason for the difference in the ability to think critically and creatively between the three groups of students in the experimental class. the difference in the increase in the ability to think creatively between the three volume 9, no 1, february 2020, pp. 93-102 99 groups of students reinforces the results of previous findings, which state that the level of mathematical ability of students also determines the improvement of high-level mathematical abilities (guseva & solomonovich, 2017; murphy, scantlebury, & milne, 2015; shodikin, 2014). furthermore, for group students on learning through an advocacy approach by presenting open-ended problems is learning that has helped them develop and demonstrate mathematical creative and critical thinking skills, even if viewed from the results are not optimal. however, that does not mean that the middle and lower learning groups do not help them develop and demonstrate mathematical creative thinking skills. this, because when viewed as a whole, students in the experimental class have improved mathematical creative thinking abilities better than students in the control class. 4. conclusion the results of the study concluded that improving students' mathematical creative thinking abilities given the treatment of an advocacy approach by presenting open-ended problems in mathematics learning was better than students who were given conventional learning treatments. also, the increase in mathematical creative thinking skills, between students in the upper, middle, and lower groups in the experimental group, did not differ significantly even when viewed from the mean of the upper group having 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(2020). developing student worksheet based on missouri mathematics project model by using think-talk-write strategy of class viii. infinity, 9(1), 81-92. 1. introduction a student can be said to think critically if the student is able to test his experience, evaluate knowledge, ideas, and consider arguments before getting justification (ismaimuza, 2011). the ways to build students’ critical thinking are developed the attitudes of students’ reasoning, be challenged, and seek the truth. garrison, anderson, & archer (2001) states that when critical thinking is developed, a person will tend to seek the truth, think openly and be tolerant of new ideas, be able to analyze problems well, think systematically, be curious, mature in think, and have crititical thinking independently. mathematical critical thinking skills are very important for students because by mastering these abilities, students are able to be realistic and able to make good decisions for themselves (jumaisyaroh, napitupulu, & hasratuddin, 2014). in addition, critical thinking skills also need to be instilled early to students so students can solve various problems related to daily life (somakim, 2011). when someone has the ability to think critically well, then he is not mailto:renywahyuni264@gmail.com wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 82 easy to believe about something without proving it first so that the information is valid and can be trusted (fithriyah, sa'dijah, & sisworo, 2016). one way that teachers can do to improve the quality of mathematics lessons is to develop a teaching material (sandiyanti, 2018). teaching material is an important tool for teachers and students. teachers will experience difficulties in providing mathematics learning to students without having complete teaching material, and students will also experience the same obstacles (nasution, 2016). teaching materials that are widely used in schools are teaching materials that only contain material that is not clearly spelled out and are not accompanied by pictures as an explanation of the material. one of the teaching materials commonly used by teachers and students is the students’ worksheet (lks). worksheets that can guide students so they can easily understand the subject matter of mathematics, where there are several functions of using worksheet is a practical teaching material, helping students to add information about the concepts learned (astari, 2017). therefore, to improve students' critical thinking skills, a worksheet is needed that includes mathematics learning material, practice questions specifically designed to facilitate the critical thinking skills of class viii students based on the missouri mathematics project model using think talk write strategy. based on the considerations above, it is necessary to compile and develop worksheet to facilitate students' critical mathematical thinking skills that are in accordance with the characteristics of junior high school students in lubuklinggau. however, the worksheets that are developing today are still fairly practical and do not emphasize the process, so they cannot demand students to be active. today, the availability of worksheets must contain a meaningful learning process and can support the achievement of mathematics learning goals that are in accordance with curriculum requirements, students' characteristics and demands for problem solving at the junior secondary level (gazali, 2016). while the development of the curriculum is currently very much looking at the activeness of students in the learning process. in learning mathematics, various learning models and learning strategies are very well applied in teaching and learning in class. related to the learning model and learning strategies used by teachers in teaching, teachers tend to choose and use learning models and learning strategies that are appropriate and in accordance with the subject matter to be delivered. one of them is the mmp learning model and ttw strategy. this is in line with the opinion of slavin & lake (alba, chotim, & junaedi, 2014) mmp learning model is a learning model designed to help teachers effectively use exercises to be able to make students get better achievements. according to nugroho, suparni, & nu'man (2012) that missouri mathematics project (mmp) is one of the structured learning models as well as the mathematics teaching structure. according to (wahyuni & efuansyah, 2018) this mmp will be more effective if it is collaborated with learning strategies that are in line with it. one strategy that can be used to support the success of student learning is the think-talk-write (ttw) strategy. the purpose of this study is to determine the feasibility of worksheets that are developed based on the missouri mathematics project learning model using think talk write strategy using research and development research methodology. 2. method the type of research used in this study is development research (development and research) using 4-d development flow. according to (tegeh, jampel, & pudjawan, 2014) research and development methods (research and development) are strategies or research methods that are effective enough to improve learning practices. the 4-d channel adopts thiagarajan, according to (trianto, 2010) this development model consists of four phases, volume 9, no 1, february 2020, pp. 81-92 83 namely the 4-d development model (define, design, develop, and disseminate). the subjects in this study were students of class viii smpn 11 lubuklinggau in the academic year 2018/2019 in even semester. define is the stage for defining and defining the requirements needed in learning development. this stage includes front-end analysis, learner analysis, concept analysis, task analysis and specifying instructional objectives. the next step is design, the design phase aims to design learning tools. four steps are taken at this stage, namely criterion-test construction, media selection in accordance with the characteristics of the material and learning objectives, format selection, and initial design. the develop phase is the stage to produce product development which is carried out through two steps, namely: expert appraisal followed by revision and developmental testing. the last stage is disseminate, this stage includes validation testing, packaging, diffusion, and adoption. the instrument used in this study was a worksheet validation sheet for material, media, and language validators to measure the validity of the students’ worksheets that were developed, as well as the practical sheets of students and teachers to see the practicality of using the student worksheets in the learning process. questionnaire assessment is developed based on an evaluation component that includes appropriateness content, appropriateness presentation, assessment critical thinking ability. the worksheet assessment using the mmp model uses the ttw strategy, feasibility of graphic, language assessment, display, presentation of material, and the benefits (table 1). table 1. indicator aspects indicators material experts appropriateness of content material compatibility with sk and kd material accuracy supporting learning material updated materials appropriateness of presentation presentation techniques supporting presentation learning presentation completeness of presentation critical thinking ability assessment the suitability of the questions with the indicators of critical thinking skills the lks assessment using the mmp model uses the ttw strategy the compatibility of lks with mmp model uses ttw strategy media experts feasibility of graphic lks size lks cover design design content lks linguists language assessment straightforward communicative conformity with the level of development of students tuning and integrated thought flow wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 84 data that has been obtained through a questionnaire by expert assessment of products and student in the form of qualitative value will be converted into quantitative values likert scale (table 2). table 2. likert scale score criteria 5 excellent 4 good 3 enough 2 less 1 very less (sugiyono, 2017) from the data that has been collected, we calculate the average by the formula: ̅ ∑ the data obtained from material expert, media expert, linguist, and student is transformed into qualitative values based on ideal assessment criteria. the ideal scoring criteria are shown in table 3 (aini, masykur, komarudin, 2018). table 3. the ideal scoring criteria average interval score category very good good enough not good use of terms, symbols or icons aspects indicators student display clarity of text image clarity eye catching image suitability of the image with the material presentation of material presentation of material ease of understanding material systematic accuracy of presentation of material clarity of sentences clarity of symbols and icons clarity of terms suitability of the example with the material the benefits ease of study interest in using teaching materials in the form of student worksheets increased learning motivation volume 9, no 1, february 2020, pp. 81-92 85 very poor 3. results and discussion 3.1. results define at the define phase, field observations and interviews with mathematics teachers of class viii smp n 11 lubuklinggau were conducted to obtain the data needed for the definition process. the process of this stage is front-end analysis, learner analysis, concept analysis, task analysis, and specifying instructional objectives. in the front-end analysis, it was found that the average mathematics critical thinking ability of students of class viii of smp n 11 lubuklinggau is still in the low category, students are not accustomed to answering non-routine questions that require higher levels of thinking. in the learner analysis, it was found that the average age of eighth grade students at smp n 11 lubuklinggau is at the age of 12-14 years, where at that age students still need a lot of guidance and assistance from their teacher. the results of the task analysis found that students of class viii had used the 2013 curriculum in accordance with government recommendations. based on the 2013 curriculum, it was found that the material that students learned was cubes and cuboids. in the concept analysis stage, students must be able to master the material about the properties of cubes and cuboids, and students are able to make nets to build cubes and cuboids. students must also be able to master the concept of cube and cuboid surface area along with the volume of cube and cuboid. students must also be able to model mathematical forms from existing problems and be able to solve problems related to daily life. whereas in specifying instructional objectives it was found that students after learning the material to build a flat side space using lks to facilitate the ability to think critically mathematics can achieve predetermined competencies. design the next phase is the design phase, the results of the defining phase serve as a source of reference in making lks to facilitate the ability to think critically mathematics. the things done at this stage are criterion-test construction, what is done is the preparation of instruments consisting of worksheet validity assessment instruments, worksheet practicality assessment instruments and mathematics critical thinking skills test instruments was done. the instrument preparation process pays attention to aspects of the content eligibility component, the feasibility of presentation, the aspects of indicators of critical mathematical thinking ability, conformity to the missouri mathematics project learning model and think talk write strategy, language feasibility, and the feasibility aspects of graphics in accordance with bsnp standards. next is the format selection which consists of steps to compile the worksheet framework and compilation of the worksheet systematics. next is designing first draft of worksheet to be developed. the following is the first draft of worksheet that was made (figure 1, figure 2, figure 3). wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 86 figure 1. first display of material 1 figure 2. first display of material 2 volume 9, no 1, february 2020, pp. 81-92 87 figure 3. first display of mmp and ttw develop a. expert appraisal at the development phase, the worksheet which had been developed was validated by 3 validators consisting of material validation by mr. idul adha, m.pd., media validation by mr. dr. dodik mulyono, m.pd., and language validation by mrs. dr. rusmana dewi, m.pd. based on the results of the validation analysis of 3, the average value obtained by the 3 validators is 3.99 (valid category). table 4. recapitulation of assessment no validator number of items score obtained average score category 1 material expert 32 128 4.00 valid 2 media expert 25 102 4.08 valid 3 linguist 11 43 3.91 valid total 68 273 3.99 valid based on table 4, the results of the recapitulation assessment above, it can be seen that the developed students’ worksheet was valid criteria in all aspects. this shows that the student worksheet that was developed is valid and feasible to be tested on students in the next phase. after the worksheet was declared valid, the worksheet was revised based on the comments and suggestions of the validators. the worksheet revision is based on the instructions and suggestions given by the validator through the validation assessment sheet. the following are comments and suggestions from the validator (table 5). wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 88 table 5. comments and suggestions of validators validator comments and suggestions material clarify the instructions on the worksheet media pay attention to the word choice, sizes and colors select images based on the context of the material bring the identity of worksheet on the cover language the language must be edited according to the command sentence function. on the cover, put the identity for what school level and class b. developmental testing after the worksheet was revised, the next step was to conduct a small group trial consisting of 6 students consisting of 2 high-ability students, 2 medium-ability students and 2 low-ability students who were not as subject of product user trials. during the trial of the small group, the assessment sought responses and comments from target product users. the results of this small group trial are used to improve the worksheet. based on the results of the calculation of small group trials, obtained 6 students who showed a positive assessment of the worksheet to facilitate students' critical thinking skills in mathematics. after conducting a small group trial, the next stage is the field test stage in the actual class. the lks trial was conducted in class viii.b of smp n 11 lubuklinggau with a total of 23 students aiming to determine students' critical thinking skills. this trial was conducted 5 times in accordance with the learning syllabus used by smp n 11 lubuklinggau. after students have completed the learning process by using worksheets to facilitate students' critical mathematical thinking skills, students are asked to fill out a practical assessment sheet on the use of worksheets against worksheets developed. after students fill out the practicality assessment sheet, the sheet is analyzed. the analysis of assessment sheet to find out the practicalities of the worksheets developed. based on the results of the practicality assessment sheet analysis, an average score 0.903 was obtained and is in the very practical category. disseminate phase the final phase of this development process is the disseminate phase, at this phase the worksheet that has been improved according to the suggestions and comments at the time of development is carried out the packaging process of the worksheet so that it can be printed and distributed. after the worksheet is printed, the worksheet is disseminated so that it can be absorbed or understood by others and can be used in other classes. lks is distributed to other grade, mathematics teachers of grade vii and ix, and the principal of the smp n 11 lubuklinggau. 3.2. discussion based on the description above, the research product obtained in the form of student worksheets based on the missouri mathematics project learning model uses think talk write strategies to facilitate students' critical thinking abilities on the cube and cuboid material. the research product is in the form of student worksheets because in worksheet, students will get material, summaries, and assignments related to the material volume 9, no 1, february 2020, pp. 81-92 89 (prabawati, herman, & turmudi, 2019). through learning by using this worksheet students are required to be active in the learning process so that the results obtained are in line with expectations, this is in line with the opinion of pariska, elniati, & syafriandi (2012) which states that through student worksheets students notice given the responsibility to complete tasks and feel must do it, especially if the teacher gives full attention to the results of their work, so students are actively involved in learning. therefore, the teacher as a presenter must be able to choose a method or approach that is appropriate to the conditions of students' abilities in the classroom, including suitability in developing teaching materials / materials to support these learning activities (gazali, 2016). the result of student worksheet based on the missouri mathematics project learning model using think talk write strategies on cubes and cuboids to facilitate the thinking skills of eighth grade students. according to wahyuni & efuansyah (2018) this mmp will be more effective if it is collaborated with learning strategies that are in line with it. one strategy that can be used to support the success of student learning is the think-talk-write (ttw) strategy. the developed lks adopted a 4-d development procedure that went through four stages, the lks was developed using the 2013 curriculum which was in accordance with government recommendations. at the junior high school stage students have been able to think abstractly, so students have been able to think formally, although they still need guidance and guidance from the teacher during the mathematics learning process in class. according to ibda (2015) the level of cognitive development at the junior high school level is at the level of the formal operational stage. so that at this stage students have been able to think abstractly. with the worksheet that suits the needs of students, so as to be able to increase students' interest and motivation in learning mathematics, where there are several functions of using worksheet is a practical teaching material, helping students to add information about the concepts learned (astari, 2017). before lks is ready to be used in learning in the classroom, worksheet must first go through the stages of product validation and development trials. based on table 4, it can be seen that the worksheets validated by three experts are all in the valid category and the average value of the 3 validators is 3.99 and in the valid category. this is in line with the opinion of haviz (2013), which states that the first aspect determining the quality of learning products is validity. after students have completed the learning process using worksheets to facilitate students' critical mathematical thinking skills, students are asked to fill out a practical assessment sheet on the use of worksheets against worksheets developed. after students fill out the practicality assessment sheet, the sheet is analyzed. the analysis of the assessment sheet to find out the practicalities of the worksheets developed. based on the results of the practicality assessment sheet analysis, an average score obtained was 0.903 in the very practical category. 4. conclusion based on the results of research and discussion above, the development of student worksheets based on the missouri mathematics project learning model using think talk write strategy has been validated by material experts, media experts and linguists with valid criteria and obtained an average score 3.99 and the results of the worksheet trial conducted at smpn 11 lubuklinggau in class viii were in very practical criteria and an average score 0.903 was obtained. acknowledgements wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 90 the authors would like to thank for the support to the directorate of research and service society, directorate general strengthening research and development ministry of research, technology and higher education that has been support our research through beginner lecturer research scheme. 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(2018). model pembelajaran missouri mathematics project (mmp) menggunakan strategi think talk write (ttw) terhadap kemampuan berpikir kritis dan kemampuan pemecahan masalah. jnpm (jurnal nasional pendidikan matematika), 2(1), 24-36. http://dx.doi.org/10.33603/jnpm.v2i1.778 https://doi.org/10.31980/mosharafa.v8i1.383 https://doi.org/10.31980/mosharafa.v8i1.383 https://doi.org/10.31980/mosharafa.v8i1.383 https://doi.org/10.31980/mosharafa.v8i1.383 https://doi.org/10.24042/djm.v1i2.2280 https://doi.org/10.24042/djm.v1i2.2280 https://doi.org/10.24042/djm.v1i2.2280 http://dx.doi.org/10.33603/jnpm.v2i1.778 http://dx.doi.org/10.33603/jnpm.v2i1.778 http://dx.doi.org/10.33603/jnpm.v2i1.778 http://dx.doi.org/10.33603/jnpm.v2i1.778 http://dx.doi.org/10.33603/jnpm.v2i1.778 wahyuni, efuansyah, & sukasno, developing student worksheet based on missouri … 92 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p129-142 129 the teachers’ experience background and their profesionalism yulyanti harisman* 1 , yaya sukjaya kusumah 2 , kusnandi 3 dan muchamad subali noto 4 1 universitas negeri padang jl. prof. dr. hamka air tawar barat, indonesia 2,3 universitas pendidikan indonesia 4 universtas swadaya gunung djati article info abstract article history: received nov 8, 2018 revised may 5, 2019 accepted june 4, 2019 based on literature review three categories of teachers are: good, very good, and excellent which are viewed from aspects of beliefs, attitude, depth of pedagogical and didactic aspects, and teacher reflection in the learning process has been obtained in previous studies. various external aspects are considered to affect teacher professionalism in learning about mathematical problem solving. these aspects need to be studied to maximize the teacher professionalism. this study will examine these external aspects, ranging from teaching experience, educational background, and experience in participating in training to improve teacher competencies. this type of qualitative research with survey methods was chosen as a research method. three teachers from three junior high schools with different clusters were selected as research subjects. each teacher is given short questions related to this. the results obtained are the experiences of the trainings that teachers follow in increasing their competence and teacher's educational background have more significant influence on teacher professionalism compared to the experience or length of teacher teaching. keywords: learning process, problem solving, teacher profesionalism copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: yulyanti harisman, departement of mathematics education, universitas negeri padang, jl. prof. dr. hamka air tawar barat, west sumatera, indonesia email: yulyyuki@gmail.com how to cite: harisman, y., kusumah, y. s., kusnandi, k., & noto, m. s. (2019). experience teacher background and their professionalism. infinity, 8(2), 129-142. 1. introduction teacher professionalism in the process of learning mathematics can be seen from various factors. factors influencing teacher professionalism according to muir & beswick (2007) are: teacher beliefs, the depth and breadth of the didactic and pedagogical aspects, and the teacher's reflection on the mathematics teaching practice in the classroom. furthermore, according to ernest (1989) things that affect teacher professionalism are attitudes towards mathematics and the process of learning mathematics itself. research on teacher professionalism in mathematics learning experiences a very exponential growth. some studies that explore and improve teacher professionalism in mathematics learning process are researches (beswick, 2007, 2012) this study examines teachers’ beliefs about mathematics, learning, and mathematics learning and discerns how mailto:yulyyuki@gmail.com harisman, kusumah, kusnandi, & noto, experience teacher background … 130 teachers' beliefs about mathematics will affect the way teachers conduct mathematics learning process in class. furthermore, research conducted by zoest, jones, & thornton (1994) has observed and categorized the breadth and depth of the teacher's pedagogical and didactic abilities in the learning process. these categories are named with content performance, understanding content, and learner interaction. in content performance, the teacher only focuses on the achievement of the material in learning mathematics without regard to students' understanding and student interaction with other students. content understanding and learner interaction focus on how students understand and construct learning material. in addition to the belief factor and the depth and breadth of didactic and pedagogical aspects, teacher reflection in the learning process is also an important aspect of teacher professionalism. reflection factor is how the teacher's reflection in the learning process also greatly influences how a professional teacher can evaluate and improve the learning process that has been done. according to beswick (2007) there are three categories of teacher types in reflecting the mathematics learning process, namely technical reflection, deliberate reflection, and critical reflection. in technical reflection, the teacher is at the stage of being only able to describe the general sequence of classroom teaching practices, often only focusing on technical aspects, without considering any experience or behaviour values. the deliberate reflections are where the teacher has been able to identify reflection more critically with the rationale or explanation for action or behaviour which is not only based on teaching facts. the critical reflections are where the teachers who have conducted reflections are able to identify, criticize, and provide explanations by including perspectives on considering others and offering alternatives for settlement. furthermore, it is expanded by research muir & beswick (2007) which discusses how teachers reflect themselves, the learning, and students in mathematics teaching and learning process. teacher’s attitude in the learning process has also been studied in research ernest (1989) which has categorized the teacher's attitude towards mathematics learning. in research ernest the teacher's attitude towards mathematics and the learning process are categorized into three categories: teachers who do not like mathematics, teachers who are in the middle of whether liking and disliking mathematics and teachers who like mathematics. ulug, ozden, & eryilmaz (2011) conducted a study of 353 students at istanbul university. this study looks at the relationship between teacher attitudes and student attitudes. the findings of the ulug et al. (2011) are that the teacher's positive attitude influences students' attitudes. the above studies only focus on the professionalism of teachers in general mathematics learning. if viewed from the learning process of problem solving, teacher professionalism in problem solving and the learning process has not been explored and studied by educational researchers. researches on this subject can be seen from several studies, namely research harisman, kusumah, & kusnandi (2018, 2019a). research harisman et al. (2019a) studied categories of teachers' beliefs in the learning process of mathematical problem solving which can be seen in table 1. volume 8, no 2, september 2019, pp. 129-142 131 table 1. category of teachers beliefs on learning process on mathematical problem solving aspect category good very good excellent teacher’s beliefs about problem solving learning the teacher considers that learning about mathematical problem solving is emphasized on performance (emphasizing to know rules and procedures without understanding) of the mathematical problemsolving process the teacher considers that learning about mathematical problem solving is emphasized in understanding the process of solving mathematical problems the teacher considers that learning about mathematical problem solving is emphasized on how students construct a mathematical problem-solving process. teacher’s beliefs about problem solving teaching the teacher considers that if given a mathematical problem then the student completes by receiving knowledge from the teacher the teacher sees that if given a mathematical problem then the student finishes by constructing his knowledge with the teacher's direction the teacher considers that if given a mathematical problem the student completes by exploring the problem-solving strategy based on their own interests teacher’s beliefs about what is the positions of mathematical knowledge to learning about mathematical problem solving teacher considers that in the process of mathematical problems, solving we must remember the definitions/ rules needed to solve the problem teacher considers that in the process of solving mathematical problems, we must understand the definitions/rules needed to solve the problem teacher considers that in the process of solving mathematical problems, we must be able to adjust and differentiate the definitions used to solve the problem the categories of teacher beliefs in the learning process of solving problems are graded from good to excellent, along with categories of how teachers reflect the learning process they do in problem solving which can be seen in table 2. the results of teacher reflection categories are contained in research harisman et al. (2018). table 2. categories of teacher reflection on the learning process of problem solving aspect aspect description categories good very good excellent teacher's reflection on the learning process of mathematical problem solving teacher reflection on how to provide understanding to understand problems in the learning process of mathematical problem solving reflecting on the process of how to provide understanding to understand the problem in the learning process of solving the problem is limited to revealing the facts that occurred in class able to reflect on the process of how to provide understanding to understand problems in the learning process of problem-solving, but cannot provide an alternative how to overcome them able to reflect on the process of how to provide understanding to understand problems in the learning process of problem-solving, and provide an alternative how to overcome them teacher's reflection on the selection of strategies in the learning process on mathematical problem solving reflecting the process of selecting strategies in learning about mathematical problem solving is limited to revealing the facts that occur in the classroom able to reflect on the strategy selection process in learning about mathematical problem solving, but cannot give an alternative how to overcome it able to reflect on the strategy selection process in learning about mathematical problem solving, and provide an alternative how to overcome it harisman, kusumah, kusnandi, & noto, experience teacher background … 132 aspect aspect description categories good very good excellent teacher's reflection on the use of strategies in the learning process on mathematical problem solving reflecting on the use of strategies in learning about mathematical problem solving is limited to revealing the facts that occur in the classroom able to reflect the process of using strategies in learning about mathematical problem solving, but cannot provide an alternative how to overcome them able to reflect on the use of strategies in learning about mathematical problem solving, and provide an alternative how to overcome them teacher's reflection on the verification of solutions in the learning process on mathematical problem solving reflecting the process of verifying solutions in learning about mathematical problem solving is limited to revealing facts that occurred in the classroom able to reflect the process of verifying solutions in learning about mathematical problem solving, but cannot provide an alternative how to overcome them able to reflect the process of verifying solutions in learning about mathematical problem solving, and providing an alternative how to overcome them reflection of teachers on students in the learning process on mathematical problem solving reflecting student behavior during the learning process on mathematical problem solving is limited to revealing the facts that occur in the classroom able to reflect student behavior during the learning process on mathematical problem solving, but cannot provide an alternative how to overcome it able to reflect on student behavior during the learning process on mathematical problem solving, and provide an alternative how to overcome it furthermore, research harisman et al. (2018) has also categorized teacher's attitude towards mathematical problem solving and the learning process on mathematical problem solving that can be seen in table 3. table 3. three categories of attitudes teachers toward mathematical problem solving and its learning process aspect category good very good excellent teachers attitude on mathematical problem solving teacher is phobia (fear) when faced with mathematical problem solving content teachers have a little phobia (fear) when faced with problemsolving content matematis teachers like, enjoy, and are interested, when faced with solving content attitude of teachers to learning about mathematical problem solving teachers look nervous in the learning process of mathematical problem solving teachers do not seem nervous in the learning process of mathematical problem solving, but they do not like totally, enjoy, or excited teacher looks to like, enjoy, and passionate in the learning process of mathematical problem solving research harisman et al. (2019b) has also categorized the extension of teacher's didactic and pedagogic in the learning process of problem solving, categorized in good, very good, and excellent which can be seen in table 4. volume 8, no 2, september 2019, pp. 129-142 133 table 4. categories of depth and breadth of pedagogy and didactic aspects of the teacher in the learning process of mathematical problem solving aspect description of aspect categories good very good excellent depth and extent aspect of didactic and pedagogic in learning about mathematical problem solving using various problem solving strategies directing students to problem solving process with strategy that has been set directing students to the problemsolving process with a variety of strategies teachers ask the students to display their own troubleshooting process by providing only help needed instructions heuristic mathematical problem solving learning students with teachers try to understand the process of solving mathematical problems in accordance with predetermined strategy students with teachers try to understand the problem-solving process toward pre-defined strategy choices students recognize the process of understanding the mathematical problem solving of the stratum self-constructed strategy creating interactions among students, teaching materials, and teachers in the learning process on mathematical problem solving one-way interaction from teachers to students in mathematical problem-solving process two-way interaction but not involving other students in mathematical problem-solving process two-way interaction by involving other students collaboratively in the process of solving mathematical problems these teacher categories influence how students behave in problem solving. research harisman et al. (2019b) examines the influence of teacher professionalism in the learning process on students' problem solving behaviour. the results of this study conclude that teacher professionalism is one of the external factors that influence student behaviour in solving problems. different teachers in the excellent category tend to generate students who behave sophisticatedly in mathematical problem solving and vice versa. the problem is what external aspects cause the teacher to be in an adequate category in solving mathematical problems. there are no studies trying to conduct a depth study to seek what aspects greatly affect the teacher category (level) in learning about mathematical problem solving. this study attempts to reveal external factors that influence teacher professionalism in the learning process of mathematical problem solving. for that reason, this research proposes research questions, namely: what external factors influence the category of teachers in mathematical and learning problem solving in terms of beli efs, attitude, depth and breadth of pedagogy and didactic behaviour, and also teacher's reflection on the process of learning mathematical problem solving. 2. method 2.1. study description three junior high school teachers were categorized using the aspects proposed by study harisman et al. (2018, 2019a, 2019b). teachers are selected from three schools that have different clusters, namely: high, medium, and low clusters. the aspects seen were the same aspects as the research, namely: teacher beliefs, teacher attitude, depth of didactic and pedagogical aspects, and teacher's reflection, which were all measured and reviewed harisman, kusumah, kusnandi, & noto, experience teacher background … 134 on the learning process of mathematical problem solving. after obtaining the teacher categories in each aspect, the same way as the one done by research, the study began by documenting the teacher's learning process of problem solving, giving questionnaires, and interviewing each teacher, as well as watching videos and giving some questions related to the learning process that had been done. after the teacher category in every aspect of professionalism was obtained, the teachers were interviewed in depth about educational background, attended training, and teaching periods. this is considered as an external factor that will affect the professionalism of each teacher towards mathematical problem solving and the learning process. this aspect was chosen based on recommendations from the muir, beswick, & williamson (2008) study. things that will become the focus of this research are what factors greatly influence the professionalism of teachers in mathematical problem solving and the learning process of mathematical problem solving. the process of categorizing teachers will not be discussed in detail, but the final category of the teacher will be displayed inasmuch as the focus of the research is the external aspect that affects the category while the categorization process only performs the imitation of the process and the instruments carried out in the research. 2.2. research subjects three volunteer teachers were selected from three junior high schools. the three teachers were given open questions related to teaching experience, educational background and attended trainings. beforehand, the three teachers were categorized according to beliefs, attitude, depth of pedagogical and didactic aspects and teacher's reflection on mathematical problem solving and learning processes on mathematical problem solving. teachers were selected from three schools with different clusters. the purpose of selecting schools with varying levels is to avoid the problem of reliability and validity that might happen in small sample sizes (patton, 1990). 2.3. instruments and data analysis technique the instruments used to categorize teachers for beliefs, attitude, and depth of pedagogical aspects were the same as the instruments used in research. after the teacher category was obtained, the main instrument in this study was to provide open questions related to teacher teaching experience, personal development training experience, and the level of education held by the teacher. questions could be improvised by researchers in accordance with research needs; this is due to the role of researchers as the main instrument in qualitative research. data were analysed by giving codes to teacher's answers. 3. results and discussion 3.1. results teachers from each school are grouped according to the rubric stated in study. each teacher was given a code that is: teachers who came from school one (t-1), teachers from school two were coded with (t-2), and teachers coming from school three were coded with (t-3). in the early stages the teacher was classified and the results were obtained as shown in table 5. volume 8, no 2, september 2019, pp. 129-142 135 table 5. the category of teacher professionalism in learning about problem solving aspects things shown by teachers first school’s teacher (t-1) second school’s teacher (t-2) third school’s teacher (t-3) teacher’s belief teacher's belief in the nature of mathematics the teacher views mathematics as a creative science that is useful in everyday life (problem-solving) (excellent). the teacher considers mathematics as a collection of facts, rules and skills that are interrelated with one another (very good). the teacher considers mathematics as a collection of facts, rules and skills that stand alone (good) teacher's belief in learning about mathematical problem solving the teacher views the learning of mathematical problem solving as it is emphasized on how students construct mathematical problem solving processes (excellent). the teacher considers learning mathematical problem solving emphasizing in understanding the mathematical problem solving process (very good). the teacher views that learning about mathematical problem solving is emphasized on performance (knowing rules and procedures without understanding) on the mathematical problem solving process (good) teacher’s beliefs in students in learning mathematical problem solving the teacher views that if given a mathematical problem, students solve by exploring problem solving strategies based on their own interests (excellent). the teacher considers that if given a mathematical problem, students solve it by constructing their knowledge with the teacher's direction (very good). the teacher considers students to receive knowledge from the teacher when solving mathematical problems (good) teacher's beliefs on mathematical knowledge for learning about mathematical problem solving the teacher considers that in carrying out mathematical problem solving processes, we must be able to adjust and distinguish the definitions used to solve problems (excellent). the teacher considers that in carrying out mathematical problem solving processes, we must be able to adjust and distinguish the definitions used to solve problems (excellent) the teacher believes that we must understand the definitions/rules needed to solve problems (very good ) teacher's response is very good positively or negatively towards certain ideas, objects, people, or situations (attitude) teacher's attitudes to mathematical problem solving the teacher likes, enjoys, and is interested, when faced with mathematical problem solving content (excellent) the teacher has little phobia (fear) when faced with mathematical problem solving content (very good ) the teacher has little phobia (fear) when faced with mathematical problem solving content (good) teacher's attitude towards learning mathematical problem solving the teacher looks like, enjoys, and eager in the learning process about mathematical problem solving (excellent) the teacher looks like, enjoys, and eager in the learning process about mathematical problem solving (excellent) the teacher is afraid when faced with solving content. the teacher does not seem nervous in the learning process about mathematical problem solving, but is not so fond of, enjoying, or excited ( very good ) depth and extent of didactic and pedagogical use of various problem solving strategies the teacher asks students to display their own problem solving process by only providing help directing students to the problem solving process with a predetermined strategy (good) directing students to the problem solving process with a predetermined strategy (good) harisman, kusumah, kusnandi, & noto, experience teacher background … 136 aspects things shown by teachers first school’s teacher (t-1) second school’s teacher (t-2) third school’s teacher (t-3) aspects in learning about mathematical problem solving with the instructions needed (excellent) mathematical problem solving learning in a heuristic manner students understand the process of understanding mathematical problem solving from strategies that have been constructed on their own (excellent) students and teachers understand the process of mathematical problem solving in accordance with a predetermined strategy (good) students and teachers understand the process of mathematical problem solving in accordance with a predetermined strategy (good) creation of interaction between students, teaching materials, and teachers in the learning process of mathematical problem solving two-way interaction with collaboratively involving other students in the mathematical problem solving process (excellent) two-way interaction but does not involve other students in the mathematical problem solving process (very good ) one-way interaction from teacher to student in the mathematical problem solving process (good) teacher's reflection on the learning process of mathematical problem solving teacher's reflection on how to provide understanding to understand problems in the process of learning mathematical problem solving able to reflect on the process of how to provide understanding to understand problems in the learning process about problem solving, and provide alternatives on how to overcome them (excellent) able to reflect on the process of how to provide understanding to understand problems in the learning process about problem solving, and provide alternatives on how to overcome them (excellent) able to reflect on the process of how to provide understanding to understand problems in learning about mathematical problem solving but it is only limited to revealing the facts that occur in class. (good) teacher's reflection on the choice of strategies in the learning process about mathematical problem solving able to reflect on the process of selecting strategies in learning about mathematical problem solving, and providing alternatives on how to overcome them (excellent) able to reflect on the process of selecting strategies in learning about mathematical problem solving, and providing alternatives on how to overcome them (excellent) able to reflect on the process of selecting strategies in learning about mathematical problem solving but it is limited to revealing only the facts that occur in class. (good) teacher's reflection on the implementation of strategies in the learning process about mathematical problem solving able to reflect on the process of implementing strategies in learning about mathematical problem solving, and provide an alternative how to overcome them (excellent) able to reflect on the process of implementing strategies in learning about mathematical problem solving, and provide an alternative how to overcome them (excellent) able to reflect on the process of selecting strategies in learning about mathematical problem solving but it is limited to revealing only the facts that occur in class (good) volume 8, no 2, september 2019, pp. 129-142 137 aspects things shown by teachers first school’s teacher (t-1) second school’s teacher (t-2) third school’s teacher (t-3) teacher's reflection on verification of solutions in the learning process of mathematical problem solving able to reflect on the verification process of solutions in learning mathematical problem solving, and provide an alternative how to overcome them (excellent) able to reflect on the verification process of solutions in learning mathematical problem solving, and provide an alternative how to overcome them (excellent) able to reflect on the process of selecting strategies in learning about mathematical problem solving but it is limited to revealing only the facts that occur in class (good) reflection on students in the learning process of mathematical problem solving able to reflect on students' behavior during the learning process of mathematical problem solving, and provide alternatives on how to overcome them (excellent) able to reflect on students' behavior during the learning process of mathematical problem solving, and provide alternatives on how to overcome them (excellent) able to reflect on the process of selecting strategies in learning about mathematical problem solving but it is limited to revealing only the facts that occur in class (good) based on the criteria in table 5, it can be concluded that the first school's teacher (t-1) tends to be in the excellent category, the second school teacher (t-2) is in the very good category, and the third school's teacher (t-3) is in good category. furthermore, external aspects that affect the categories of the three teachers will be displayed. these aspects are reviewed from the length of teaching, training experience, as well as teacher education background. the following are the results of interviews for the three teachers. interview footage of the first school teacher (t-1) below is the footage of interview with the first school teacher (t-1) researcher : how long have you been teaching? t-1 teacher : 20 years researcher : what was your last education? t-1 teacher : the last is a master's degree in education at a leading public university in the city of bandung researcher : did you pay the institution individually? t-1 teacher : i was granted by the government researcher : how about trainings? t-1 teacher : there were many. school level training, teacher competence training. there was teacher competence training on certain subject recently. researcher : was it obligatory? t-1 teacher : yes it was, especially when the one who conducted it was the education authorities harisman, kusumah, kusnandi, & noto, experience teacher background … 138 based on the interview, t-1 teacher has 20 years of teaching experience and has a master education background and often attends professional training. hereafter, the interview results of t-2 teacher is presented as follows: interview footage of the second school teacher (t-2) below is the footage of interview with the second school teacher (t-2) researcher : how long have you been teaching? t-2 teacher : 15 years since 2003 researcher : is there any intention to continue your education to a higher degree? t-2 teacher : there was. however, there was an accident where my house was burnt down by a fire, so i dropped it. researcher : so you used to study in a higher degree? t-2 teacher : i had already studied for one semester, but due to the accident, i quitted. researcher : where did you study? t-2 teacher : i had my bachelor's degree at one of the private universities in bandung on 2007. researcher : how about now? t-2 teacher : i need to take care of my children first. researcher : does school offer you a help? t-2 teacher : no, i used my own wallet. there is, if we submit a request to the official, but it was a complicated process. researcher : do you attend trainings conducted by school? t-2 teacher : rarely, but i usually attend competence exam. it was jogja which conducted it recently. i was challenged to do better researcher : thank you. based on the interview, t-2 teacher has 15 years of teaching experience, an undergraduate education background, and a desire to continue the master degree. she rarely attends professional training. hereafter, the interview results of t-3 teacher is presented as follows interview footage of the third school teacher (t-3) below is the footage of interview with the third school teacher (t-3) researcher : how long have you been teaching? t-3 teacher : i have been teaching here for 24 years. i have actually taught for 30 years but 24 years were spent in this school. researcher : you seem to be very experienced. does it mean that you have attended lots of olympiads and trainings? t-3 teacher : olympiad training was once attended in 2007, but i have never volume 8, no 2, september 2019, pp. 129-142 139 attended the improvement training researcher : why? t-3 teacher : there used to be seniority level, i used to go to the makkah also. there were some obstacles. i used to be a trainer for an olympiad, but it might be due to luck. researcher : is there any intention to continue your education to a higher degree? t-3 teacher : no, i will be retiring soon. researcher : why don’t you try to continue? t-3 teacher : it's even difficult for me to make paper. i intend to attend a two-year education at one of the public universities in bandung. i have never taken a placement test and i was given a scholarship by the government. after that, i continued my studies at the undergraduate level. everything is free. based on the interview footage, t-3 teacher has 30 years of teaching experience, an undergraduate education background, and no desire to continue the master's degree. she rarely attends professional training. 3.2. discussion the discussion is stated in table 6. table 6. classification of each teacher no teacher’s origin teacher’s category on mathematical problem solving and mathematical problem solving learning teaching experience level of education training experience 1 first school teacher (t-1) excellent 20 years master degree often 2 second school teacher (t-2) very good 15 years bachelor degree (have a strong intention to continue) rare 3 third school teacher (t-3) good 30 years bachelor degree (doesn’t have a strong intention to continue) almost never these results are in accordance with expert opinion, that is muir et al. (2008) external factors that influence student behaviour are teacher professionalism in the process of learning mathematical problem solving. furthermore, research zsoldos-marchis (2015) conducted research in elementary schools that the attitude of an elementary school teacher towards mathematics influences the attitudes of their students. research beswick (2012) also saw the relationship between beliefs possessed by teachers towards students whom they teach. in research, there is a consistency between the practice of learning and students' perceptions of the beliefs held by the teacher, although not always, it is always consistent. harisman, kusumah, kusnandi, & noto, experience teacher background … 140 furthermore, research opolot (2014) also stated based on the results of his research that the professional factors of teachers greatly influenced students' achievement in science and mathematics classes in uganda. the implication of these researches is that teacher should be given the widest opportunity to continue education and trainings that support their professionalism at the school and education level. this is very related to the student achievement in mathematical problem solving. 4. conclusion the findings of this study are external factors such as the level of education and experience in participating in training are determinants in the professionalism of teachers in the learning process about mathematical problem solving. the duration of teaching is not a professional determinant of a teacher in the learning process of solving mathematical problems. the above is important to know by decision makers because teachers who are professional in problem solving and problem solving learning are closely related to students' mathematical problem solving behavior (harisman et al., 2019b). references beswick, k. 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(2015). changing pre-service primary-school teachers’ attitude towards mathematics by collaborative problem solving. procedia-social and behavioral sciences, 186, 174-182. https://core.ac.uk/download/pdf/82755761.pdf https://core.ac.uk/download/pdf/82755761.pdf https://core.ac.uk/download/pdf/82755761.pdf https://link.springer.com/article/10.1007/bf03217261 https://link.springer.com/article/10.1007/bf03217261 https://link.springer.com/article/10.1007/bf03217261 https://www.sciencedirect.com/science/article/pii/s1877042815023605 https://www.sciencedirect.com/science/article/pii/s1877042815023605 https://www.sciencedirect.com/science/article/pii/s1877042815023605 harisman, kusumah, kusnandi, & noto, experience teacher background … 142 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p69-80 69 mathematics communication mistakes in solving hots problems lelly nur rachmawati, yus mochamad cholily, zukhrufurrohmah* universitas muhammdiyah malang, indonesia article info abstract article history: received aug 8, 2020 revised jan 10, 2021 accepted jan 11, 2021 student mistakes in communicating mathematical ideas are still widely practiced. therefore, it is essential to analyze students' mathematical communication errors in solving mathematical problems so that learning planning can be better. this study aims to describe students' mathematics communication errors in solving higher-order thinking skills in linear algebra and matrix subject. the type of research is a qualitative descriptive study. they were 155 students as subject research. the data analysis started by collecting students' answers and then grouped them according to mathematics communication skills criteria. later identified and analyzed the errors made by students of each mathematics communication criteria. the results showed that mathematical communication errors on the indicators of writing mathematical situations were concept errors and principle errors. the declaring idea's mathematical communication error is a concept error, a principle error, and an operation error. furthermore, mathematical communication errors on the indicator state that solving-problem using the language itself is a concept error and operator error. keywords: high order thinking skill, mathematical communication, mathematics mistakes copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: zukhrufurrohmah, department of mathematics education, faculty of teacher training and education universitas muhammdiyah malang, jl. raya tlogomas 246 malang, east java 65144, indonesia email: zukhrufurrohmah@umm.ac.id how to cite: rachmawati, l. n., cholily, y. m., & zukhrufurrohmah, z. (2021). mathematics communication mistakes in solving hots problems. infinity, 10(1), 69-80. 1. introduction learning and communication are integral parts. kleden (2016) emphasized that in learning activities there was an exchange of information in the form of knowledge and experience between lecturers and students, students and students, also between students and teaching materials, so mathematical communication skills were very much needed. firdaus and aini (2019) added that learning activities carried out by students were not limited to memorizing formulas and mastering calculations, but also learning through mathematical communication. kleden (2016) emphasized that with excellent mathematics communication, students can convey ideas or thoughts appropriately to convince themselves and others. the importance of student mathematical communication is currently a problem that must be considered (angraini, 2019; argarini, yazidah, & kurniawati, 2020; astuti & https://doi.org/10.22460/infinity.v10i1.p69-80 rachmawati, cholily, & zukhrufurrohmah, mathematics communication mistakes … 70 leonard, 2015; nuraida & amam, 2019; prabawanto, 2019; rosita, nopriana, & silvia, 2019). therefore, the development of students' mathematics communication skills in problem-solving must master at the lecture level (fatimah, 2012). it will be useful in social life and good problem-solving results for students (ruswana, 2019). excellent communication skills are essential for students because, through symbolic language, students can communicate mathematically accurately and accurately (novianti, 2017). student's ability to solve problems can be influenced by students' written mathematics communication skills through their answer sheets (ardina & sa’dijah, 2016). umar (2012) emphasized that communication and problem-solving skills are related and essential to have so that mistakes not occurred. therefore, it is crucial to develop student mathematics communication. the importance of having mathematics communication skills is not in line with the facts which show that many students make mistakes in communicating their ideas (pratiwi, 2015). most of them do not master the concepts but only memorize them (ruswana, 2019). students make a lot of mistakes in mathematics communication skills. therefore mathematics communication conducted through tests will determine the students' ability to master the material (subekti et al., 2016). based on this information, mathematics communication skills are related to mathematical errors made by students. according to ardiawan (2015), mathematics errors are systematic errors that occur when the answers are written differently from the real solutions. firdaus (2019) said that there were some mistakes when solving matrices about understanding the question of the story, using formulas and concepts, and doing calculations. besides, wijaya and masriyah (2013) said that there were several types of mistakes made, namely concept errors, principle errors, and operating errors. concept errors are caused by students when misinterpreting or using terms so they cannot understand a given material/problem (widodo, 2013; widodo et al., 2020). principal error is a mistake made because of the inability of students to connect/link several concepts in the right relationship (astuty & wijayanti, 2013). then, astuty and wijayanti (2013) said the operation error was an error made due to incorrect calculation. at the same time, zukhrufurrohmah and kusumawardana (2019) research states that the types of mathematical errors can be in the form of skills, concepts, and techniques. but in this study, the mathematics errors that be analyzed were concept errors, principle errors, and operating errors. agree with asviangga et al. (2018) that one of the problems that can lead to mathematics communication skills is a matter of high-level thinking. rahayuningsih and jayanti (2019) said that improving the quality of learning can obtain by solving higherorder thinking skills (hots). thus when students do problem-solving activities can use a different approach, it will be possible to do so that a high-level thinking process occurs (rahayuningsih & jayanti, 2019). hots questions are questions that require you to use higher-order thinking skills (pratiwi et al., 2017). the aspects of the hots problem are analyzing, evaluating, and creating (rahayuningsih & jayanti, 2019). based on some of these aspects in this study hots problem indicators in measuring mathematics communication are students can: 1) analyze the mathematical situation by breaking down the information obtained; 2) evaluate by testing against an idea by the procedure to clarify a statement, and 3) create with writing a summary or theory to conclude. according to the explanation, this researcher is interested in describing mathematics communication errors that students have when completing hots questions in linear algebra and matrix subjects. volume 10, no 1, february 2021, pp. 69-80 71 2. method this research is qualitative descriptive research aiming to describe the written mathematics communication errors made by students in completing the hots problem on matrix material. the study begins by conducting field observations with the data obtained is that students make some mistakes in conveying the idea of solving a given problem. based on this problem, instruments were then arranged in the form of hots-style test questions to lead to errors and mathematics communication of students. the test questions were validated by two mathematics education lecturers who experts in teaching linear algebra and matrix subjects. the test was given to the students through the lms.umm.ac.id platform. students collect their answers by uploading handwritten photos or sending typed results to lms.umm.ac.id through their respective accounts. hots problems that valid were served in table 1. table 1. research instruments number of question hots aspect mathematics communication indicator on the answer to the problem mathematical error that appears in problem 1. find matrix 𝐴, matrix 𝐵, matrix 𝐶, so 𝐴𝐶 = 𝐵𝐶 and 𝐴 ≠ 𝐵! analyze understanding the mathematical situation clearly, students write that the matrix 𝐴𝐶 = 𝐵𝐶 and matrix 𝐴 are different from matrix 𝐵 concept errors, principle errors and operating errors evaluate writing ideas clearly, students will test by multiplying the two matrices to find out whether 𝐴𝐶 = 𝐵𝐶 and served matrix 𝐴 are different from matrix 𝐵 concept errors, principle errors and operating errors create stating the results of problemsolving using their language, students write the conclusion that matrices a and b are different matrices so that 𝐴𝐶 = 𝐵𝐶 and 𝐴 are different from 𝐵 concept errors, principle errors and operating errors 2. suppose there is a matrix 𝐴 with the following components: 𝐴 = [ 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ] if 𝐴2 = 𝐴𝐴, 𝐴3 = 𝐴𝐴𝐴,𝑎𝑛𝑑 analyze by understanding the mathematical situation clearly, students write down the information obtained from matrix 𝐴, matrix 𝐴2 = 𝐴𝐴, matrix 𝐴3 = 𝐴𝐴𝐴 and 𝐴𝑛 = 𝐴𝐴…𝐴⏟ 𝑎𝑠 𝑚𝑎𝑛𝑦 𝑎𝑠 𝑛 concept errors, principle errors and operating errors evaluate writing out ideas clearly, students test by calculating the matrices 𝐴2and 𝐴3 whether they produce a zero-matrix or not, then clarify that the results concept errors, principle errors and operating errors rachmawati, cholily, & zukhrufurrohmah, mathematics communication mistakes … 72 number of question hots aspect mathematics communication indicator on the answer to the problem mathematical error that appears in problem 𝐴𝑛 = 𝐴𝐴…𝐴⏟ 𝑎𝑠 𝑚𝑎𝑛𝑦 𝑎𝑠 𝑛 , is it possible that 𝐴𝑛 produces a zero matrix? provide detailed explanations of the answers! of the matrix 𝐴2 and 𝐴3 are not zero-matrices, and then test for the student matrix 𝐴𝑛 use each idea to find out whether the matrix 𝐴𝑛 will produce a zero matrix create using their language, students write the possibility of the matrix 𝐴𝑛 to produce a zeromatrix concept errors, principle errors and operating errors the result of 155 students work are grouped according to good mathematics communication skill or less mathematics communication skill. the good mathematics communication skills are given when students answer fulfill three indicators of mathematics communication. otherwise, students have less mathematics communication skill. data analysis techniques in this study were carried out through 3 stages: data reduction, data presentation, and concluding. the data reduction stage is carried out by grouping the answers of the two questions based on similar answers into several types of the four classes. this is done to facilitate the analysis based on indicators that have been determined. then the data obtained are presented in tables and diagrams. and the next step is to conclude the result based on data gathered. 3. results and discussion 3.1. results data obtained from the four classes were 155 student answers. figure 1 shows that 148 students have less mathematics communication in the question 1 or first question, and seven students who have excellent mathematics communication. judging from the number of answers shows that mathematics communication possessed by students is still lacking. only 4.5% of the total amount has excellent mathematics communication. it shows that students cannot communicate their ideas well so that they have difficulty in problem-solving. figure 1. number of mathematics communication questions 1 7 148 0 100 200 a m o u n t o f s tu d e n t' s a n sw e r mathematical communication skills question 1 good less volume 10, no 1, february 2021, pp. 69-80 73 whereas in figure 2, the number of answers to second question or question 2 does not differ from question 1. the number of students with communication is more or less compared to students with excellent communication. but for the second question, there is no difference with the distant numbers, namely 85 students with less mathematics communication and 70 students with excellent communication. it means that students are more able to communicate their ideas in the second question than the first question. figure 2. number of mathematics communication questions 2 figure 1 and figure 2 shows that students with good mathematics communication skills are minimal. it can happen because students were making many mistakes when solving problems, such that it did not match the mathematics communication indicator. table 2 shows that most students made mistakes when stating their problem-solving problems by using their language. these mistakes can be taken because students get used to calculating without looking back on the problem. another possibility that causes students to have less mathematics communication skills understands the multiple matrices such that students cannot write their ideas clearly. the weakness of understanding the problems is also one of the factors that caused students to make mistakes to determine the solution steps. the percentages of each student's mathematics communication criteria were presented in table 2. table 2. percentage of mathematics communication of students each indicator no. mathematics communication indicator question 1 question 2 average good less good less good less 1 understand the mathematical situation clearly 84.52% 15.48% 83.23% 16.77% 83.86% 16.13% 2 write ideas of problemsolving clearly 76.77% 23.23% 73.55% 26.45% 75.16% 24.84% 3 state the results of problemsolving using your language 6.45% 93.55% 58.06% 41.94% 32.26% 67.75% table 2 shows that not all students meet the three mathematics communication criteria in solving hots problems. the third criterion of mathematics communication, declare the answer to solving hots's problem using their language, obtain the most significant percentage of 93.55%. this percentage is the highest compared to the other two 70 85 0 20 40 60 80 100 a m o u n t o f s tu d e n t' s a n sw e r mathematical communication skills question 2 good less rachmawati, cholily, & zukhrufurrohmah, mathematics communication mistakes … 74 indicators. it happens because students are incomplete in writing conclusions of problemsolving. the highest rate in the second question for the category lacking was also found in the indicator, stating the solving results using their language that is 41.94%. most students did not write the conclusions of the calculation that they obtain. although students are good at understanding mathematical situations or writing their ideas, it did not mean that students' mathematics communication was excellent. each rate of mathematics communication skills showed that not all students fulfilled all criteria of mathematics communication. students in problem-solving with problems may make mistakes by mathematics communication written owned. the mistakes made by students in completing problems are presented in table 3. table 3. forms of student mathematical errors mathematics communication indicator types of mistakes made by students concept principle operation understand the mathematical situation clearly in question 1, it is wrong to determine matrix 𝐴,𝐵,𝐶 in problem 2, the concept of an identity matrix and matrix multiplication is wrong not found write ideas of problemsolving clearly in question 2, it is wrong to apply the concept of multiplication to the 𝐴𝐴 matrix in question 1, it is wrong to use ideas, so there is no relationship between the matrix 𝐴 𝐵 𝐶 in question 2 it is wrong to form the matrix 𝐴𝑛 because it does not find a relationship with the matrix 𝐴2 and 𝐴3 not found state the results of problem-solving using your language in question 1, it is wrong to conclude that 𝐴𝐶 = 𝐵𝐶. in question 2, it is wrong to conclude that 𝐴𝑛 will probably produce a zero-matrix not found in question 1, it was wrong to calculate the 2𝑥2 matrix multiplication. 3.1.1. mathematical errors when understanding mathematical situations clearly on the indicators of understanding the mathematical situation clearly, students made mistakes, namely misconceptions and principles, but did not make operational errors. in the second question, students did not make misconceptions when understanding mathematical situations. students created a mistake in the first question. the mistake made by students were wrong in determining the a, b, c matrices. these errors resulted in students not getting ac = bc. the answers of students who make misconceptions when understanding the mathematical situation are clearly shown in figure 3. figure 3 shows that students only determine matrix c and then conclude that ac is the same as ab, while matrix a and matrix b are not explained as components. volume 10, no 1, february 2021, pp. 69-80 75 𝐶 = [ 1 2 3 4 ] 𝐴𝐶 = 𝐵𝐶 𝐴 = [ 1 2 3 4 ] = 𝐵[ 1 2 3 4 ] figure 3. misconceptions when understanding mathematical situations in the first problem students make a principles mistake in the form of incorrectly linking the concept of the identity matrix and matrix multiplication. figure 4 is the answer to students who earned a principles error when they did not understand the mathematical situation correctly. students make a mistake when multiplying the matrix and stating that the matrix a2 is the identity matrix. 𝐴2 = [ 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 ]= matrix a still produces an identity matrix (fixed matrix). the answer is the matrices does not produce a zero matrix, because the matrix 𝐴𝑛 still produces an identity matrix because the multiplication result of the matrix is the number of columns. figure 4. mistakes in principle when understanding mathematical situations clearly 3.1.2. mathematical mistakes when writing out problem-solving ideas clearly students make misconceptions, principles, and operations when writing problemsolving ideas on the second problem. but in the first question, no students made mathematical mistakes. in the second problem, students created a misconception, namely not applying the concept of multiplication to matrix a. students wrote the conclusion that an produced a null matrix, but did not test the multiplication against a4 matrix. figure 5 shows students' mistakes, not doing the calculation concept of matrix a, so that they do not write ideas. yes, 𝐴𝑛 yields a zero matrix explanation: suppose 𝐴4= matrix 0 because 𝐴3 = [ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ] figure 5. misconceptions when writing clear problem-solving ideas rachmawati, cholily, & zukhrufurrohmah, mathematics communication mistakes … 76 students made a principles mistake in the second problem by not doing the multiplication concept on the matrix a3. students only perform calculations on matrix a2 so that they do not find the relationship between the a2 and a3 matrices in finding the an matrix. so, it concludes without checking whether an yields a null matrix (figure 6). figure 6. principle mistakes when writing clear problem-solving ideas 3.1.3. mathematical errors when expressing problem solving results in his language students make misconceptions and operational errors when clearly stating the results of solving problems. the mistake made by students when reporting the results of solving the problem was that the multiplication matrix did not recognize ranks so that an could not be a zero matrix. in figure 7 showed that students wrote that an could not be a null matrix because matrix a is not a zero matrix. figure 7. misconceptions when expressing problem solving results in his language operation errors made by students when stating the results of solving the problem in the first problem are wrong to conclude that ac = bc. students get the same results, but the calculations that have been done are not correct. in figure 8, students state that the results of problem-solving do not match the calculations obtained. 𝐴 = [ 2 3 2 4 ]𝐶 = [ 2 −4 −2 4 ] → 𝐴𝐶 = [ −4 8 −2 4 ] 𝐵 = [ 1 3 0 1 ]𝐶 = [ 2 −4 −2 4 ] → 𝐵𝐶 = [ −4 8 −2 4 ] it proved that the both multiplication (above), the matrix 𝐴𝐶 = 𝐵𝐶 have the same result. figure 8. operating errors when stating problem-solving results in his language matrix operation does not recognize the power form. the power referred to in matrix operation is the repeated multiplication of a matrix with the matrix itself. the condition of a matrix to be raised is that it must be a square matrix or a square matrix. thus, the exponent of the square matrix itself is the sum of the powers. if the matrix is nonzero, then 𝐴𝑛 is unlikely to produce a null matrix later. for example number 1, if number 1𝑛 then whatever n will not change from 1 to 0. volume 10, no 1, february 2021, pp. 69-80 77 3.2. discussion the data of figure 1 and figure 2 show that the student's mathematics communication ability was needed to be increased. this finding was in line with firdaus and aini (2019) research, which found that students' communication is still in the low category. this research also shows that most students did not fulfill the three indicators of mathematics communication, which indicated their mathematics communication skills need to be improved. firdaus and aini (2019) also found that students' weakness in mathematics communication can be caused by student mathematics communication indicators that have not been achieved. moreover, syafina and pujiastuti (2020) found that only students with good mathematics ability can fulfill all mathematics communication indicators. this research found that students' ability to state the problem-solving results using their language is the most difficult mathematics communication indicator for students to fulfill. students' answers do not give reasons with their language style, so it does not state the conclusions of the given problem. this phenomenon is also found in mirna's (2018) research. besides, some students still have difficulty writing ideas and knowing the mathematical situation clearly. so that in this study, students made mistakes in concepts, principles, and operations. students' types of errors are in line with zulfah's (2017) research that students make mistakes in the form of concept, process, and principles errors. student's mistakes influenced the ability of mathematic communication measurement in each indicator. in other words, mathematics communication ability related each other with students' mistakes (zulfah, 2017). afifah et al. (2018), in their research, stated that the types of mistakes made by students were concept errors, operation errors, and principles errors. firdaus (2019) research shows that in solving the matrix problem, the type of error made is concept and operation error. likewise, with this study that students also carried out misconceptions and operations. more specifics, based on table 3, most students made conceptual mistakes while solving the problems given. this finding can be considered to point out that students may not be able to communicate their idea well because they did not understand the problem. this possibility inline with abidin et al. (2017) research concluded that a misconception caused another five kinds of students' mistakes. these conceptual mistakes also affect principles mistakes of student's work (mirna, 2018). for example, in figure 4, students' misconception about the definition of identity matrix then applied this wrong idea to analyze the situation. albeit a little, students also make procedural mistakes when they calculate or simplify the algebra form. students also found this mistake in parwati and suharta (2020) research that procedural mistakes are still made frequently by students when solving algebra problems. 4. conclusion students' mathematics communication errors in completing hots problem on matrix material are concept errors, principles errors, and operating errors. these errors can occur in every student mathematics communication indicator. in the first mathematics communication indicator that understands the mathematical situation clearly, mistakes made are concept errors and operating errors. the second indicator of mathematics communication is writing ideas clearly; the mistakes made are concept errors and principles mistakes. as for the third mathematics communication indicator that states the problem-solving results using their language, the mistakes made are concept errors and operating errors. rachmawati, cholily, & zukhrufurrohmah, mathematics communication mistakes … 78 references abidin, s. n. z., zain, s. m. m., hamzah, h. h. m., & abd rahim, n. z. 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(2017). analisis kesalahan peserta didik pada materi persamaan linear dua variabel di kelas viii mts negeri sungai tonang. jurnal cendekia: jurnal pendidikan matematika, 1(1), 12-16. https://doi.org/10.15642/jrpm.2019.4.1.1-10 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p69-80 69 optimization of guided discovery learning models to increase students' interest in mathematics nurhayani *1 , raden rosnawati 2 , tuslikhatun amimah 3 1,2,3 universitas negeri yogyakarta article info abstract article history: received jan 21 2020 revised jan 29, 2020 accepted feb 2, 2020 student interest in learning is a very important factor in determining student success in learning mathematics. various attempts were made by educators and educational researchers to increase student interest in learning. this research is a classroom action research model by kemmis and mc taggart that aims to describe the application of guided discovery learning in optimizing students' interest in learning mathematics. the increase in students' interest in learning mathematics is also supported by the results of student achievement. the research data consisted of students' interest in learning mathematics, learning achievement data, and observations of learning outcomes. data on learning interest in mathematics is obtained through a questionnaire, data on learning achievement is obtained through tests and data on the results of observations of learning achievement are obtained through observation sheets during learning. in general, the results of the study showed that the average student interest in learning mathematics at 83.93 reached the good category. the completeness of student achievement test results reached 83.87% of students achieving the minimum completeness criteria with an average student score of 85.61. the percentage of teacher and student learning outcomes respectively at 83.80% and 76.91% reached the good category. therefore it can be concluded that the guided discovery learning model can be applied to optimize students' interest in mathematics learning especially by paying attention to the results of reflections from this study. keywords: guided discovery learning, learning interest, mathematics education copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: nurhayani, department of mathematics education, universitas negeri yogyakarta, jl. colombo no.1, karang malang, depok, sleman, yogyakarta 55281, indonesia. email: nurhayaniyhaniuny@gmail.com how to cite: nurhayani, n., rosnawati, r., & amimah, t. (2020). optimization of guided discovery learning models to increase students' interest in mathematics. infinity, 9(1), 69-80. 1. introduction education is very important in the survival of life. education can make the individual into a person of knowledge and character. education cannot be separated from school, because in school there is a teaching and learning process that can be a provision of children's education. one of the subjects studied at school is mathematics. the role of mathematics increase in line with the development of science. someone who understands mailto:nurhayaniyhaniuny@gmail.com nurhayani, rosnawati, & amimah, optimization of guided discovery learning models … 70 and can do mathematics has the opportunity to increase significantly so that the y can make their future. this opinion is in line with nctm (2000) which revealed that someone who understands mathematics will be able to increase opportunities and choices to make their future. however, the reality shows that there are still obstacles or problems faced by students in learning, especially learning mathematics. one of the problems currently facing our education world is the problem of poor understanding of mathematics in the learning process. in the learning process, students have less interest in learning mathematics. this can be seen from observations made by researchers in one of the schools namely in man 1 yogyakarta, especially in class xi mipa 2. researchers observed the lack of students who have math textbooks that can be used to study mathematics. student notes are also incomplete. when asked about the material they have recorded, not all students are able to explain back to the researcher the purpose of their notes. this also shows the lack of student attention to mathematics. in addition, student involvement in the learning process is still lacking so that student interaction is not achieved properly. this causes learning to be less meaningful and easy for students to forget. this is supported by the results of students’ interest questionnaire before implementing guided discovery learning that shown in table 1. table 1. results of questionnaire students' interest before implementing guided discovery learning criteria early mathematics learning interest very good 0% (0 student) good 22.58% (7 student) quite good 58.06% (18 student) poor 12.90% (4 student) very poor 6.45% (2 student) from the table 1, we obtained information on interest in learning mathematics before implementing guided discovery learning model shows that there is not yet a student who achieved very good criteria, 22.58% or 7 students included in the good category, 58.06% or 18 students included in the quite good category, 12.90% or 4 students included in the poor category and 6.45% or 2 students included in the very poor category. so that obtained an average for initial interest of 75.93 which is included in the category is quite good. this reflects the lack of student interest in learning so that learning achievement still low. the results of pre-research observations show that learning achievement is still low and does not meet the minimum completeness criteria (mcc) set by the school. it can be shown in table 2. table 2 data of learning achievement before implementing guided discovery learning description criteria mean 26 standar deviation 19.459 ideal minimum score 0 minimum score 0 ideal maximum score 100 maximum score 70 volume 9, no 1, february 2020, pp. 69-80 71 variance 378.667 minimum completeness criteria 76 from the table 2 obtained information on students’ learning achievement before implementing guided discovery learning model shows that the mean of students’ learning achievement is 26 with minimum and maximum score respectively 0 and 70. according the table, it can be concluded that there were no students or 0% of students who achieved the minimum completeness criteria (mcc=76). wibowo (2017) revealed that interest is an impulse in a person to give more attention to an object. furthermore, henriksen, dillon, & ryder (2015) say that interest illustrates a mindset that is characterized by the need to give selective attention to something that is important to someone such as an activity, purpose, or subject. this is in accordance with the expression elliot (astutik, 2017) who revealed that interest is the same thing and is closely related to curiosity or interest where the characteristics are expressed by the relationship between a person and certain activities or objects. students who have an interest in a particular subject give greater attention to the subject. this can be interpreted that if students have a high interest in mathematics it will facilitate students in learning mathematics, because their attention will tend to learn mathematics. based on the description above, it can be concluded that interest is an attitude of awareness of each individual to do something that moves themselves because there is a sense of pleasure and interest to achieve the desired goal. the level of student interest in learning can be seen and measured by: 1) feelings like pleasure (krapp & prentzel, 2011), 2) interests marked by reading and learning mathematics, learning without coercion, and recording mathematics lessons (paszkowska-rogacz & yıldız, 2015) , 3) attention that is in the form of focus in learning mathematics (vulperhorst, wessels, bakker, & akkerman, 2018) and 4) student involvement which includes activeness in participating in the learning process and doing assignments (mumba, mbewe, & chabalengula, 2015) learning outcomes that are often seen are learning achievements. nitko & brookhart (2011) revealed that achievements are students' knowledge, skills and abilities that have been developed as a result of learning. the high interest in learning mathematics students will support increased student learning achievement. the same thing was expressed by herzog, ehlert, & fritz (2019) that interest has a strong positive relationship with success related to mathematics. the correlation between interests and achievements is quite high because the more people who are attracted to an object the more knowledge they obtain. thus, learning mathematics should be able to stimulate students to gain high interest in learning mathematics and good learning achievement. therefore, solutions are needed to realize these goals, such as the application of appropriate learning models so that students' interest in learning can increase and train students to be actively involved in the learning process. so students can build active interaction each other and eith teachers in order to build their knowledge. one model solution that allows to increase student interest in learning is guided discovery learning models. in discovery learning, students must determine their own model used to solve problems as said by alfieri, brooks, aldrich, & tenenbaum (2011) that “discovery learning occurs whenever the learner is not provided with the target information or conceptual understanding and must find it independently and with only the provided materials”. guided discovery is a learning model that emphasizes the activeness and involvement of students in learning. this is in line with what was expressed by khasnis & aithal (2011) that guided discovery combines two strategies, namely learning to work alone and learning to work in groups in an effort to make learning more efficient for all students. furthermore khomsiatun & retnawati (2015) revealed that guided discovery nurhayani, rosnawati, & amimah, optimization of guided discovery learning models … 72 involves a process that comes from one's own experience so that students will get as many opportunities as they can to discover for themselves the knowledge they need to master. the teacher's guidance can be done directly or poured in the form of student activity sheets. mulyana, rusdi, & vivanti (2018) view that guided discovery will provide an opportunity for students to learn about how to find facts, concepts, and principles based on the experience they are doing directly. students freely develop their ideas and knowledge in learning, but if there are obstacles the teacher has a role to provide assistance or guide students to find the right concepts. this is consistent with the opinion of tran, nguyen, bui, & phan (2014) which states that, in guided discovery, teacher give problem, provide context, necessary tools and students have opportunities to discover, solve problem. teacher here plays a role as an encourageing, assistant man to ensure that students do not have troubles or do not perform their surveys, experiments. according to arends (2012) there are several steps involved in guided discovery learning, namely preparation and explanation of the discovery process, problem presentation, formulation of hypotheses, data collection to test hypotheses, formulation of test results and conclusions, as well as reflection on learning activities. meanwhile, according to agustyarini & jailani (2015), there are four stages of guided discovery namely, formulating questions, building procedures and gathering information, using procedures and information obtained in the second step, analyzing and evaluating the discovery process that has been carried out. furthermore, sanjaya (2008) mentions the steps of guided discovery learning methods including orientation, formulating problems, formulating hypotheses, collecting data, testing hypotheses and formulating conclusions. framework for thinking about the relationship between the steps of guided discovery learning and indicators of interest in learning look like the figure 1. figure 1. the linkages of the steps of guided discovery learning volume 9, no 1, february 2020, pp. 69-80 73 and aspects of student interest in learning the steps of guided discovery learning models allow it to be used to optimize the aspects that exist in students' interest in learning. like the orientation phase that is intended to foster an atmosphere of learning that is responsive can bring joy or excitement to students before starting learning. this is in line with what was expressed by krapp & prentzel (2011) that the feeling of pleasure is one indicator of interest in learning, so that a high interest score indicates that students are happy when learning mathematics. the stage of formulating problems and hypotheses can stimulate students' interest and attention to read, focus on learning mathematics and be active in learning without coercion paszkowska-rogacz & yıldız (2015). then the stages of collecting data and testing hypotheses allow students to feel happy and can encourage students to learn and practice more to solve the given problem. this stage encourages students to interact with assignments so as to increase student interest in learning. this is supported by the statement of habig et al. (2018) which says that the assignments given to students create interactions between assignments and students that are able to increase student interest in learning. next to the final stage, namely formulating conclusions can stimulate students to focus, record all lessons learned and be active in taking part in expressing opinions or answers obtained by mumba et al. (2015). thus, the relationship between these variables is expected to increase students' interest in learning mathematics. 2. method this research is a class action research conducted in class xi mipa 2 man 1 yogyakarta which is located at jl. c. simanjuntak no. 60, terban, kota yogyakarta. this research was conducted in mathematics with trigonometry material in odd semester 2019/2020 school year. this research was conducted in two cycles. the purpose of classroom action research is an effort to improve teaching practices through the provision of actions in class that begin with lesson plans followed by actions and observations in class as well as reflections on these actions. the implementation of this study was designed to follow the kemmis and mc taggart planning models, namely, planning, implementing actions, observing and reflecting. the next cycle is based on the results of reflection. the data in this study consisted of learning achievement data, students' interest in learning mathematics, and observations of learning outcomes. data from observations of the implementation of learning contain activities during the teaching of mathematics obtained through observation sheets when implementing actions and analyzed descriptively during reflection. learning achievement data was obtained through tests on trigonometric material and students' mathematics learning interest data were obtained through a questionnaire with four indicators namely feeling, attraction, attention and involvement. validity test used in this study uses the product moment correlation formula:          2 2 2 2 xy n xy x y r n x x n y y            xy r = correlation coefficient betwen variable x dan variable y n xy = the number of multiplications between variable x dan variable y nurhayani, rosnawati, & amimah, optimization of guided discovery learning models … 74 x = number of variables x y = number of variables y from the results of the validity test using spss 21 as many as 31 respondents who answered 25 questionnaires from the total questionnaire given to respondents included in the valid category so that there were no questionnaires that needed to be deleted (a) as shown in the table 3. table 3 validity test results n % cases valid 31 100 excluded a 0 0 total 31 100 based on the table 3, it can be concluded that the learning interest questionnaire used in this study is valid. as for the reliability coefficient estimation used the alpha croncbach formula with the following formula. 2 1 2 1 1 i p yi x p p                 information:  = reliability sought p = number of questionnaires tested 2 1 i p yi   = number of variance scores per questionnaire 2 x  = total variance the reliability test results are shown in the table 4. table 4 reliability test results cronbach’s alpha n of items 0.823 31 based on the results of the reliability test using spss 21 (table 4) as indicated in the table above, obtained a significant alpha cronbach value of 0.823. this means that the learning interest questionnaire used is included in the reliable category. there are several indicators of success used in this study, including indicators of success in learning interest, indicators of success in learning achievement and indicators of learning accomplishment. this study was successful if the average student interest in learning was included in either category with an interval value (x) of 83.3 < x 100, students who achieved mastery learning achievement reached a minimum of 75%, and the implementation of learning reached a good category with a percentage (p) which is 65% < p 85%. 3. results and discussion 3.1. cycle i volume 9, no 1, february 2020, pp. 69-80 75 the first cycle carried out four meetings which began from 7, 8, 14 and 15 october 2019. there are two core materials that students learn about simple trigonometric equations and trigonometric equations in the form of quadratic equations. based on the data obtained, observations of the implementation of guided discovery learning by the teacher in the first cycle are in the good category with a percentage of 77.14%. this shows that the implementation of guided discovery learning by the teacher in the first cycle has reached the predetermined indicator that is 75%. the observation of the implementation of guided discovery learning by students in the first cycle is in the quite good category with a percentage of 60%. this shows that the implementation of guided discovery learning by students in the first cycle has not reached the predetermined indicator that is 75%. the results of the student interest in learning mathematics questionnaire at the end of the first cycle are shown in table 5. table 5 results of questionnaire students' interest in cycle i criteria mathematics learning interest cycle i very good 0% (0 student) good 38.7% (12 student) quite good 51.61% (16 student) poor 9.67% (3 student) very poor 0% (0 student) from the table 5, information obtained on the results of the interest in learning mathematics after the implementation of guided discovery learning in cycle i, there were no students in the excellent category, amounting to 38.7% or 12 students included in the good category, amounting to 51.61% or 16 students included in the good enough category, amounting to 9.67 or 3 students fall into the unfavorable category, and there are no students whose interest in learning mathematics is included in the very poor category. in order to obtain an average student interest in learning for cycle 1 of 79.35 which is included in the quite good category. one indicator of success in this study, said to be successful if the average student interest in learning is included in either category with an interval value (x) that is 83.3 < x ≤ 100. because the average value of students 'interest in learning for cycle 1 is 79.35, the results of students' interest in learning show that they have not been successful or have not met the specified success indicators. the percentage of learning achievement after the guided discovery model applied in cycle 1 was 51.61% or 16 students reached mcc and 48.38% or 15 students who have not yet reached the mcc with an average student score of 71.29. indicators of success from the results of learning achievement are said to be successful if the student score reaches the completeness of learning achievement reaches a minimum of 75%. however, because the results of learning achievement in cycle 1 showed only 51.61% or 16 students reached mcc, the results of student achievement showed not successful or did not meet the specified success indicators. the learning implementation data reached a good category with a percentage (p) that is 65% < p ≤ 85%. in the learning process that has been carried out, it is stated several things that have not been carried out in accordance with the mindset chart designed by the researcher at the outset. the teacher starts the lesson by praying and checking student attendance. the teacher then gives apperception to students and conveys the material to be studied as well as a brief explanation that can lead students to solve the problems given in the worksheet. nurhayani, rosnawati, & amimah, optimization of guided discovery learning models … 76 in the learning process students learn by group settings. there are 6 groups formed consisting of 5-6 students per group. this group division is done randomly. each group is given a worksheet that contains problems related to trigonometric equations. at the orientation stage, namely the granting of apperception, the majority of students still did not show excitement because the responses given by students were still lacking. students also do not record fully what is conveyed by the teacher. then in the stage of formulating the problem and formulating a hypothesis, only some members of the group are actively involved to understand the problem given which shows that students' interest is still lacking. this is in line with what was delivered by rautiainen, mäensivu, & nikkola (2018) which revealed that students who pay attention to lessons and are actively involved in the lessons indicate that these students have high learning interest and vice versa. when learning in class, most students are still confused to develop their way of thinking to solve the problems presented. therefore, the teacher gives limited direction, but apparently students are still having difficulties. this causes the direction of the teacher still dominates the learning process compared to the activeness of students in developing their way of thinking. at the stage of formulating conclusions, the teacher tries to encourage students to present their findings in front of the class, but student participation is still lacking. the student representatives who wrote the answers on the board, were still constrained in explaining what they had written. in addition, other students have not fully focused on listening to what their peers say. this indicates that the stages of the guided discovery model have not fully stimulated students' interest in learning. based on the results of cycle 1, it is necessary to do further learning in cycle ii. 3.2. cycle ii the second cycle carried out four meetings which began on 28, 29 october, 7 and 11 november 2019. in cycle ii, the material students learned was analytic trigonometry. based on the data obtained, the results of students' interest in learning mathematics at the end of the second cycle are shown in table 6. table 6. results of questionnaire students' interest in cycle ii criteria mathematics learning interest cycle ii very good 9.67% (3 student) good 41.93% (13 student) quite good 48.38% (15 student) poor 0% (0 student) very poor 0% (0 student) based on the table 6 obtained information about the percentage of interest in learning mathematics after the application of guided discovery learning in cycle ii, namely 9.67% or 3 students achieving very good criteria, 41.93% or 13 students achieving good criteria, 48.38% or 15 students achieving the criteria quite good, and already none of the students achieved the criteria for being either poor or very poor. the conditions of students' interest in learning cycle i and cycle ii as a whole are shown in figure 2. volume 9, no 1, february 2020, pp. 69-80 77 figure 2. conditions of student interest in cycle 1 and cycle 2 from the figure 2, it appears that the conditions of student interest in learning individually in cycle i and cycle ii. although not all students experience an increase in learning interest from cycle i to cycle ii as happened to the first, third, twelfth, thirteenth and other students, but overall the average student interest in learning for cycle ii has reached 83.93. this shows that the results of students’interest in learning have reached the specified indicators of success. the conditions of students for learning achievement results are shown in the figure 3. figure 3. conditions of students’ learning achievement in cycle 1 and cycle 2 figure 3 show that not all students have increased learning achievement. this is shown by the first, third, ninth, twelfth, and other students, but overall the percentage of learning achievement obtained by students after the guided discovery model has been applied in the cycle ii has reached 83.87% or as many as 26 students have achieved mcc and the remaining 16.12% or 5 students have not yet reached the mcc with an average student score of 85.61. this shows that the mastery of learning achievement has reached the specified indicator. observations on the implementation of guided discovery learning by the teacher in the second cycle are in the good category with a percentage of 83.80%. this shows that the implementation of guided discovery learning by the teacher in the second cycle has 0 20 40 60 80 100 120 pd 1 pd 3 pd 5 pd 7 pd 9 pd 11 pd 13 pd 15 pd 17 pd 19 pd 21 pd 23 pd 25 pd 27 pd 29 pd 31 students' interest in cycle i and cycle ii students' interest in cycle i student interest in cycle 2 0 50 100 150 pd 1 pd 3 pd 5 pd 7 pd 9 pd 11 pd 13 pd 15 pd 17 pd 19 pd 21 pd 23 pd 25 pd 27 pd 29 pd 31 students' learning achievement in cycle i and cycle ii students' learning achievement in cycle i students' learning achievement in cycle 2 nurhayani, rosnawati, & amimah, optimization of guided discovery learning models … 78 reached the specified indicator that is ≥ 75%. observations on the implementation of guided discovery learning by students in the second cycle are also already in the good category with a percentage of 76.19%. this shows that the implementation of guided discovery learning model by students in the second cycle also has reached the specified indicator that is ≥ 75%. based on the data obtained, the average learning interest of students is 83.93 which is included in either category, the percentage of student achievement is 83.87% or as many as 26 students have succeeded in achieving the minimum completeness criteria, and the percentage of student and teacher learning outcomes respectively 83.80% and 76.91% have achieved good criteria. in process learning, the teacher opens the lesson by praying and checking student attendance. the teacher then gives apperception to the students and tries to ask some questions about the material that has been given in cycle i. after that the teacher then gives motivation and advice to be more earnest in learning, because understanding the material taught will really help them to understand the next material , considering that trigonometry material is a fairly dense and continuous material. in contrast to during the first cycle, this time the students in the class began to respond by giving questions to the teacher, the interaction was already more visible. group settings are no longer done in this second cycle. the teacher tries to use power point media to make it easy to present problems that will be solved by students. the teacher emphasizes that each student will be called randomly to present the results of their work. the class conditions began to be active which indicates student interest in learning is better than before. this is in line with what was expressed by rautiainen et al. (2018) who revealed that students who pay attention to lessons and are actively involved in learning indicate that these students have a high interest in learning. the teacher then asks each student to prepare a notebook and asks students to write down and formulate problems and hypotheses of the problems presented. students seem more excited and interact with each other to exchange opinions. stage of collecting and testing hypotheses, the teacher goes around observing students. some students who experience obstacles start to build interaction with the teacher by confirming their answers, those who ask questions they don't understand. after the work time is up, the teacher checks by asking students who have finished working to raise their hands. in this case there are still some students who have not been able to solve the problems given, especially the male students. the teacher then provides assistance to students who have not yet finished and provokes other students who dare to write and explain their answers without being appointed in front of the blackboard. some students began to dare to raise their hands and come forward to present the results of their work. after that the teacher asks other friends to respond and ask questions they don't understand. at the end of the lesson the teacher corrects if there are errors and gives reinforcement to the concepts of the material that have been learned so that the understanding of the concepts obtained by students is not wrong. this is in accordance with the opinion of kirschner, sweller & clark (2006) who view that the learning process must be guided by the teacher so that students get the right understanding of concepts. the teacher appreciates any progress made by students by asking other friends to give applause to friends who dare to present their work in front of the blackboard. students also looked happy, especially those who managed to solve the problem and who were given an appreciation by their friends for daring to present the results of their work. according the explanantion above, it can be concluded that in cycle ii the overall indicators of success from this study have been achieved. volume 9, no 1, february 2020, pp. 69-80 79 4. conclusion after carrying out learning for two cycles, in cycle i we obtain average students’ interest is 79.35 which is included in the quite good category and have not met the specified success indicators. the average student interest in learning for cycle ii has reached 83.93. this shows that the results of interest in learning have reached the specified success indicators. the results of learning achievement in cycle 1 showed only 51.61% or 16 students reached mcc, it showed not successful or have not met the specified success indicators. in cycle ii, the results of learning achievement has reached 83.87% or as many as 26 students have achieved mcc and the remaining 16.12% or 5 students have not yet reached the mcc with an average student score of 85.61. this shows that the mastery of learning achievement has reached the specified indicator. observations on the implementation of guided discovery learning by the teacher and students in cycle i rescpectively are 77.14% and 60%. this shows that. it has not met the specified success indicators. in cycle ii, observations on the implementation of guided discovery learning by the teacher and students in cycle i rescpectively are 83.80% and 76.19%. this shows that the implementation of guided discovery learning model by teacher and students in cycle ii also has reached the specified indicator that is ≥ 75. according the result of this research, it appears that the interrelationships between the guided discovery learning model and learning interest, it is found that the steps in the guided discovery learning model can be used to optimize aspects of student interest in learning. so it can be concluded that the application of guided discovery learning can be used to optimize students' interest in learning mathematics. references agustyarini, y., & jailani, j. 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(2017). pengaruh pendekatan pembelajaran matematika realistik dan saintifik terhadap prestasi belajar, kemampuan penalaran matematis dan minat belajar. jurnal riset pendidikan matematika, 4(1), 1–10. https://doi.org/10.21831/jrpm.v4i1.10066 https://doi.org/10.21831/jrpm.v4i1.10066 https://doi.org/10.21831/jrpm.v4i1.10066 https://doi.org/10.21831/jrpm.v4i1.10066 https://doi.org/10.21831/jrpm.v4i1.10066 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p203-216 203 pisa-like problems using islamic ethnomathematics approach muhammad win afgani*, retni paradesa universitas islam negeri raden fatah palembang, indonesia article info abstract article history: received nov 19, 2019 revised apr 8, 2021 accepted apr 11, 2021 the study aimed to produce pisa-like mathematics problems with the islamic ethnomathematics approach that were valid and practical. a development study with formative evaluation was used as the method in this study with 32 9th-grade students as the subjects at one of the junior high schools in palembang, south sumatra province, indonesia. there are five phases: selfevaluation, expert review, one-to-one, small group, and field test. interviews, questionnaires, and tests were used in this study as the instruments to collect the data. the results showed that three experts from the expert review phase assess that 77.78% agree that six pisa-like mathematics problems meet the validity criteria. for the practicality criteria, three students from the one-toone step set about 77,78% agree, three students from the small group phase assessed about 61,11% strongly agree, and 26 students from the field test phase considered 61,33% agree. this result is supported by the average test result that was classified as a low category. this showed that the results obtained are not optimal because students still did not understand the problems and had difficulty solving them. keywords: islamic ethnomathematics, mathematics problems, pisa copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: muhammad win afgani, department of mathematics education, faculty of tarbiyah science and teacher training, universitas islam negeri raden fatah palembang jl. prof. k. h. zainal abidin fikri km 3,5 kemuning, palembang, south sumatra 30126, indonesia email: muhammadwinafgani_uin@radenfatah.ac.id how to cite: afgani, m. w., & paradesa, r. (2021). pisa-like problems using islamic ethnomathematics approach. infinity, 10(2), 203-216. 1. introduction minister of education and culture regulation no. 22 of 2016 (mecri, 2016) emphases that the purpose of learning mathematics at primary and secondary education levels is that students are able to: (1) have mathematical knowledge, (2) use reasoning, (3) solve problems, (4) communicate ideas with symbols, tables, diagrams, or other media to clarify the situation or problem and (5) have an attitude of appreciating the usefulness of mathematics. mathematical learning objectives are also formulated by the national council of teachers of mathematics, but to realize it is not easy (mckinney & frazier, 2008). the results of the program for international student assessment (pisa) and trends in international mathematics and science study (timss) show that middle-level students in indonesia still do not have high mathematical abilities. https://doi.org/10.22460/infinity.v10i2.p203-216 afgani & paradesa, pisa-like problems using islamic ethnomathematics … 204 indonesian students have difficulty solving pisa problems because of its problems are very demanding of reasoning and problem-solving ability while they usually solve a mathematics problem that emphasizes mastery of basic mathematics skills. this is in line with the results of oktaviana et al. (2018), damayanti and yunianta (2018), and aprianti and kesumawati (2019) study which reported that students' mathematical problem-solving ability is not optimal yet, likewise their mathematical reasoning ability according to handini et al. (2015), napitupulu et al. (2016), and cahyani (2019) study. the abilities are not optimal, according to afgani (2018) showed that the students' mathematical understanding scheme has a problem. this means that mathematical knowledge schemes that students have learned are not intertwined. pisa problems also use a global context that is not close to the daily lives of students in indonesia. as a result, indonesian students cannot imagine the relevance of the problems to their mathematics understanding, because the use of local cultural contexts, such as jambi (charmila et al., 2016; muslimahayati, 2019), lampung (putra et al., 2016), palembang (utari, 2017), north sumatra (hasanah, 2017), can help students understand the mathematical content contained in the problems. therefore, to familiarize indonesian students' interest and accustom to solve mathematical problems in the form of reasoning and problem solving, it is necessary to apply innovation in developing pisa-like mathematical problems with contexts that are close to student life. meanwhile, mardhiyanti et al. (2011) reported that pisa-like mathematics problems have positive potential effects on students' mathematical ability. the involvement of the context in pisa problems is in line with the ethnomathematics approach, but the context under consideration is the culture inherent in the lives of students. with ethnomathematics approach, giving pisa problems to students, besides to develop mathematical reasoning and problem-solving ability or mathematical literacy (kehi et al., 2019), aims to also introduce the culture that exists around students, so students can better appreciate and preserve the culture. one of the cultures that are around students is islamic culture, because ethnomathematics in indonesia relates to it, such as mosque architecture in kudus culture (masamah, 2018), as well as mosques in palembang has unique architecture, and also influenced by other cultures. this means that cultural acculturation in mosque buildings shows that islamic ethnomathematics can be given to a group of multicultural students because it is positioned only as a context of mathematics problems (kehi et al., 2019). islamic culture is a consideration to be integrated into the development of the problem because the majority of the population in palembang is muslim, so are students in schools who are the subject of this study. therefore, the additional mission in this research is specific to introduce islamic culture. besides that, the presence of nonmuslim students in schools requires researchers to develop mathematical problems with contexts containing islamic elements without offending them and they do not mind to solve the problems. so, this study aimed to produce pisa-like mathematics problems with the islamic ethnomathematics approach that were valid and practical based on the considerations for 9th-grade students of junior high school. the introduction presents the purpose of the studies reported and their relationship to earlier work in the field. it should not be an extensive review of the literature. use only those references required to provide the most salient background to allow the readers to understand and evaluate the purpose and results of the present study without referring to previous publications on the topic. volume 10, no 2, september 2021, pp. 203-216 205 2. method in this study, the method used a development study by van den akker (1999). there are five phases that are, self-evaluation, expert review, one to one, small group, and field test. the five phases are called formative evaluation. before the phases are conducted, the preliminary evaluation conducted firstly. this reference is from tessmer (1993) and zulkardi (2002). in a preliminary evaluation, characteristics of the pisa problem, ethnomathematics approach, literature review of islamic culture in palembang, and junior high school curriculum in indonesia were analyzed, and then mathematics problems were developed based on the framework of pisa. in the first phase of formative evaluation, self-evaluation was conducted to assess the first prototype of pisa-like mathematics problems by the researcher to find the obvious error of words, sentences, and mathematical language. the second phase was an expert review. in this phase, the first prototype was reviewed by experts in mathematics education that are, lecture and teacher. the experts validated the content, construct, and language of the mathematics problems. the instrument used an open and closed questionnaire. for a close questionnaire, the likert scale was used to collect the data with nine items and four assessments, that are, strongly agree (sa), agree (a), disagree (d), strongly disagree (sd). after that, the open questionnaire was used to verify the reason for their assessment. after the first prototype was revised and become the second prototype, it was tried to students in one to one phase. they were 9th-grade students of junior high school in palembang. in this phase, the second prototype was tried to see the practicality according to tessmer (1993) by 3 students with a representative background in mathematical ability that are, low, middle, and high. they were given a chance to solve the problems individually. like instruments in the expert review phase, one to one phase used a questionnaire with six items and added an interview. the interview was used to identify students' understanding and difficulty in the problems. after the second prototype was revised and become the third prototype, it was tried to small group phase with 3 students. in this phase, practicality about the problems was continued to be observed and used the same instruments like one to one, but the students solve the problems in a group and they were given a chance to discuss it. this phase would produce the fourth prototype. the prototype experimented in a field test with 26 of 9th-grade students of junior high school. in the field test, the instruments used questionnaires and tests to evaluate the practicality. the test was analyzed by three indicators that are, no answer, irrelevance answer, and relevance answer with consecutive scores 0, 1, and 2. for conclusion, the average score was converted by using formula total score divided ideal score, and then multiplied 100. the result was categorized after that. 3. results and discussion 3.1. preliminary evaluation phase in this phase, characteristics of the pisa problem, ethnomathematics approach, literature review of islamic culture in palembang, and junior high school mathematics curriculum of 2013 in indonesia were analyzed so that the first prototype with 6 mathematics problems has resulted. it covered one problem with quantity content, two problems with space and shape contents, one problem with change and relationships, one problem with uncertainty content, and one problem with relationship content. the first problem regarding the quantity content related to plane figure concept and quantity of people that occupy an area with certain conditions. the narrative of the problem is as follows (see figure 1). afgani & paradesa, pisa-like problems using islamic ethnomathematics … 206 muslims are required to pray five times a day and night. the prophet muhammad saw strongly recommended that men who were already pubescent do it in the mosque. estimate the maximum number of adult congregations a mosque that has an area of 200 m2 of worship can accommodate when praying together? explain your answer! figure 1. the first problem based on junior high school mathematics curriculum of 2013, area concept is learned by junior high students in 7th grade, while islamic ethno that is related to quantity content was praying together in the mosque building (see figure 1). hereinafter, the problem with space and shape content related to the pyramid, and the third problem with the same content related to the circle (see figure 2). the typical architectural style at the great mosque of palembang is the roof structure pattern of the main building which is pyramid and consists of three levels. how to sketch the roof of the main mosque, when viewed from above? the roof of the great mosque of palembang is in the form of a pyramid with a row of golden ornaments in the shape of spearheads like the style of the roofs of chinese houses, even the minarets of the mosque are in the shape of a pagoda with metal decorations in the shape of a crescent moon and a star at the top. if the crescent moon ornament is sketched from an intersection of two circles with a radius ratio of 0.8, what is the minimum distance from the center of the two circles to form a crescent moon? figure 2. the second problem islamic ethno that related to the pyramid was the roof of the great mosque, whereas the intersection of two circles related to the crescent ornament on its roof. pyramid and circle subject is learned by junior high students in 8th grade, while the intersection is learned in sets subject in 7th grade. the third problem with change and relationship content related to exponent or power of numbers that are in formula of population growth projection. the narrative of the problem is as follows (see figure 3). figure 3. the third problem volume 10, no 2, september 2021, pp. 203-216 207 exponent or power of numbers also related to the degree of algebra form that is learned by junior high school students in 7th grade. islamic ethno that is involved in the problem was related to cultural pilgrimage activities after doing prayers where they are pilgrims to the great mosque which has grown from year to year. for uncertainty content, the problem that was developed related to data presented in a chart. a line diagram or chart is learned by junior high school students in 7th grade. the islamic ethno related to the problem is a culture of preparation wedding that is held by people in palembang (see figure 4). figure 4. the fourth problem while the last problem, it covered quantity content and related to the number pattern. mathematics subject relates to number pattern is learned by junior high school students in 8th grade. islamic ethno related to the problem is the habits of muslims in palembang when performing prayers using prayer beads repeatedly (see figure 5). figure 5. the fifth problem afgani & paradesa, pisa-like problems using islamic ethnomathematics … 208 this preliminary phase produced the first prototype of pisa-like problems with the islamic ethnomathematics approach to be reviewed by the researcher and the experts in the expert review phase. 3.2. self-evaluation phase in this phase, pisa-like mathematics problems that have been designed were evaluated and reviewed independently by the researcher. the result assumed that the first prototype of the pisa-like mathematics problem fulfills content and construct validity based on suitability with the problems characteristic according to pisa framework, ethnomathematics approach, junior high school mathematics curriculum of 2013, and there were not found an obvious error in words, sentences, and mathematical language. 3.3. expert reviews phase in this phase, the first prototype of pisa-like mathematics problems was reviewed by three experts to observe the validity of the problems relate to pisa characteristic framework, ethnomathematics, and compatibility with the curriculum used in junior high school. they are dak, bas, and yp. dak is a lecturer in padjajaran university, bandung that has a doctorate in mathematics education. the second expert, bas is a mathematics education lecturer at the university of pgri semarang that has a doctorate too. based on the background of both lectures, the researcher set them as the experts to validate content and construct of the problems via whatsapp. while yp, she is a teacher of mathematics subject matter at one of the public junior high schools in palembang that has a master's degree in mathematics education. yp has experience in teaching for 22 years. because of that background, the researcher also set yp as an expert. they reviewed the content and construct validity of pisa-like mathematics problems are in table 1. table 1. expert reviews for the first prototype of pisa-like mathematics problems no. reviewer review 1 dak • for the first problem, the ethno aspect is lacking. if the direction is to islamic ethnomathematics, it is not specific enough to mention the area/region, for example, the scope is narrowed, where the mosque or prayer location. if the ethno domain in arab (the center of islam), then you should mention the location in arab, but if in indonesia, for example, in aceh or demak. if this, it is too general and it leads to thematic problems or mathematical connections (combining math with islamic religion education). it can also be added like this: the students in pesantren a (the name of the pesantren in the research area that is well known, but which is located not in the city, in the suburbs and still thick customs, as in east java, for example gontor), then give a little narration about the unique habits that are ethnic which was carried out at the pesantren, for example ceremonies making red and white porridge when celebrating the birthday of the prophet muhammad (if in east java, there are habits like that). the problem can be developed again, which is dug up from the culture that is usually done by the students (not only about the size of the mosque, can be to the maximum number of al-quran that must be provided if there are so many students who enter per semester), which represents "culture" here is the habitual activity of the students who "fuse" with the local culture (the community around the pesantren). volume 10, no 2, september 2021, pp. 203-216 209 no. reviewer review • for the second problem, the ethno isn't deep enough. just because of the word "typical architecture of the great mosque of palembang", so it's still lacking. the architecture of the great mosque of palembang is influenced by what? • for the third problem, is it palembang ethno or china? the ethno is obscure. • for the fourth problem, this is more contextual. • for the fifth problem, this seems appropriate, because palembang ethno exists and islam also exists. "suku" is the name of a unit of weight typical of palembang?. ethnomath in palembang interesting too. • for the sixth problem, if this is too islamic and the ethno is not compatible yet. this is more of a historical islamic contextual or this problem can be completed with specific tribal names. ethno must bring up "cultural uniqueness" and the definition of culture here is different from routinity. 2 bas the problem of relationship content (no. 6) and quantity content (no. 1) are okay, the problem of uncertainty content is okay (no. 5), the problem of change and relationship (no. 4) is less related to islam and can still be used in context and need to state the validity of the formula, it must be explored again. problem no. 2 and 3 need to add images. 3 yp the problems are already quite good and under the local cultural content, but some writings are not under the rules of the indonesian language, for example, the name of the place that still uses lowercase letters, as in problem no. 6, palembang is written with palembang. overall, the problems are good. from their review, the first prototype was revised and become the second prototype. hereinafter, the three experts assessed again the validity of the problems that had been developed through a questionnaire. the results of the data from the questionnaire can be seen in table 2. table 2. the validity of the second prototype according to experts review no reviewer rating sa a d sd 1 dak 22.22% 77.78% 0.00% 0.00% 2 bas 0.00% 100% 0.00% 0.00% 3 yp 33.33% 55.56% 11.11% 0.00% average 18.52% 77.78% 3.70% 0.00% from the validity result in table 2, the second prototype of problems was valid in content and construct which about 77.78% of three experts assessed agree that pisa-like mathematics problems that were developed fulfill valid criteria. in the next phase, one to one phase, the second prototype of problems experimented with three students with different cognitive ability that is low, middle, and high. afgani & paradesa, pisa-like problems using islamic ethnomathematics … 210 3.4. one to one phase in this phase, the researchers tested the second prototype problem to see its practicality to three students of 9th grade with three abilities, namely high, medium, and low. the researcher assigned them to solve the problems individually. the results of interviews with teachers obtained information that students who have the mathematical high ability are nb, the medium ability is mr, and low ability is ad. the assessment of the three students in this phase regarding the practicality of the problems can be seen in table 3. table 3. the practicality of the second prototype on one to one phase no student rating sa a d sd 1 nb 16.67% 83.33% 0.00% 0.00% 2 mr 0.00% 66.67% 33.33% 0.00% 3 ad 0.00% 83.33% 16.67% 0.00% average 5.56% 77.78% 16.67% 0.00% from the results of the questionnaire (see table 3), about 77.78% of three students considered agree that the second prototype problems met the practicality criteria. also, the three students gave comments on the problems they solved. their comments are presented in table 4. table 4. student comments on the second prototype problem no student comments 1 nb • the problems are too many explanations, so they must be understood more deeply. • the contents of several problems have not been studied. • the problems have not led to the imagination. 2 mr • the problem that is difficult for me is part of the circle ratio and the increasing population of mosque users. • the contents of the problems are less interesting because there are too many narratives, preferably lots of pictures, and the sentences are concise. • the contents of the problems are quite confusing because there are problems related to the content in 8th grade, so it has been forgotten, it should be following the current material only. 3 ad • the contents of the problem are not interesting to solve because i forgot how to solve them. • the meaning of the problems is not confusing, but the way to solve them is confusing. from the comments of the three students, mr had difficulty understanding an intersection of two circles with a radius ratio of 0,8. he sketched the intersection as one circle sliced into two halves. to guide his understanding by interview, researcher gave an analogy of intersection of two sets. when he was asked about an intersection of two sets, he knowed it and could give an example. his explanation was sketched on figure 6. volume 10, no 2, september 2021, pp. 203-216 211 figure 6. student sketches of an intersection of two circles figure 6 show that mr could sketch of an intersection of two sets, that are, a and b, but could not relate yet the concept of an intersection of two sets with two circle, so he could not solve the problem. this mean that the problem need to be revised, because his misconception about an intersection of two circles. he suggested that the problems should add a figure and reduce the narrative. so, not only mrs’ comment but also nb and ad, the second prototype problem was revised to become the third prototype. the next step was testing the third prototype problem to the small group phase on three different students from before. 3.5. small group phase in the small group phase, the researchers tested the third prototype problem to see its practicality to three students of 9th grade. to get it, the same as in the one to one phase, the researcher asked the teacher for help. the teacher determined ada, ndp, and at. the student who has a high mathematical ability was ada, the medium ability was ndp, and the low ability was at. at this phase, the three students were allowed to discuss how to solve each given problem. the assessment of the three students in this phase regarding the practicality of the problems can be seen in table 5. table 5. the practicality of the third prototype on small group phase no student rating sa a d sd 1 ada 16.67% 83.33% 0.00% 0.00% 2 ndp 66.67% 33.33% 0.00% 0.00% 3 at 100% 0.00% 0.00% 0.00% average 61.11% 38.89% 0.00% 0.00% afgani & paradesa, pisa-like problems using islamic ethnomathematics … 212 from the results of the questionnaire (see table 5), the third prototype problems was practical because about 61.11% of the three students stated strongly agree. also, the three students gave comments on the problems they solved (see table 6). table 6. student comments on the third prototype problem no student comments 1 ada the problems presented are under the material being studied, but indeed the material is little developed, so the level of the process is more difficult than usual. 2 ndp the problems given are understandable. the questions discussed are better accompanied by pictures so that the contents in the problems are interesting to discuss and include problems about pray with prayer beads. 3 at great, the problem is very easy to understand. from the comments of the three students and the results of observations as they tried to solve them, the third prototype question was revised to become the fourth prototype. the prototype also revised due to they could not solve all the problems, but one of interesting answers of them was the answer when they solved a problem about a muslim recites dhikr 1000 times at a time. in solving it, ada more active than ndp and at. discussion occurred between ada and ndp, while at just watched. in her explanation, ada used a different strategy from nb's answer in the one to one phase. if nb divided 1000 dhikr with 33 beads, then she had passed 30 groups of tasbih. since the tasbih was divided into 3 groups, he had done one complete cycle of 10 times and had been doing tasbih 990 times. meanwhile, the next 10 dhikr occurred in the first group. while ada’s method, she divided or compared 1000 to 100 times of dhikr. because the result was 10, according to ada, the 10 beads were still in the first group. the ada answer could be interpreted as ad's answer in the one to one phase, which is like a person who recited 100 times during 10 times where 99 beads had passed one prayer beads round and the next 1 beads returns to the first group of beads. from the two answer strategies, this showed that the problem has made students creative in finding ways to solve problems, without having to use standard mathematical formulas. according afgani and paradesa (2019), when student found strategy to solve mathematical problem by him/herself, that mean that s/he had constructed and improved her/his mathematical knowledge. the next step was testing the fourth prototype problems to the field test phase on several students in one class. 3.6. field test phase at this phase, the researcher tested six problems of the fourth prototype to 26 students’ junior high school of 9th grade. the problems were given simultaneously to students with 80 minutes. before the problems were given, the researcher identified the religion of all students. this was done to consider the existence of non-muslim students if she feels objected to being involved in learning. from the results of identification, there was one non-muslim student. when she was asked about the objections, she answered wanting to stay involved and did not mind. next, they were assigned to solve the problems individually. after they have worked with the time limit that has been determined, a questionnaire regarding the practicality of the problems was given to students. the following volume 10, no 2, september 2021, pp. 203-216 213 is an evaluation of the practicality of the fourth prototype problems according to the students which can be seen in figure 7. figure 7. the practicality of the fourth prototype on field test phase figure 7 show that 61.33% of 26 students agreed that the problems developed had met the practicality criteria. also, they provided comments on the assessment given to the problems developed. some of their comments were as follows: a. the problem can be solved but it takes a very long time. b. the material provided is easy to understand and is related to mathematical material that has been studied. c. the problems provided are interesting by containing interesting stories as well. d. the problems given are easy, medium, some are difficult, and are related to everyday math material in grade 9. e. the material in the problem should be discussed first in front of the class so that i can smoothly solve the problem. the problems given are rather complicated and it seems that i rarely get problems like that. f. all the problems given are interesting to solve but some are difficult. g. this problem is more interesting to discuss than the usual problem, but without explanation this problem is complicated, so i am lazy to do it. from some of the students' comments, the problems that had been developed meet three difficulty levels, namely easy, medium, and high. this means students understood the problems given when they are easy to find a solution and students did not understand the problems when they are difficult to find a solution. they also complained about too much narration. this showed that narratives that are too long can eliminate the attraction to find mathematical solutions. also, they realized that it is difficult to solve the problems because they were rarely trained in pisa questions, but they also found the problems given interesting, because they contained contexts related to the islamic culture around them. this finding is following the results of research by nizar et al. (2018) who reported that the integration of daily contexts that are close to students' lives into mathematical problems can make students actively discuss and assist them in solving them. their interest was also expressed from their curiosity about the answers to the problems given where they hoped that the problems would be discussed together in mathematics learning. the results of research at this phase, the conclusions that could be drawn are problems that have been developed until the fourth prototype has not met the optimal practicality criteria because 31.34% of students gave negative responses. this finding was 7.33% 61.33% 22.67% 8.67% 0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% sa a d sd afgani & paradesa, pisa-like problems using islamic ethnomathematics … 214 not much different from the results of susanti and suparman (2018) who reported that 27.5% of students gave negative responses to the problems in the worksheet related to ethnomathematics. this means that pisa-like problems using the islamic ethnomathematics approach need to be further developed. student negative responses are still quite large showed the schema of students' understanding of mathematical problems contained in the problem also had problems. this is evident from the results of the test data which showed the average score of students is 38.46 with the standard deviation is 13.14. on a scale of 0100, the score is included in the low category. this result was consistent with hasanah's research results (2017). she reported that the mathematical abilities of junior high school students in north sumatra who were the subject of her research in solving problems of the pisa model with the local cultural context were still in the low category. this means students who understand the problem, may not be able to find a solution, because in terms of problem-solving strategies according to polya, understanding the problem is only the first step of the four steps to solve the problem to arrive at a re-examination of the strategy taken. 4. conclusion based on the result of this study, we concluded that pisa-like mathematics problems that had been developed using islamic ethnomathematics approach were valid and practical for 9th grade students of junior high school, but still needs to be developed continually, because about 77.78% of three experts assessed agree that the problems that were developed fulfill valid criteria. in terms of practicality criteria, about 77.78% of three students on one to one phase considered agree, about 61.11% of the three students on small group phase stated strongly agree, and about 61.33% of 26 students agreed that the problems met the criteria. this result is supported by the average test result that was classified as a low category, which is 38.46 of 100. from this conclusion, we recommend to develop it not only in the form of problems directly but also in the form of worksheets. this will guide and get them used to solve pisa mathematics problems. acknowledgments the author would like to thank the institutions of research and community service of raden fatah state islamic university which have given research funding, so that authors could finish it without any significant obstacles. last but not least, authors render thanks and give respect very grateful to all other individuals who have supported authors to writing this paper. references afgani, m. w., & paradesa, r. 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(2002). developing a learning environment on realistic mathematics education for indonesian student teachers. doctoral dissertation. enschede: university of twente. https://doi.org/10.22342/jme.7.2.3542.117-128 https://doi.org/10.1088/1742-6596/1088/1/012063 https://doi.org/10.25134/jes-mat.v4i1.909 https://doi.org/10.15294/kreano.v7i1.4832 https://doi.org/10.19109/jpmrafa.v3i1.1444 https://doi.org/10.1007/978-94-011-4255-7_1 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p133-146 133 profile of elementary school teacher in concept understanding of geometry samuel igo leton*, kristoforus djawa djong, irmina veronika uskono, wilfridus beda nuba dosinaeng, meryani lakapu universitas katolik widya mandira, indonesia article info abstract article history: received mar 3, 2020 revised aug 1, 2020 accepted aug 7, 2020 students need teachers with a deep understanding of mathematical concepts to improve their mathematical knowledge and achievement. the observation results of several elementary school teachers showed that they still have a lack of understanding of the geometry concepts. this research is an exploratory study with a qualitative approach that aims to describe the performance of elementary school teachers in understanding the concepts of triangles and squares. the participants in this study were elementary school teachers across soe city district. a description test deals with the geometry concept of twodimensional shapes that were implemented to determine the most appropriate teachers to participate in the study. thirty-three teachers were then selected based on this preliminary test results. in-depth interviews were also conducted with the participants. the data analysis showed that the participants had a lack of understanding of the concept of two-dimensional shapes and necessary arithmetic skills. moreover, the data suggested that the participants held various perceptions regarding their understanding of certain concepts based on their experience in teaching the mathematical concept. based on these results, some programs are recommended to improve professionalism and pedagogical competencies, such as a refresher training program for basic mathematical material and training in teaching aids used. these programs are expected to help prepare elementary school teachers in teaching mathematics. keywords: understanding of mathematical concepts, two-dimensional geometric shapes, arithmetic skills, elementary school teachers copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: samuel igo leton, department of mathematics education, universitas katolik widya mandira jl jend ahmad yani no.50-52, kupang, east nusa tenggara 85225, indonesia email: letonsamuel@unwira.ac.id how to cite: leton, s. i., djong, k. d., uskono, i. v., dosinaeng, w. b. n., & lakapu, m. (2020). profile of elementary school teacher in concept understanding of geometry. infinity, 9(2), 133-146. 1. introduction mathematical understanding is an essential component of the mathematical skills needed for anyone to learn mathematics successfully (stylianides & stylianides, 2007). one of the expertise expected to be achieved when studying mathematics is understanding mathematical concepts (mwakapenda, 2004). this is observed through the ability to: (1) convey their understanding of the learned concepts; (2) explain the interrelationships between concepts; and (3) apply the concepts flexibly, accurately, efficiently, and precisely in problem-solving (depdiknas, 2003). understanding mathematical conceptsalso refers to https://doi.org/10.22460/infinity.v9i2.p133-146 leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 134 the ability to: (1) explain the concept; (2) implementthe concepts in various situations; and (3) develop some consequences of the existence of a concept that can explain the interrelationships between concepts and its application or algorithms flexibly, accurately, efficiently, and precisely in problem-solving (duffin & simpson, 2000). one’s mathematical ability is greatly influenced by his understanding of mathematical concepts and principles learned. the ability to understand mathematical concepts for elementary school teachers has received much attention in the last few decades. understanding mathematical content refers to the basic mathematical knowledge possessed by these teachers. mastery of mathematical concepts is essential because it facilitates teachers in teaching mathematics so that students can easily understand it. mastery of mathematical concepts is a necessary construct that can support or hinder student learning progress (philipp et al., 2007). lack of mastery of mathematical concepts possessed by elementary school teachers can lead to mathematics misconceptions among students as well asno improvement or changes in learning (thames & ball, 2010). thus, a good mastery of mathematical concepts can assist the development of mathematical connection capabilities between various ideas, help to understand how mathematical ideas are interrelated to one another so that a comprehensive understanding is possible, and the ability to use mathematics in contexts outside mathematics (korn, 2014).therefore, the deeper the mathematical knowledge possessed by teachers, the better it is to communicate mathematical concepts, models, and representations with students (philipp et al., 2007). mastery of mathematical concepts greatly affect student learning outcomes (sowder, 2007). to improve mathematical knowledge and achievement, students need teachers with deep mathematical knowledge. grover and cannor (2000) suggests that content knowledge is a key factor of effective teaching. they found that teachers not only understood mathematics, but also concepts that would support effective teaching (grover & connor, 2000). the observation data on how elementary school teachers teach mathematics suggests that there is still plenty of teachers who do not understand the concept of the area of a rectangular area very well. these teachers only memorize the formula for the area of a rectangular area that is length × width without knowing the concept of the area. even when they were asked whether the area of a rectangle be written as width × length, they firmly believe that it is incorrect because the correct formula is length × width and not the other way around. it seems they were trapped in believing that a concept is considered correct because it is widely accepted as a habit instead of according to the mathematical concept. they tend to hold the belief that the mathematical concepts possessed for teaching mathematics are based on their experience in mathematics classes that have been built up from time to time and that are considered valid (ball, lubienski & mewborn, 2001). they accept more of what they read or learn as truth without daring to question the reason behind the formula. this fact shows that not all elementary school teachers master the basic mathematical concepts correctly. this can be detrimental to the students they teach. the results of uasbn data analysis of 2017/2018 and 2018/2019 for mathematics of 504 elementary schools in the south-central timor regency, were obtained an average of 58.19 and 52.70 with the lowest scores of 13.37 and 20.83. concerning the uasbn results and observational data, we believe that there is a need to enhance teachers' professionalism and pedagogical competencies in the form of training/course. this should be done by first mapping out the mathematical concepts understood by the teachers. this mathematical training/course needs to be related to the development of basic mathematical concepts and pedagogy (thames & ball, 2010). in this paper, we examine the application of basic mathematical concepts by elementary school teachers regarding two-dimensional shapes. to our knowledge, there are no in-depth studies to explore the mastery of mathematical concepts by elementary volume 9, no 2, september 2020, pp. 133-146 135 schoolteachers regarding two-dimensional shapes. this study seeks to identify findings related to the mastery of mathematical concepts possessed by elementary school teachers for professionalism and pedagogical competencies improvement program to meet the needs of developing mathematics in elementary schools. therefore, this study focuses on addressing the following questions: what are the participants' understanding of the concept of the area of a triangle, the area of a rectangle, and how they teach the concept; also, basic arithmetic skills related to solving given problems. 2. method this research is an exploratory study with a qualitative approach that aims to obtain a depiction of the performance of elementary school teachers in understanding the concept of geometry. the research focuses on understanding the geometrical concepts held by elementary school teachers. exploration was carried out in detail on the capabilities of each teacher in answering questions related to the concept of area which consists of the area of a rectangle and the area of a triangle; how they teach the concept; also, basic arithmetic skills related to solving given problems. the subjects in this study were elementary school teachers across soe city district which were purposively selected. the participants were selected through a test. the test consists of 18 items related to the geometry concept of twodimensional shapes. the test results were then analyzed and grouped into several groups of answers as follows (table 1). table 1. the rubric scoring for mathematical representations teacher answer group code description based on the definition of area l1 the area of a two-dimensional shape is the number of square units that cover the entire surface of the twodimensional shape l2 the area of a two-dimensional shape is related to the length, width, (and height) of the two-dimensional shape itself. l3 the size of an area is the number of sides of the twodimensional shape itself. l4 the area of a two-dimensional shape is related to ribs and lines l5 unclear answer l6 no answer is given based on the definition of square and rectangle p1 squares and rectangles are explained based on the characteristics of their shapes p2 squares and rectangles are explained based on their perimeter p3 squares and rectangles are explained based on the size of all of their four sides p4 no clear answer is given lp1 a = l x w as stated by the experts leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 136 teacher answer group code description based on the area of a square lp2 a = l x w because it is the formula they got since they were in school lp3 a = l x w because the length must be written down first lp4 both are correct because they correspond to the commutative nature of multiplication lp5 no clear answer is given based on the understanding of the rectangular area formula lp1 a = l x w as stated by the experts kp2 answer that l x p is the formula to calculate the area of a rectangle based on the understanding of the relationship between square and rectangle pp1 states that square is considered as a rectangle pp2 states that square is not considered as a rectangle based on the analysis of the test results, 33 teachers who were considered able to contribute to providing data to attain the research objectives were then selected as the participants. in-depth interviews were also conducted to ascertain their understanding of the concept of area, how they teach these concepts, and basic arithmetic skills related to solving the given problems. since the researchers acted as the main instrument, the questions presented during interviews were open-ended; thus, the questions can be improvised according to research needs. the analytical model used in this study is an analysis based on the ideas generated from the teachers that refer to the correct mathematical concepts or definitions that have been prepared previously. some recommendations then are provided based on the results in regard to the competency development program for elementary school teachers in supporting better mathematics learning. 3. results and discussion 3.1. results data analysis in this study consists of the analysis of the data collected from the test result and semi-structured interviews of 33 participant teachers on their understanding of the concept of a two-dimensional shape. this is divided into three parts: the concept of the twodimensional area consisting of the area of a rectangle and the area of a triangle; how they teach the concepts; also, basic arithmetic skills related to solving given problems. in connection with understanding the concept of two-dimensional area, the teachers were asked to explain what they understand about the two-dimensional area. the results then are grouped in table 2 as follows. volume 9, no 2, september 2020, pp. 133-146 137 table 2. teacher answer group based on the definition of area group number of teachers percentage l1 3 9% l2 20 61% l3 2 6% l4 3 9% l5 4 12% l6 1 3% group l1 was a group that was able to precisely explain the concept of twodimensional area asthe number of square units that cover the entire surface of a twodimensional shape. 9% of the total number of the participant teachers were in this group. some teachers in this group used pictures to explain the answers, as shown in figure 1. figure 1. a teacher’s understanding of the concept of area in group l1 based on figure 1, one of the teachers from group l1 explained that the twodimensional area could be obtained by calculating the number of square units that can be formed as seen in figure 1. thus, the area of a rectangle with 5 cm in length and a width of 3 cm can be calculated using the concept. however, the participant teachers appeared to apply this concept only when dealing with two-dimensional shapes of square and rectangle. when they were asked about the area of other types of two-dimensional shapes, they were not able to explain the related mathematical concept. group l2 wasthe group that explained the area of two-dimensional shapes using the length, width, (and height) of the shapes as mentioned earlier. there were 61% of the total number of participant teachers in this group. the teachers in group l2 explained area as a measure of the area obtained by calculating the length and width or the length (base) and height of the two-dimensional shape and operate it using specific predetermined formulas. they were able to quickly determine the area of a rectangle that was 5 cm long and 3 cm wide but failed to explain why the answer was correct. the mathematical concept of area is understood procedurally, which is to remember a formula that can be used to calculate the area of the two-dimensional shape and how to apply it without understanding why the formula is correct (figure 2). figure 2. a teacher’s understanding of the concept of area in group l2 leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 138 group l3 consisted of teachers who explained the concept of the area of twodimensional shapes based on the number of sides of the shape. there was 6% of the teachers included in this group. using a rectangle, the teacher in group l3 explained the area of the two-dimensional shape as the sum of the four sides of the rectangle. based on this understanding, the area of a rectangle measuring 5 cm long and 3 cm wide was calculated by adding up the two lengths and widths equal to 16 cm2. however, they were confused when they realized that the results were different if the calculation was done using the rectangular area formula. from the explanation given, it wasclear that the teachers in the l3 group were unable to conceptually distinguish the area from the perimeter of a twodimensional (rectangular) shape. like group l2, they remembered the formula but did not understand why itwas conceptually correct. group l4 was a group of teachers who explained the concept of a two-dimensional shape area related to ribs and lines. 9% of the total participant teachers fell under this category. the teachers in this group understood the area of two-dimensional shape as the area formed by the intersection of line segments. they argued that when the line segments intersect, a space formed in the middle of the intersection is a two-dimensional area. based on the understanding that a line is a set of points, a two-dimensional shape such as a rectangle is considered as a set of all points from four lines facing each other. while the area of a twodimensional shape, according to this group, is the area bounded by these lines (figure 3). figure 3. a teacher’s understanding of the concept of area in group l4 group l5 was a group of teachers who did not have a clear answer to the question. the teachers who were classified in this group were those who did not provide a clear response. 12% of the total participant teachers were included in this group. the responses collected from this group, including: (1) to obtain the area of a two-dimensional shape we must measure and calculate how much volume is in the two-dimensional shape; (2) the area of a two-dimensional shape is the size of an object known by tracing its full shape; (3) the area of a two-dimensional shape is a shape withthe sides unit of lengths; (4) the area of a two-dimensional shape is a shape with the same length and is parallel. elementary school teachers' understanding of square and rectangular concepts the first analysis is carried out on the concepts of square and rectangle. the teachers' answers are grouped in table 3. volume 9, no 2, september 2020, pp. 133-146 139 table 3. teacher answer group based on definition of square and rectangle group number ofteachers percentage p1 17 51.5% p2 3 9.1% p3 11 33.3% p4 2 6.1% as presented in table 3, group p1 was the group with the highest percentage (51.5%). they explained the concept of square and rectangle based on the characteristics of these two shapes. a rectangle was understood as a two-dimensional shape which has two pairs of sides of equal length, all four angles are right-angled, and has two diagonals of the same length. meanwhile, a square was understood as a two-dimensional shape formed by four equal lengths and four right angles. they were able to name the characteristics of twodimensional shapes precisely but did not correctly understand the concept of the length and width of the rectangle. the results of our interviews with the teachers in this group showed that they generally understood length as the longer side, and width as the shorter side. we argue that there was a misunderstanding that length is the longer side, and the width is the shorter side (figure 4). figure 4. length and width of a rectangle sized when rotated group p2 wasthe group that explains the meaning of square and rectangle using the concept of perimeter. they described a rectangle as a two-dimensional shape that has a length and width, with a perimeter of two times the length and twice its width, while a square is a two-dimensional shape with the same length and width (sides) and the perimeter is four times the length of the sides. the participant teachers then were shown a picture of parallelogram whereits four sides have the same length and asking them to determine the type of the shape. they said that it was a parallelogram and they improved their descriptionof the square by adding that on a square the angle was 90°. the next image to be determined was a square that had been rotated 45°. almost all teachers in this group classified the picture as a rhombus and not square because they thought the shape was different from the shape of a typical square. we argue that the participant teachers generally did not have a sound understanding of the relationship between a square and a rhombus that the two is not necessarily a square. group p3 wasthe group that stated that the meaning of squares and rectangles are explained based on the size of all of their four sides (33.3%). they described rectangles as two-dimensional shapes that have long sides and wide sides of different sizes; the longer side is called the long side, and the shorter side is called the wide side, and the square is a two-dimensional shape that has four same and congruent lines (sides). during the interview, most of them explained that what was meant by same was that the four sides were both in the form of a straight line and the four sides have the same length. to further explore their understanding of the concept of squares and rectangles, two images were presented. the first was a parallelogram that had the same length on all four sides; the second imagewas a leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 140 rectangle that had the same side length as the parallelogram side. the participants were asked to determine whether the two shapes were congruent. based on their understanding of the congruent concept, they generally classified the two structures as two congruent shapes. when they were asked to elaborate their answer on howthe shapeswere congruent, they said because the two shapes have the same side length. we argue that the teachers in this group did not understand well the concept of the congruency of two two-dimensional shapes (figure 5). figure 5. a teacher’s answer when classifying the congruency of two two-dimensional shapes group p4 wasa group of teachers thatprovided answers that were not following mathematical concepts. in general, they explained the concept of squares and rectangles as a two-dimensional shape that has angles, angles, sides, and segments. some believed that a rectangle was a two-dimensional box-like shape, while a square was a two-dimensional shape that has the same length and sides. the researchers agreed with this answer; the teachers in this group knew which pictures or objects are square or rectangle. they were also able to mention the elements of a square or a rectangle. this suggested that this group had a lack of understanding in regard to the concepts of two-dimensional shapes. their understanding was limited to images, drawings, and concrete objects in the shape of a square and rectangle. we argue that, in general, they did not have a strong understanding of the mathematical concepts of squares and rectangles both their definitions and properties. elementary school teachers' understanding of the concept of rectangular area the subsequent analysis focuseson how teachers understand the concept of a rectangular area. the teachers were asked to determine which formula is the correct one to calculatea rectangular area: l x w or w x l . the responses are grouped in table 4. table 4. teacher answer group about the area of a rectangle group number of teachers percentage lp1 18 54% lp2 4 13% lp3 2 6% lp4 8 24% lp5 1 3% volume 9, no 2, september 2020, pp. 133-146 141 group lp1 was the group of teachers that stated a = l x w is the correct formula to calculate the area of a rectangle instead of a = w x l. they argued that this was following the existing formula. 54% of the total teachers were in this group. one of the teachers in this group argued: "... the correct formula is a = l x w because the experts have determined it since long time ago and no one can change it". this shows that this group had a lack of understanding of the rectanglearea, whichwas still limited to memorizing the formula without understanding why the formula is correct or where it came from. the second group (lp2) were the teachers who believed that a = l x w is the correct formula to calculate the area of a rectangle because it is the formula they learnt since they were in school. there were 13% of teachers who were included in this group. similar to lp1, lp2 group teachers also did not understand the concept of the rectangular area. their understanding was still limited so that they were not able to give a proper reason for the rectangular area formula. group lp3 consisted of teachers who believed that a = l x w is the correct formula to calculate the area of a rectangle because they thought the length must be written down first. there are 6% of teachers who were included in this group. they assumed that the correct formula is a = l x w because this is the order of how it is written on the school textbooks. when calculating the area of a rectangle, the length is written down before the width. group lp4 consistedof the teachers who stated that the rectangular area formula is a = l x w = w x l because it is based on the commutative nature of integer multiplication where a x b = b x a. 24% of teachers were included in this group. group lp5 consisted of teachers who did not provide clear answers. one of the teachers explained that the area of the rectangle is a = l x w because a rectangle has two pairs of ribs of the same length and parallel also four right angles. this particular teacherdid not have a well understanding of the concept of the two-dimensional area; thus, he only mentioned the characteristics of rectangles according to his understanding to answer the questions given. concerning the answers above, we argue that in general, the participant teachers did not understand the concept of the area of a two-dimensional shape, causing them to be shackled by a habit that has been used and considered correctsince long ago which is a = l x w and not the other way around. this was evidenced by a large number of participants who cannot properly interpret the questions given as follows: an abcd rectangle with ab = 5 cm and bc 10 cm. the area of the rectangle = length × width = 5 cm × 10 cm = 50 cm2. we classified the answers from the teachers in table 5. table 5. teacher answer group group number of teachers percentage kp1 5 15% kp2 28 85% group kp1 was a group of teachers who stated that the solutionwas correct. as can be observed in table 5, there were five out of 33 teachers who agreed that the statement was correct. these five teachers were interviewed to explore their arguments regarding the statements further. four of these teachers argued that the solution wascorrect because it was following the formula of length x wide. their reasoning was correct but only limited to known formulas, without defining in more detail. moreover, one of the teachers argued that the statement was correct since ∠ab is the length and ∠bc is the wide. this confirms that the teacher cannot distinguish an angle from a length of a line; hence he assumed that the angle and length of a line were the same things. leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 142 group kp2 consisted of teachers who believed that the solution was incorrect. the interview results showed that most of them had similar reasoning towards the statement that the length is 10 cm and the width is 5 cm . therefore, the rectangle area = length × width = 10 cm × 5 cm = 50 cm2. this suggested that the teacher understood the concept of the rectangular area is length × width. they understand that the length of the rectangle is longer than the width. this concept has been accepted for long and considered correct. it has been considered correct not because it follows a mathematical concept where the size of the sides (length and width) on a rectangle only states a dimension of a two-dimensional shape but as a custom. elementary school teachers' understanding on whether a square can be categorized as a rectangle? the analysis of teachers responses on "whether a square can be categorized as a rectangle?” can be observed in table 6. table 6. teacher answer group group number of teachers percentage pp1 6 18% pp2 27 82% group pp1 was a group of teachers who correctly answered that a square can be categorized as a rectangle (18% or six out of 33 teachers). these six teachers provided different reasons for their answers in the follow-up interview. some of them believed “... because a square is a rectangular two-dimensional shape that has four parallel and equal sides and four right angles …”. some teachers provided incomplete reasons, “ ... because they have four sides.” in addition, some teachers gave an inconsistent argument as well, saying that a square is a rectangle while stating that a square has four different sides of length and width. group pp2 was a group of teachers who stated that a square could not be categorized as a rectangle. 27 out of 33 teachers who did not approve the statement provided several different arguments in the follow-up interview. for one, a square cannot be categorized as a rectangle because the formula to calculate the perimeter of a square is 4s. while to calculate the perimeter of a rectangle we use 2 x l and 2 x w; some teachers also suggested that the formula for the area of a rectangle is different from that of a square as in asquare = s x s while arectangle = l x w. some teachers also provided arguments based on differences in the size of the four sides. it appeared that their answers were based on the differences in area, perimeter and side size of the two shapes. the participants did not understand that the perimeter concepts of the two shapes are the same; the only difference is the symbols used. none of the participants provided any explanation based on the properties possessed by a square and a rectangle. they did not know that both a square and a rectangle share the same characteristics; thus, a square can be categorized as a rectangle. these findings confirm that the participants did not understand well the properties of a square and a rectangle, hence they cannot provide arguments that are considered as mathematically correct regarding the relationship between a square and a rectangle. volume 9, no 2, september 2020, pp. 133-146 143 teachers' understanding of the concept of area of a triangle the results of the analysis of the questions related to their understanding of the concept of the area of the triangle area, obtained information that in general they provide answers based on the formula to calculate the area of a triangle that has been widely accepted, to find the area of a triangle we must multiply the base of the triangle by its height. during the follow-up interview, all of the participants were able to correctly mention the formula to calculate the area of a triangle which is a= 1 2 b x h. to further explore their understanding, the participants were then asked why the formula is as stated above. several answers were given included this formula is part of a rectangle; this formula = the area of the rectangle, divided by two; and this formula is a derivative formula of a rectangle. these answers were rooted from the same idea: a rectangle that is divided in two using one of its diagonals create two triangles; hence the area of a triangle is equal to half of the area of a rectangle. next, the participants were asked to explain how to teach the concept of the area of a triangle. most of the participant teachers believed that it is sufficient to teach the concept by making the pupils memorize the formula. they argued that memorizing formulas does not cause any problem because students memorize them in multiplication operations of the numbers. we argued that this is a misperception on the teachers’ side in regard to the understanding of mathematical concepts, where understanding the concept is critical compared to only memorizing formulas. this may occur due to a lack of understanding of the triangle area from the teacher side. thus, this concept is difficult to be correctly taught to the students. this is evidenced by a large number of participants who cannot solve the problems given below (figure 6). figure 6. samples of teacher answers related to the area of a triangle from the sample answers in figure 6, it appeared that the participants could not solve the problem correctly. the results of the interview data analysis showed that in general, it was difficult for the participants to determine the height of an obtuse triangle. the difficulty was caused by the lack of understanding of the concept of height in a triangle. they were accustomed to giving questions related to the image of an acute triangle or a picture of a right triangle so that when solving problems using an image of an obtuse triangle, they generally cannot solve the problem correctly. 3.2. discussion the results of the data analysis from the test and the semi-structured interview provide a depiction of the performance of elementary school teachers in understanding the concept of the geometry of the two-dimensional shape. in general, the findings of this study indicate the low mastery of the concept of two-dimensional shapes such as square, rectangle, and triangle as well as the low basic numeracy skills possessed by elementary school teachers in solving mathematical problems. the participants argue that heavy teaching workload is leton, djong, uskono, dosinaeng, & lakapu, profile of elementary school teacher … 144 the cause of their lack of mastery. when a particular concept is not well mastered, the teacher tends to skip the lesson. they have varying degrees of mathematical proficiency and in many cases, maybe lacking knowledge bases (cai & wang, 2010;vistro-yu, 2013). based on the findings, the researchers would like to propose some recommendations to be considered in order to establish a program to enhance professionalism and pedagogical competence for elementary school teachers. the researchers believe that there is a need for a refresher training program in regard to mathematical concepts. the elementary school teachers lack an understanding of the concept of two-dimensional shapes needs to get attention. teachers lack understanding can severely affect teaching and learning in the classroom since they are the guide for students in learning. they must have sufficient mathematical knowledge to teach mathematics (kasoka, jakobsen & kazima, 2017). the more in-depth the mathematical knowledge of a teacher, the better they can communicate mathematical concepts, models, and representations to their students (philipp et al., 2007). based on the results of this study, the researchers suggest that teachers need to deepen their understanding of mathematical concepts through competency improvement programs, especially in regard to their professionalism. professionalism in this sense refers to the ability to master the mathematical content knowledge, which is the foundation of mathematics teaching that must be possessed by teachers (ball, thames & phelps, 2008). the knowledge about the concept of a two-dimensional shape needs to be improved from only memorizing formulas to understanding the formulas and the rules or to know the characteristics of a twodimensional shape. teachers with insufficient content knowledge will find it challenging to explain mathematical concepts, provide models, and build relationships between concepts (reid & reid, 2017) and difficult in developing hots-oriented problems (dosinaeng, leton & lakapu, 2019). therefore, there is a need for government-facilitated programs provided by the education office or by the teacher associations such as subject teacher group or teacher working group to improve their mathematical knowledge. through this process, groups of teachers in the association can be actively involved and work together to strengthen their mathematics teaching skills and thus positively affect students' mathematics achievement (reid, 2013). the training that teachers join in improving their competence, as well as their educational background, have a more significant influence on teacher professionalism than their teaching experience or the length of teaching time (harisman, kusumah, kusnandi & noto, 2019). these training programs are expected to help prepare teachers to teach mathematics to elementary school students. 4. conclusion the mastery of basic mathematical concepts is the foundation of knowledge that elementary school teachers need to have. the findings of this study suggest that the mastery of the concept of two-dimensional shapes and the teachers’ experience in teaching mathematics at the elementary school level is still very low and limited to what is provided in the textbook. apart from the low level of basic arithmetic skills and their experience in teaching mathematics, elementary school teachers need to take refresher training programs to improve their professional and pedagogic competencies. this recommendation stems from the fact that the teachers generally do not master the basic concept of two-dimensional shapes even though they are teaching at the elementary school level which supposed to be the ones who are responsible for building the mathematical foundation for students. the researchers believe that when these refresher training programs have been implemented, at minimum the issues related to the low mastery of the mathematical concepts can be volume 9, no 2, september 2020, pp. 133-146 145 overcome and ultimately can improve the quality of mathematics learning in the elementary school level. references ball, d. l., lubienski, s. t., & mewborn, d. s. 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(2013). cross-national studies on the teaching and learning of mathematics : where do we go from here ?. zdm, 45(1), 145–151. https://doi.org/10.1007/s11858013-0488-4 https://www.iejee.com/index.php/iejee/article/view/289 https://www.iejee.com/index.php/iejee/article/view/289 https://scholarworks.umt.edu/tme/vol4/iss1/8/ https://scholarworks.umt.edu/tme/vol4/iss1/8/ https://scholarworks.umt.edu/tme/vol4/iss1/8/ https://pubs.nctm.org/view/journals/tcm/17/4/article-p220.xml https://pubs.nctm.org/view/journals/tcm/17/4/article-p220.xml https://doi.org/10.1007/s11858-013-0488-4 https://doi.org/10.1007/s11858-013-0488-4 https://doi.org/10.1007/s11858-013-0488-4 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p183-196 183 relationship between statistical literacy and mathematical representation of students through collaborative problem solving model johannis takaria1*, wahyudin2, jozua sabandar3, jarnawi afgani dahlan2 1universitas pattimura, indonesia 2universitas pendidikan indonesia, indonesia 3institut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received mar 14, 2020 revised sep 5, 2020 accepted sep 6, 2020 the purpose of this study was to find out the relationship between statistical literacy and mathematical representation of students as pre-service elementary school teachers through the collaborative problem solving (cps) model. the relationship between statistical literacy and mathematical representation was analyzed by using a product-moment correlation with a sample of 35 students of elementary school teacher education study program at one of the state universities in ambon city. the results showed that there was a positive and strong relationship between statistical literacy and mathematical representation with a correlation value of 0.66. this relationship means that if students have good mathematical representation abilities, statistical literacy abilities are also getting better. exploration of statistical literacy and mathematical representation abilities can be facilitated by using the cps learning model. the cps learning model can facilitate student learning as a structure in mathematical thinking so that statistical literacy and mathematical representation abilities of students can be explored through the transformation of ideas among students. the cps learning model aspects were implemented in high and very high categories while the indicators were at rating-2 and rating-3. keywords: statistical literacy, mathematical representation, cps learning model copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: johannis takaria, department of mathematics education, universitas pattimura jl. ir. m. putuhena, poka, tlk. ambon, maluku 97233, indonesia email: johannistakaria007@gmail.com how to cite: takaria, j., wahyudin, w., sabandar, j., & dahlan, j. a. (2020). relationship between statistical literacy and mathematical representation of students through collaborative problem solving model. infinity, 9(2), 183-196. 1. introduction thinking skills are important to be developed in statistical and mathematical learning. hendriana et al. (2019) stated that mathematical thinking skills are the focus of mathematical learning in schools. takaria and talakua (2018a) stated that the abilities of students in statistical learning must be supported by good mathematical abilities so that https://doi.org/10.22460/infinity.v9i2.p183-196 takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 184 students can be actively involved in facing and solving various challenging statistical problems. statistics in the school curriculum is an integrated material in mathematics. best and khan (1995) stated that statistics are part of mathematical techniques, related to the collection, organization, analysis, and interpretation of numerical data. mastery of statistical concepts requires students to have good mathematical abilities. in higher education, statistics are taught separately and not integrated into mathematics, but mathematics is crucial in understanding statistics. takaria (2015) showed a gap between the importance of statistics and the abilities of students. it was found that the statistical abilities of students as pre-service elementary school teachers at one of the state universities in ambon city had not yet achieved the expected results. this is caused by the lack of basic skills in statistics and mathematics. there is a lack of basic statistical knowledge possessed by students, so that showed a phobia of mathematics and statistical anxiety that had an impact on learning interest (garfield, 1995; garfield & ahlgren, 1988; tishkovskaya & lancaster, 2010; verhoeven, 2006). garfield (1995) stated that in statistical learning, students are not empowered to apply aspects of statistical knowledge in solving general problems from certain contexts. based on the researcher's experience as a teaching lecturer in educational statistics courses and basic concepts of mathematics, it was found that problems with basic statistics and mathematics knowledge is also a problem for students who are pre-service elementary school teachers in the study location. in the basic ability of mathematical compute operations, students made a frequency distribution table to determine classes by using the sturgess rule k = 1 + 3.3 log n. students showed wrong basic calculation skills, where students added 1 + 3.3 first then the results were multiplied by log n. this showed that students did not understand the level in mathematical operations, 3.3 log n was completed first, then the result was added with 1. other findings showed that students calculated the average by not using the procedure correctly. students are directly calculated without writing the equations and statistical symbols first. the basic ability of analysis and interpretation is important to be improved, especially the ability to understand statistical information, read and write (graphs and tables), and interpret correctly. the main thing that is often overlooked by students is not writing the title of tables and graphs. statistical and mathematical learning requires literacy. fatmanissa and sagara (2017) stated that literacy is the ability to read and write that is used when reading and understanding a problem to be written in a mathematical model. literacy has been developed in various fields, one of which is statistical literacy, although there are still many problems with statistical knowledge. lack of statistical knowledge due to low statistical literacy abilities. the lack of statistical literacy abilities is due to the inability to apply it in everyday life (gal, 2002; tishkovskaya & lancaster, 2010; verhoeven, 2006). schield (1999) stated that statistical literacy is more than just the use of numbers, but individuals must be able to understand what is affirmed, think critically about statistical arguments, and have inductive reasons for those arguments. research conducted by watson (2003) found that statistical literacy is important and becomes part of the curriculum. according to him, several factors that contribute to the development of statistical literacy in schools are due to: 1) the expectation to participate as citizens in accessing information related to data; 2) the importance of abilities and skills in every possible decision making on data. basic skill and is important to use in understanding statistical information or research results. these skills aim to organize data, compile and display tables, and work with different volume 9, no 2, september 2020, pp. 183-196 185 data representations. statistical literacy also includes an understanding of concepts, vocabulary, symbols, and includes an understanding of probability as a measure of uncertainty (ben-zvi & garfield, 2004; callingham & watson, 2017). this problem is inversely proportional to the importance of statistical literacy. the reading and interpreting statistical reports requires statistical literacy abilities which include sufficient knowledge and understanding of calculations, statistics, understanding of literacy in general, utilize quantitative data for data presentation and make summary reports on personal or professional assignments (chick & pierce, 2013; gal, 2002; ben-zvi & garfield, 2004; watson, 2006). the study on statistical learning stated that statistical learning through problem-solving can improve student skills, especially when they interact directly with data (garfield, 1995; garfield & ben-zvi, 2007; marriott et al., 2009). statistical literacy is the ability to critically interpret and evaluate statistical information in argument-based data on various media channels and their ability to discuss it (gal, 2002). aoyama and stephen (2003) stated that statistical literacy is the ability to extract qualitative information from quantitative and make new information from qualitative and quantitative data. bidgood et al. (2010) stated that statistical literacy requires skills in problem-solving namely, abilities in reading, writing, listening, and speaking. statistical literacy is defined as the ability to read, write, understand, interpret, analyze at a basic level, and interpret data through the abilities possessed and can understand and present information in the form of tables, graphs and statistical symbols in various media. statistical literacy also plays a role in minimizing errors that occur in activities with data, so that data users can overcome the problems encountered. various media are used to improve the statistical literacy of students, namely electronic media, print media, internet, journals, and various other statistical literacy media. media literacy can be used as information in statistical learning, so as to increase the abilities of students in reading and writing statistically. statistical literacy abilities need to be supported by good mathematical abilities. one of them is mathematical representation. the representation can be used as a tool through diagrams, graphs, tables, and symbols in expressing mathematical abilities for problemsolving, communication (learning by oral, written, drawing, graphic, and concrete concepts), and see the relationship to a mathematical problem (bal, 2014, nctm, 2000; takaria & talakua, 2018a). ainsworth (2006) stated that representation is a way to interpret what is captured and interpreted through an image, on the screen or in words where someone can say whatever they want to say. mathematical representation in statistical learning is the ability to convey ideas or mathematical ideas in various forms (tables, graphs, symbols, the meaning of words, and mathematical equations) of what is seen or observed through statistical information obtained and can interpret information. mathematical representation in statistical literacy learning for students is important, because students can convey mathematical ideas in various forms (tables, graphs, symbols, meaning of words, and mathematical equations) from something seen/observed through statistical information obtained on various media and the student's ability to give meaning to this information (takaria & talakua, 2018b). statistical literacy and mathematical representation are two capabilities that synergize with each other and contribute to the learning process. the relationship between statistical literacy and mathematical representation can be facilitated with the collaborative problem solving (cps) learning model. collaborative learning involves intellectual efforts to seek understanding, solutions, meanings, and produce a product based on mutual agreement (van den bossche et al., 2006). takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 186 the selection of the cps learning model is based on the idea that this model is a form of group learning to form students into individuals who are strong in problem-solving. the cps learning model requires skills in problem-solving and managing differences, which are implemented through a collaborative process. the cps learning model consists of five stages namely, engagement, exploration, transformation, presentation, and reflection (ngeow, 1998). the cps learning model is used to analyze the relationship between statistical literacy and mathematical representation of students as pre-service elementary school teachers. 2. method 2.1. method and sample this study used a correlational analysis method. the purpose of this study was to analyze the relationship between statistical literacy and mathematical representation of the abilities of students with the cps learning model. the sample of this study was 35 students of the elementary school teacher education study program enrolled in statistical education at one of the state universities in ambon city, maluku province, indonesia. this study used a purposive sampling technique. this technique was used by researchers based on several considerations: (1) researchers expected the study to be carried out well and effectively; 2) researchers analyzed the problem according to the objectives to be achieved; and 3) students passed mathematical education i and ii as prerequisite courses before enrolling statistical education. 2.2. instrument an instrument is a tool used to measure the variables to be studied. therefore, research instruments need to be prepared appropriately, so that the data collected comprehensively can answer the problem and research objectives. the instruments used were statistical literacy and mathematical representation tests, and non-test instruments were in the form of observation and interview guidelines. 2.3. data analysis data analysis used a product-moment correlation. the pearson product-moment correlation coefficient is widely used in social science research as a correlational technique between two variables (x and y) and also in accordance with various univariate and multivariate methods (smithson, 2000; walker, 2017). to analyze the feasibility of the cps learning model, the feasibility rating model was used. table 1 presents the feasibility rating of the cps learning model that refers to the new teacher project (takaria, 2015; takaria & talakua, 2018b). table 1. the feasibility rating of the cps learning model rating indicator 3 all indicators are implemented 2 half or most indicators are implemented 1 more than half of the indicators are not implemented 0 all indicators are not implemented the ratings in table 1 were analyzed using the percentage qualifications in table 2 adapted from (linnusky & wijaya, 2017) volume 9, no 2, september 2020, pp. 183-196 187 table 2. percentage qualifications of learning model feasibility qualification category k ≥ 90% very high 80% ≤ k < 90% high 70% ≤ k < 80% fair 60% ≤ k < 70 low k < 60% very low 3. results and discussion 3.1. results 3.1.1. correlational analysis the correlational analysis is a technique for looking at linear relationships between two or more variables. based on that definition, the analysis statistically tests the relationship between statistical literacy and mathematical representation abilities with the cps learning model. the analysis was used to see the relationship between statistical literacy and mathematical representation. the hypothesis was “there is no relationship between statistical literacy and mathematical representation” (h0) and the working hypothesis was (h1) “there is a relationship between statistical literacy and mathematical representation”. based on the classical assumption test, it was obtained that the data were not normally distributed so that the transformation of the data with square root transformation was carried out. after the data of statistical literacy and mathematical representation were transformed, the data were normally distributed (see table 3). the homogeneity test showed that both data were homogeneous (see table 4). table 3. normality tests of statistical literacy and mathematical representation ability significance normality test kolmogorovsmirnov decision statistical literacy (sig.) 0.054 normal mathematical representation 0.136 table 4. homogeneity tests of statistical literacy and mathematical representation ability significance homogeneity test levene test decision statistical literacy (sig.) 0.230 homogenous mathematical representation 0.164 after the assumptions are fulfilled, a correlational test is carried out using the pearson correlation test with the test criteria, if sig. greater than 0.05 then h0 is accepted and h1 is rejected. table 5 show the results of correlation testing. takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 188 table 5. correlational test of statistical literacy and mathematical representation pearson correlation sig. decision 0.661 0.000 h0 is rejected table 5 showed that the value of sig (0,000) was smaller than 0.05 so that h0 was rejected. these results conclude that there was a significant relationship between statistical literacy and mathematical representation. the pearson correlation value obtained was 0.661, if confirmed by the test criteria, the relationship between statistical literacy and mathematical representation was in a positive and strong relationship. this relationship showed that if students have good mathematical representation abilities, statistical literacy abilities are getting better. strengthening statistical literacy and mathematical representation of students was facilitated through the use of the cps learning model and supported by a good mathematical disposition so that there would be solutions for statistical problems. figure 1 showed the relationship between statistical literacy and mathematical representation with the cps learning model in mathematical problem solving. figure 1. relationship between statistical literacy and mathematical representation with the cps learning model 3.1.2. collaborative assessment the collaborative assessment of students was conducted by using the rubric of feasibility assessment, both in groups and individually. for group assessment, the aspects assessed were: 1) group formation (gf) with several indicators namely, independence in group formation, group division according to ideal group division criteria, and division of roles of each individual in the group; 2) idea construction and transformation (ict) with several indicators namely; mutually building ideas, interactions, and the ability to express opinions or ideas; 3) presentation of results (pr) with several indicators namely, effective use of time, represent the results of collaboration well, and be able to answer questions from other groups; 4) group reflection (gr) with several indicators namely, group reflection on weaknesses when exploring and transforming ideas, reflection on the results of presentations, and reflection when arguing. based on the observation results through an observation sheet on the collaboration process to observe and record various processes in accordance with the observation statistical literacy conceptual knowledge ▪ mathematical abilities ▪ statistial abilities ▪ communication skills ▪ reasoning ability (bidgood et al., 2010; gal, 2002) disposition ▪ motivation ▪ criticism mathematical representation statistical information collaborative problem solving model statistical learning media as learning resource problem solving solution volume 9, no 2, september 2020, pp. 183-196 189 guidelines provided, the aspects of collaboration were in very good and good qualifications, with a rating that all indicators were implemented (rating-3) and half or most indicators were implemented (rating-2). table 6 presents the rating and qualification ratings for each cps aspect. table 6. rating and qualification of cps aspects aspect cps group cps group percentage of average qualification 1 2 3 4 5 6 7 1 2 3 4 5 6 7 rating (rg) qualification (%) gf 3 3 3 3 3 3 3 100 100 100 100 100 100 100 100 ict 2 3 2 2 3 3 3 75 86.7 71 75 76.7 83.3 86.7 79.2 pr 2 3 2 3 3 3 3 66.7 83.3 79.3 83.3 83.3 86.7 90 81.8 gr 3 2 3 3 3 3 3 83.3 90 86.7 86.7 90 90 90 83.3 through the observation sheet it can be observed that the lecturer: 1) developed cognitive conflict through several inducement questions during the collaboration to explore insights from students more deeply; 2) commented and examined the results of group work for each meeting after collaborating; and 3) used assessment rubrics in the form of percentages and records of the feasibility of collaboration to assess student activities both individually and in groups. the results of observations through the feasibility observation sheet, assessment rubric, and important notes during the collaboration process were obtained: (1) overall, the use of the cps learning model was carried out properly and in accordance with the learning steps; (2) at the beginning of the meeting (1st), students were less flexible in collaborating; (3) only a few students were aggressive in expressing opinions (1st & 2nd meetings), other students were not having the courage in conveying ideas; (4) constructed ideas were not in accordance with the expectations of the concept of collaboration; (5) in arguing, only a few students that were proactive in defending the ideas conveyed; (6) individuals in the group had difficulty in reaching an agreement, due to various opinions that must be united (1st meeting); (7) some students tended to emphasize personal ego in the argument (1st meeting). based on these problems, the lecturer as a facilitator, motivated and directed students in accordance with the principles and objectives of collaboration, so that the implementation of the collaboration can run well at the next meeting, students who had the low ability and were being aggressive in conveying their ideas. students were more flexible in collaboration, which was demonstrated through the exploration and transformation of ideas. decision making on collaboration results was no longer a group problem, where each individual appreciated the ideas conveyed by peers. radical obtrusiveness was also diminishing. reflection on collaboration was very useful for students, where students individually or in groups reflected the weaknesses they had at the time of collaboration. reflection is a process that is needed by students when collaborating, this makes students know their weaknesses and strengths. reflection can be in the form of attitudes when collaborating and also towards collaborated concepts. at the end of the meeting, there were several students sharing with the lecturer to discuss issues that were still being considered. figure 2 presents the ladder of reflection between students and lecturers. takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 190 figure 2. ladder of reflection in students based on the findings, it can be explained that students who have good statistical literacy and mathematical representation abilities will drive them to become strong individuals in the field of statistics so that they have better goals and expectations in statistical problems oof society. figure 3 shows the goals and expectations of statistical literacy and mathematical representation abilities. the confrontation results of collaboration in the minds of students students hypothesize collaborated results sharing with lecturers convey information from the results of collaboration and interpretation obtain an explanation from the lecturer & are advised to look for related references obtain the solution volume 9, no 2, september 2020, pp. 183-196 191 figure 3. goals and expectations of statistical literacy and mathematical representation abilities 3.2. discussion statistical literacy is the ability students must have in critically interpreting and evaluating statistical information contained in various media. students with good statistical literacy abilities can be actively involved in addressing statistical problems in society. to complete statistical literacy abilities, mathematical representation is needed, related to the ability to represent statistical problems in various forms through the representation of tables, graphs, mathematical equations, symbols, interpretation of words, and other mathematical representations. the relationship between statistical literacy and mathematical representation explicitly through testing obtained a product-moment correlation index of 0.66, which was in strong and positive category. the results showed that students with good mathematical representation abilities will have an impact on good statistical literacy abilities. mathematical representation abilities owned by students in statistical learning has a purpose to help students in presenting creative ideas on challenging statistical problems, views on the importance of statistical literacy skills, mathematical representations, and attitudes of students as preservice elementary school teachers ability that is exposed ▪ mathematical abilities ▪ statistical abilities ▪ think logically, critically, and creatively ▪ analysis & interpretation statistical media literacy: ▪ internet ▪ television and print media ▪ journals ▪ textbook ▪ billboards and other media. supported by mathematical representation abilities: ▪ visual representation ▪ representation of mathematical equations ▪ symbol representation ▪ interpretation of wordskata statistical literacy ability are required: ▪ speak and write ▪ communicate ▪ arguing ▪ proficient the importance of statistics ▪ statistics plays an important role in various activities ▪ important for the final project & research supported by attitude: ▪ communicating ideas ▪ self confidence ▪ appreciate opinion ▪ cooperate ▪ self reflection. ▪ tough in mathematics & statistics ▪ can compete in the global era ▪ become a professional elementary school teacher ▪ further study ▪ answering problems in the community ▪ strong self concept statistics goals and expectations takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 192 namely: 1) can present information from tables into the graphical form or vice versa represent graphs in tabular form; 2) can write symbols and interpret terms; 3) use mathematical procedures correctly in solving statistical problems; 4) make arguments mathematically in contextual situations related to statistics; 5) can use variables, make equations and calculations; and 6) can use mathematical representation in other forms to solve statistical problems. based on figure 1, students need to be given a reinforcement of statistical literacy abilities on the basis of strengthening mathematical knowledge, statistics, communication skills, and reasoning. statistical literacy abilities need to be supported by mathematical representation abilities and dispositions, in this case, motivation and critical attitude. through statistical literacy skills, students can find a variety of statistical information that can be used as learning resources and facilitated through the cps learning model to find solutions to statistical problem-solving. bidgood et al. (2010) stated that the two major components needed in statistical literacy are: 1) the knowledge components namely; literacy skills, mathematical knowledge, statistical knowledge, communication skills, and reasoning; 2) disposition components namely; attitude in evaluating, constructing, recognizing, challenging and communicating ideas. both components should be owned by students in responding to statistical information in various media ladder of reflection in figure 2 shows the existence of student ideas that are not channeled and become problems for students, but in the process of reflection, the problem can be overcome. reflection is a process that is needed by students for what they do when collaborating, this makes students know their weaknesses and strengths. according to jonassen and rohrer-murphy (1999), learning activities are important activities but are not enough to interpret the learning, but must reflect the learning experience so that learning is more meaningful. the results of identification through a series of questions to several students who performed sharing showed that students were hesitant to convey their ideas during the process of transformation and interpretation of work in front of the class. this is due to the lack of confidence in conveying the ideas. the problems experienced by students can be sought for solutions through reinforcement and direction given by the lecturer. the steps taken are directing thought processes in constructing creative ideas through structured experiences during collaboration and students are encouraged to be more confident in conveying ideas. according to figure 3, it can be seen that students who have statistical literacy and mathematical representation abilities have an impact on their future prospects, where they are tough in the field of statistics, can compete, become professional teachers, further study, able to answer statistical problems in society, and has a strong statistical self-concept. statistical information is the main source to improve literacy and representation skills. to understand, analyze, and interpret the information, several literacy skills are required namely; speaking and writing skills, communication, argumentation, and representation skills. attitudes in communicating ideas and self-confidence, are the main indicators and are important for students to have. attitudes need to be supported by abilities of mathematics, statistics, logical thinking, critical, creative, analysis and interpretation. attitudes and abilities possessed will make students a formidable individual in finding solutions to challenging statistical problems in society. the results of observations on the strengthening of statistical literacy and mathematical representation through the use of the cps learning model showed that all aspects of collaboration were carried out in good and very good qualifications. the volume 9, no 2, september 2020, pp. 183-196 193 feasibility indicator of the model was at rating-2, with criteria for half or most of the indicators that were implemented and rating-3 of all indicators were implemented. the cps learning model was feasible because the lecturer was able to apply the model according to stages of collaborative learning. the results of interviews with several students obtained information that, the cps learning model was effectively used in the learning process. the problem was that collaboration time needed to be added. another response showed that collaborating can facilitate students to understand statistical material with the help of statistical literacy learning media. by collaborating, students are trained to construct ideas individually in understanding statistical information (determining main ideas, seeing relationships and differences), and can present information in tables or graphs. related to anxiety, students had diverse opinions. students stated that lectures by collaborating can minimize anxiety. lectures on statistics with statistical literacy media for students can help them in writing a bachelor thesis, especially the ability to read graphs, tables, statistical symbols, and be able to describe them. the results of this study expect a reformation of the statistics learning paradigm for students which is oriented towards increasing the ability to analyze and interpret data from a statistical problem through the use of statistical literacy media in learning. statistical literacy and mathematical representations also need to be maximally developed for elementary school students through the use of information literacy-based media that contains contextual statistical problems. 4. conclusion based on the results, it can be concluded that there was a strong relationship between statistical literacy and mathematical representation of students as pre-service elementary school teachers facilitated through the cps learning model. overall, the qualification aspects of the cps learning model had an increase in high and very high qualifications, while the feasibility of the model indicators was at rating-2 (half or most of the indicators were implemented and rating-3 (all indicators were implemented). references ainsworth, s. 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(2006). statistics education in the netherlands and flanders: an outline of introductory courses at universities and colleges. icots-7 conference proceedings, 60115-2828. walker, d. a. (2017). jmasm 48: the pearson product-moment correlation coefficient and adjustment indices: the fisher approximate unbiased estimator and the olkinpratt adjustment (spss). journal of modern applied statistical methods, 16(2), 540-546. https://doi.org/10.22237/jmasm/1509496140 watson, j. m. (2003). statistical literacy at the school level: what should students know and do. the bulletin of the international statistical institute, berlim, 54, 1-4. watson, j. m. (2006). issues for statistical literacy in the middle school. icots-7 conference proceedings. iase, salvador (cd-rom). 1-6. https://doi.org/10.21831/jk.v2i2.18768 https://doi.org/10.30870/jpsd.v4i2.3852 http://iase-web.org/documents/papers/icots8/icots8_c193_tishkovskay.pdf http://iase-web.org/documents/papers/icots8/icots8_c193_tishkovskay.pdf http://iase-web.org/documents/papers/icots8/icots8_c193_tishkovskay.pdf https://doi.org/10.1177%2f1046496406292938 https://www.stat.auckland.ac.nz/~iase/publications/17/3a4_verh.pdf https://www.stat.auckland.ac.nz/~iase/publications/17/3a4_verh.pdf https://www.stat.auckland.ac.nz/~iase/publications/17/3a4_verh.pdf https://doi.org/10.22237/jmasm/1509496140 http://iase-web.org/documents/papers/isi54/3516.pdf http://iase-web.org/documents/papers/isi54/3516.pdf http://iase-web.org/documents/papers/icots7/6c1_wats.pdf http://iase-web.org/documents/papers/icots7/6c1_wats.pdf takaria, wahyudin, sabandar, & dahlan, relationship between statistical literacy … 196 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p213-222 213 relationship between misconception and mathematical abstraction of geometry at junior high school gida kadarisma*, nelly fitriani, risma amelia insititut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received aug 21, 2020 revised sep 20, 2020 accepted sep 22, 2012 this study aims to examine the misconceptions that often occur in junior high school students on the concept of geometry based on abstraction level. the research method is qualitative with a case study design. subjects in this study are 27 students of the 3rd grade of junior high school students, who had to learn all the concepts that will be appeared on the test. material that will be given on the test of this research is the concept of triangle, quadrilateral, flat side geometry and curved side geometry. this research takes a place at one of the junior high schools in cimahi. the instrument in this study is a diagnostic test (to find out the types of students’ misconception), mathematical abstraction tests (to determine the level of abstraction) and interview rubrics. misconceptions produced by students are closely related to students’ mathematical abstractions, the higher the level of abstraction ability, the more students away from misconceptions. the topic taken in this study is the topic of basic geometry, the results can be a source of information about the types of misconception that often occur in students, and how the solution so that these misconception do not re-occur. keywords: abstraction, geometry, misconception copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: gida kadarisma, departement of mathematics education, institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, west java 40526, indonesia. email: gidakadarisma@ikipsiliwangi.ac.id how to cite: kadarisma, g., fitriani, n., & amelia, r. (2020). relationship between misconception and mathematical abstraction of geometry at junior high school. infinity, 9(2), 213-222. 1. introduction geometry is a part of mathematics, geometry has an important role to several things, for instance by studying geometry, it will increase logical thinking and the ability to make proper generalizations; better understanding on arithmetic, algebra , and calculus; getting a further learning; accelerating the mental development of students (novita et al., 2018). but unfortunately, geometry is one the topic of mathematical which experiencing problems. the performance of students in geometry reportedly very bad and is already supposed to be a concern for teachers of mathematics, the parents and the government (adolphus, 2011). poor performance is indicated by the wrong answers that are often made by students. the types of error that are often made by students are very important to know, classify and examine the causes, so that learning can be more effective. based on this , https://doi.org/10.22460/infinity.v9i2.p213-222 kadarisma, fitriani, & amelia, relationship between misconception and mathematical … 214 researchers want to identify the type of error that made the students of junior high school and to investigate the possibility of the cause of the error, the reliable hypothesis emerge that the misconceptions is associated with the process of students’ abstraction. misconceptions is a mistake in understanding the concepts or errors in interpreting the concept meaning (ay, 2017). misconception can also be interpreted by the cognitive structure inherent in a person but deviates from the actual conception. misconceptions experienced by a person in learning mathematics will have a long impact on the subsequent understanding of mathematics. because concepts in mathematics are not solitary, but are interrelated with one another. one basic concept of mistake, will lead someone to make other mistakes (kadarisma, 2016). when someone is having misconception, it is different with what we called as "nescience". when a student experiences a misconception , in fact he has gone through the process of processing the information, only the way that might be wrong, some are memorizing immediately or some are wrong in interpreting so it leads to misconception. if the misconceptions happen during process of constructing the concept / structuring the cognitive experience , then the concept that is accepted will not be completely perfect and continuation for other concepts will be hampered. if it is not corrected immediately, it will become a serious problem. the process of constructing the concepts is a process of mathematical abstraction. so between these two things certainly have a close relationship. it can be concluded that if a person experiences a misconception, there is a problem in the abstraction process experiences problems or even does not have this ability. so as to avoid misconceptions happening to students, the abstraction process conducted by students must be improved/ developed. the ability of abstraction are in different levels from one to another or even there is student who does not have the ability. fitriani (2018) modifies the levels of abstraction and indicators based on research that has been done by (battista, 2007; goodson-espy, 1998; hong & kim, 2016; nurhasanah, 2018), among which are: perceptual abstraction/ level 1 (getting to know the properties of mathematical objects based on the use of physical objects, recognizing previous experiences related to the problem being faced); internalization/ level 2 (representing the results of thought in the form of mathematical symbols, words, pictures, or diagrams; interiorization/ level 3 (organizing (collecting, compiling, developing, and coordinating) concepts into new understanding or new knowledge), and second level of interiorization/ level 4 (generalizig new knowledge in a different context). based on this theory, students experiencing misconceptions do not have the ability to mathematical abstraction or perhaps at the level of beginner/ pre levels of abstraction. when that happens, students have a tendency to solve problems with routine procedures (hendriana, prahmana, & hidayat, 2018), so perhaps a teacher/ teaching materials do not facilitate students to do so, but tend to directly provide formulas and do not coordinate the concepts with one to another. misconceptions include understanding or thinking which is not based on true information (kusmaryono et al., 2020). according to dayanti, sugiatno, & nursangaji (2019), there are three types of misconceptions commonly done by students; classificational misconception, correlational misconceptions and theoretical misconceptions. researchers are interested in analyzing the extent of misconceptions experienced by students in terms of the level of abstraction. according to this, researchers can recommend things that are considered capable to avoid students from misconceptions. volume 9, no 2, september 2020, pp. 213-222 215 2. method the research method of this research is qualitative with a case study design. the research subjects in this study are 27 students of 3rd grade of junior high school who had learned all the concepts that would be presented in the test, namely the concept of triangles, quadrilateral, flat side geometry and curved side geometry. the location in this study is in one of the junior high schools in cimahi. the instrument in this study is set of question of diagnostic tests (to find out the types of difficulties that faced by students), mathematical abstraction tests (to determine the level of abstraction) and interview rubrics (as a form of data triangulation). the research procedures are: 1) students are given a test of mathematical abstraction (aims to find out the basic level of the students), 2) students are given a special diagnostic test on the topic of geometry that has been explain previously (aims to find out the types of misconceptions / errors that are faced by students) and conduct interviews with some students who experiencing misconceptions. data processing procedures: 1) analyzing the abstraction level of students, 2) examining the misconceptions of the students, 3) in depth analysis on the relation between the levels of abstraction and misconceptions done by students, 4) concluding the phenomenon. 3. results and discussion 3.1. results in this study, we obtained some data to be analyzed, including the percentage of students experiencing misconceptions based on their level of abstraction, (see table 1). table 1. level abstraction and types of misconceptions that are produced mathematical abstraction level misconception type (%) theoretical classification correlational level 1 (19 people) 55.56 62.96 70.37 level 2 (5 people) 7.41 3.70 level 3 (2 people) level 4 (1 person) based on table 1, it appears that the subjects in this study are divided into several levels of abstraction ability, ranging from level 1 to level 4. there are 4 students at level 3 and 1 student at level 4. these two level show a limited number of students; it happens because the characteristics of students taken are from students with a basic to average level of ability. students at levels 3 and 4 did not experience any misconception (see table 1). they managed to answer the diagnostic test well. students at levels 3 and 4 discover no errors in determining the elements in building space or getting up flat, they are also capable determine the relationship between concepts (between flat and shape geometry), and they are able to explain problems such as height in a triangle. in contrast to students at level 1, 55.56% of students experiencing theoretical misconception, 62.96% experiencing classificational misconception and the largest is 70.37% in a correlational misconception. we can say that, students at that level experienced a lot of misconceptions. while at level 2 there are those who experience misconceptions but kadarisma, fitriani, & amelia, relationship between misconception and mathematical … 216 the percentage is not significant. there is a suspicion that the higher level of students’ abstraction, the bigger possibility they will avoid any kind of misconception (see table 1). furthermore, the students’ error during the diagnostic test. researchers try to analyze several possible reasons for errors, the result of the analysis leads to the low ability of mathematical abstraction (see table 2). table 2. types of misconceptions and forms of errors committed by students types of misconceptions mistake made possible reason theoretical error in determining the formula for surface area and volume of space do not understanding the concepts of area and volume, students only memorizing the formulas. error in understanding some formulas for solid geometry confusion using formulas because they do not understand the concept of volume, do not understand the relationship between shapes, students only memorizing the formulas error determining height in triangle students do not understand high definition well and lack of mastery of prerequisite material mistakenly determined edge on brsl does not have well understanding on the definition of edge, has weak spatial ability mistakenly determine the diagonal plane and the diagonal plane on the cube has weak spatial ability classification student error in classifying the types of triangles incomplete understanding of the concept correlational error in determining the relationship between the concept of prism with the concept of a cube, cuboid, or cylinder low ability to visualize, low ability to construct mathematical ideas error in determining the relationship between the concept of the pyramid with the concept of cones low ability to visualize, low ability construct mathematical ideas error in determining the relationship between quadrilateral shapes low able to visualize the following will examine some of the results of errors that have been made by students. first is classificational misconception, it happens first because students are not volume 9, no 2, september 2020, pp. 213-222 217 able to classify the types of triangles. if they being asked the types of triangles, students were only able to answer the isosceles triangle, right triangle, equilateral triangle, and scalene triangle. as shown in the sample of student’s answers in figure 1. figure 1. sample of student’s answers figure 1 show that an example of the answers of students who experience classificational misconceptions. students cannot answer the types of triangles thoroughly. when teaching concepts about triangles, both the teacher and the textbook convey the concepts separately. begin with the concept of equilateral triangle, isosceles triangle and scalene triangles, then focus on formulas to find area and circumference. after that, the review a little about the triangle based on the angle is given. this has become less balanced. everything is given directly by the teacher and textbooks, not through the use of physical objects, and students are not given the opportunity to represent their observasion. so that the form of coordination between concepts is definitely not happening. students do not experience a process of abstraction in finding the concept and experiencing a classificational misconception. furthermore, in the term of theoretical conception, students experience confusion in determining the height of a triangle if they are asked about the area of a triangle, as in figure 2. figure 2. sample of student’s answers students experience misconceptions when working on these problems. generally, they know that the concept of height in the triangle is a vertical line from the top to the base, there they do not understand the definition of height in depth (see figure 2). if student examines the diagnostic problem that is given well, then the number shown will be triple phytagoras, so that between the shortest sides forming an angle of 900 means that there is a concept of base and height to determine the area of a triangle. students experiencing other theoretical misconceptions and correlational misconceptions, this includes errors in explaining mathematical facts and also connecting concepts with one another (see figure 3). kadarisma, fitriani, & amelia, relationship between misconception and mathematical … 218 figure 3. sample of student’s answers figure 3 show that students experience confusion in using the formula between the area of surface and the volume of solid geometry. in addition, students also feeling confuse in using the formulas of volume of solid geometry in solving a given problem. they have difficulty in memorizing all of these formulas, the demands given require students to memorize everything (not to understand it). this is reinforced by the results of the interview with one of the students (s1), following the interview excerpts conducted. .............................................. t : can you solve the problem? s1 : sure ma'am, it just sometimes i forgot the formula, hehehe ... t : why is that? s1 : there are so many formulas, sometimes it is like being switched, even i already memorizing it. .............................................. students do this is caused by several factors, one of which is that teachers teach concepts directly and the target of their learning is students must be able to work on the problems given. the books are used is also have a similar objective, the books contain a formula that ends with exercises. both the teacher and the book that were designed, were only oriented towards the final results/ final test, and paid little attention to the occurrence of the concept formation. in addition, the material conveyed tends to stand solitarily, not connected one to another, students do not recognize that the cube, cuboid, even the cylinder is included in the prism. pyramid and cone was never be related one another. when students are asked whether a square is a rectangle, whether a rectangle is a parallelogram, whether the rhombus is a parallelogram, students generally answer no. this has triggered the emergence of correlational misconceptions. students' understanding that they all stand alone (have each formula which is different and not related to each other (not have interconnected concepts). this is reinforced by the results of interviews with other students who experienced misconceptions (s2), following is part the interview: .............................................. t : why do you think that formulas of volume of the shape are many? s2 : yes there are indeed many, there are cube volumes, there are cuboid volumes, there are prism bolts, not to mention area of the square, the area of rhombus and others, it make me dizzy, it is too many and it so hard to memorize t : do the shapes you mentioned earlier have different formulas? s2 : obviously different, ma'am, so i have difficulty in memorizing it t : is there no relation between formula one and the other formula? s2 : there is no ma'am volume 9, no 2, september 2020, pp. 213-222 219 t : are there the same properties between one shape and another, for example a square with rectangle? s2 : (students just being quite and look confused ...) .............................................. 3.2. discussion the results showed that there were students from various levels of abstraction who experienced different misunderstandings (see table 1 and table 2). as the level rises, students generally do not experience significant misconceptions. these results are in line with research conducted by fitriani, suryadi, & darhim (2018) that the low mathematical abstraction ability of junior high school students is highly correlated with the level of mathematical ability that is indeed in the lower category, so that the tendency when the level of abstraction is medium or low, they more often experience mathematical misconceptions. furthermore, researchers have described several types of errors that have been made by students when working on diagnostic test. the types of error made are categorized based on the types of misconceptions that have been mentioned previously, the errors generated are in line with the results of research conducted by ozerem (2012). based on the diagnostic tests, it turns out students at the low level of abstraction experience these 3 types of misconceptions; theoretical, correlational, and classificational misconceptions. first is classificational misconception, this misconception happen because students are not able to classify the types of triangles. generally, students answer the teacher's question about the types of triangles, but the answers they give are less comprehensive. these results are in line with research that has been done by sanapiah & juliangkary (2017), that the understanding of the concept of a triangle of students have not fulfilled in the stages of classifying the types of triangles. after analyzing the textbooks and conducting the classroom observations, it turns out that learning designs that happen is not using a physical objects, and students are not given the opportunity to represent themselves about what they observed, then the material structure is not proportional. so that the form of coordination between concepts is definitely not happening. students do not experience an abstraction process in discovering the concept and experience a classificational misconception. then students experience theoretical misconceptions, they experience confusion when determining the height of a triangle when being asked about the area of a triangle. the results of these students' answers are in line with the results of research conducted by hutagalung, mulyana, & pangaribuan (2020). this misconception occurs to students, because in general they are usually given examples of routine problems. students are never got a good definition, they are never get the opportunity to construct a definition. students are only given an explanation, which includes the height or base of a triangle through basic examples. submission of concepts is very much avoided from the process of abstraction, students are not invited to recall previous experiences relating to the concept of phytagoras and their properties which can further strengthen the concept of the base and height of the triangle, the instructional process avoids direct physical observation, students are not directed to represent what is observed in a geometrical drawing, and at the end, students are not able to coordinate the concept of phytagoras with the height / base of the triangle so that they experience a classificational misconception to solve the area of the triangle. students experience other theoretical misconceptions and correlational misconceptions, this includes errors in explaining mathematical facts and also connecting kadarisma, fitriani, & amelia, relationship between misconception and mathematical … 220 concepts with one another. students experience misconceptions because they have to memorize all formula of geometry. there are several incorrect orientations. the process of abstraction in geometry is not happen. according to syahbana (2013), the volume of a geometry can be recognized by tracing the shape of the base. in essence, a geometrical structure derives from the broad structure of the base which forms the height of the geometric structure. if there are other irregular spaces, try to form a sketch so that the shape of the base can be recognized. assuming that this geometry has undergone a transformation in form from its normal form, it is necessary to trace which shape is the base and determine which height. with this process of abstraction, the learning to convey the concept of the volume of geometry will avoid misconceptions. the material conveyed by the teacher regarding rectangular also about geometry should be designed to be interconnected, when the teacher designs learning well, the learning objectives will be achieved (kadarisma et al., 2019), so students do not experience confusion in constructing concepts, and the coordination between concepts will be built up, the abstraction process is very instrumental in it. if this is applied, the students of geometry ability will be better and students will not face misconceptions. from the analysis, we can see some factors that ultimately make students experience misconceptions. based on this, it appears that students are very isolaeded from the process of abstraction. to overcome this, students should be directed to be able to recognize the properties of geometry by using physical objects or discover it directly, then students must re-know previous experiences related to the concept being faced (for example when studying space construction, review back to the concept of a flat build, the concept of congruence, the concept of alignment, etc.), do not let the students have limited understanding about the prerequisite material. furthermore, students must be able to represent the results of their thoughts / observations in the form of drawings (specifically geometry), lest students only see, without being able to pour what they see, as a form of analysis carried out on the results of their observations. when drawing, students indirectly do mathematical modeling, students feel which parts of the structure are congruent and analyze the properties of the observations as outlined. next, students is asked to construct the concept of construction process (put it in the student worksheet), then direct students to be able to develop what they have captured (can be by giving other cases or open ended cases), and coordinating other concepts that similar so that they become a new understanding or new knowledge that is more comprehensive. based on the data obtained, it appears that students who experience misconceptions are students who come from the lowest level of abstraction. for students with high levels of abstraction, they generally do not experience misconceptions. visible relationship between the two, then it becomes a recommendation for researchers so that the process of abstraction occurs to students, thus avoiding students from misconceptions. in order for students to have good mathematical skills, the mathematics teacher as a supporting factor must also have good mathematical skills (hidayat, 2017). in addition, this abstraction process is closely related to the level of students’ geometrical thinking, the abstraction process that is sharpened makes students develop a level of geometrical thinking so students are expected to have a higher level of knowledge in geometry and certainly avoid misconceptions. 4. conclusion based on research that has been done, it can be concluded that students experience various types of misconceptions. having analyzed that occur on student misconceptions are closely linked to the ability of abstraction that is owned by the students where the better the volume 9, no 2, september 2020, pp. 213-222 221 ability of abstraction, the more students are protected from misconception. the topic taken in this research is the topic of basic geometry, the results can be a source of information for middle school teachers about the types of errors that often occur in students. references adolphus, t. 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(2013). alternatif pemahaman konsep umum volume suatu bangun ruang. edumatica, 3(2), 1-7. https://doi.org/10.22460/infinity.v9i1.p15-30 https://doi.org/10.21831/jrpm.v5i1.16836 http://repository.upi.edu/49513/ http://repository.upi.edu/49513/ http://repository.upi.edu/49513/ https://doi.org/10.1016/j.sbspro.2012.09.557 http://elpsa.org/proceeding/index.php/ec17/article/view/10 http://elpsa.org/proceeding/index.php/ec17/article/view/10 https://online-journal.unja.ac.id/edumatica/article/view/2662 https://online-journal.unja.ac.id/edumatica/article/view/2662 infinity infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 25 etnomatematika masyarakat pengrajin anyaman rajapolah kabupaten tasikmalaya oleh: mega nur prabawati program studi pendidikan matematika, universitas siliwangi megarafaadzani@gmail.com abstrak sebagian masyarakat sering tidak menyadari telah menerapkan ilmu matematika dalam kehidupannya. kecenderungannya adalah mereka memandang bahwa matematika hanyalah suatu mata pelajaran yang hanya dipelajari dan diperoleh dibangku sekolah. padahal tidak hanya itu, matematika sering digunakan dalam berbagai aspek kehidupan, misalnya dalam mengukur, mengurutkan suatu bilangan, dan masih banyak lagi yang lainnya. keberadaan etnomatematika kerajinan anyaman ini dapat digunakan sebagai sumber belajar dan tentu saja dapat membuat siswa ataupun masyarakat lebih memahami bagaimana budaya mereka berhubungan dengan matematika. kata kunci : etnomatematika, kerajinan anyaman. abstract some people often do not realize has been applying mathematics in their life. the tendency is they looked at that math is simply a subjects who only studied and obtained in college. but not only that, the math is often used in various aspects of life, such as in measure, sort of a number, and much more. the existence of the etnomatematika woven crafts can be used as a source of learning and of course can make students or the community better understand how their culture related to mathematics. keywords: ethnomathematics, woven crafts i. pendahuluan matematika dianggap sebagai sesuatu yang netral dan terbebas dari budaya (culturraly-free). seperti yang diungkapkan oleh rosa dan orey (2011) bahwa “mathematics always taught in scholl as a culturaly free subject that involved learning supposedly universally accepted facts, concept and content”. matematika dipelajari di sekolah sebagai mata pelajaran yang tidak terkait dengan budaya yang secara umum pembelajarannya maliputi fakta-fakta, konsep, dan materi. matematika juga dianggap sebagai ilmu pengetahuan yang sempurna dengan kebenaran yang objektif dan dirasakan jauh dari realitas kehidupan sehari-hari. menurut d’ambrosio (gerdes, 1996:912) pada masa sebelum dan diluar sekolah hamper semua anak di dunia telah menjadi “matherate” artinya, mereka mampu mengembangkan kemampuan untuk menggunakan bilangan, menghitung, dan menggunakan beberapa pola inferensi. tetapi, seorang individu yang dengan sempurna telah mampu mengunakan bilangan, operasi, bentuk geometris, dan gagasan, ketika disekolah dihadapkan pada pendekatan yang sama sekali baru dan formal melalui fakta¬-fakta. sebagai akibatnya, terbentuklah penyumbatan psikologis yang tumbuh sebagai penghalang antara perbedaan model-model numerik yang dipelajari di sekolah dengan pemikiran geometris yang sudah dipelajarinya dari kehidupan nyata sebelum atau diluar sekolah, sehingga tahap awal infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 26 pendidikan matematika memberikan pengaruh pada anak rasa kegagalan, ketergantungan, bahkan kehilangan kemampuan matematisyang telah dimiliki pada masa pra sekolah. hal tersebut menunjukkan bahwa pembelajaran matematika disekolah terlepas dari kehidupan nyata yang kaya akan budaya nyata dan peradaban. namun penelitian-penelitian yang telah dilakukan menunjukkan bahwa terdapat keterkaitan antara matematika dan budaya. menurut clements (karnilah, 2013:2) salah satunya dapat dilihat dari hasil pertemuan-pertemuan international community of mathematics education yang menyatakan bahwa permasalahan yang terkait dengan budaya mau tidak mau akan mengelilingi proses belajar mengajar matematika, bahkan mengelilingi pula semua bentukbentuk matematika. turmudi (2012:5) menyatakan bahwa matematika berurusan dengan gagasan, matematika bukan tanda-tanda sebagai akibat dari coretan pensil, bahkan kumpulan benda-benda fisik berupa segitiga, namun berupa gagasan yang direpresentasikan oleh benda-benda fisik. sehingga menurut turmudi terdapat tiga sifat utama dari matematika. pertama, matematika sebagai objek yang ditemukan dan dicptakan manusia. kedua, matematika itu dicptakan bukan jatuh dengan sendirinya, namun muncul dari dari aktivitas yang objeknya telah tersedia, serta dari keperluan sainsdan kehidupan keseharian. ketiga, sekali diciptakan objek matematika memiliki sifat-sifat yang ditentukan secara baik. pembelajaran matematika sekolah saat ini banyak mengadopsi dari pembelajaran matematika negara luar yang dianggap lebih maju. seperti yang ditulis dalam artikel yang berjudul budaya pengaruhi ilmu matematika pada harian umum pikiran rakyat 14 januari 2010, pengadopsian kurikulum ini tidaklah salah hanya saja pendekatan budaya setempat dalam pembelajaran mateatika perlu diterapkan juga agar penguasaan siswa lebih sempurna. menurut hemat peneliti, indonesia dengan keragaman budayanya sudah seharusnya memasukkan nilai-nilai budaya setempat ke dalam pembelajaran matematika, agar matematika tidak dianggap sebagai ilmu pengetahuan yang jauh dari realitas kehidupan. hal ini dikarenakan dalam aktivitas budaya terdapat ide-ide matematis yang dianggap sebagai hal yang penting dalam pembelajaran matematika. ii. pembahasan a. etnomatematika pada abad 19-an sudah dikenal beragam istilah dengan kata awal ethno yang mengalami perluasan makna, powell (francois, 2010:1518) ethno diatikan sebagai sebagai suatu konsep yang mengacu pada kelomok etnis, kelompok nasional, kelompok ras, kelompok professional, kelompok dengan dasar filosofis atau ideologis, kelompok social dan budaya. beragam kajian mengenai ethno telah dikenal seperti ethnomusicology, ethnobotany, ethnopsychology. jika ethnoscinece dimaknai sebagai kajian scientific berkaitan dengan fenomena-fenomena teknologi yang berkaitan langsung dengan latar belakang social, ekonomi, dan budaya. ethnolanguage dimaknai kajian bahasa dalam hubungannya dengan keseluruhan budaya dan kehidupan social, sehingga dengan analogi yang sama ethnomathematicsdimaknai sebagai kajian matematika (ide matematika) dalam hubungannya dengan keseluruhan budaya dan kehidupan social (gerdes, 1996:916); (gerdes, 1997:343). infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 27 di akhir tahun 1970-an dan di awal tahun 1980, pertumbuhan kesadaran akan aspek social dan budaya dari matematika dan pendidikan matematika mulai muncul dikalangan matematikawan. dukungan – dukungan terhadap hal tersebut dapat ditemui dalam berbagai pertemuan-pertemuan internasional para matematikawan, pendidik matematika, dan pembuat kebijakan politik pendidikan dimana keobjektifan social terhadap pendidikan matematika secara tulus dipertimbangkan. salah seorang yang berperan penting dalam melakukan inisiatif-inisiatif tersebut yakni ubiratan d’ ambrosio seorang matematikawan brazil dan guru matematika. pada periode itu d’ambrosio meluncurkan ethnomathematical program sebagai metodologi untuk melacak dan menganalisis proses produksi, pemindahan, penyebaran, dan pelembagaan pengetahuan (matematika) dalam berbagai macam system budaya, yang kemudia d’ambrosio dikenal sebagai intellectual father of the ethnomathematical program. d’ambrosio membedakan matematika akademik yang biasa diajarkan dan dipelajari disekolah dengan etnomatematika yang digambarkan sebagai matematika yang dipraktekkan diantara kelompok budaya yang dapat diindentifikasi seperti masyarakat, suku, kelompok buruh, anak-anak dari kelompok tertentu dan kelas professional. menurut beliau, mekanisme sekolah menggantikan praktekpraktek tersebut dengan praktek-praktek lain yang setara yang telah memperoleh status matematika, yang telah diambil alih dari bentuk aslinya dan kembali disusun menurut versi suatu sistem. kemudian munculah pertanyaan “haruskah kita memasrahkan kurikulum sekolah dan menggantinya dengan etnomatematika? jelaslah tidak”. d’ambrosio menyatakan bahwa kurikulum matematika sekolah semestinya memasukan etnomatematika sedemikian hingga memfasilitasi siswa untuk mendapat pengetahuan, umtuk memahami, dan menyelaraskan pengetahuannya dengan praktek-praktek (budaya) yang telah dikenal (gerdes, 1996:912-913). dari segi etimologis, d’ambrosio (peard, 1996:42) awalnya mendefinisikan ethnomathematics sebagai praktik matematika yang dilakukan oleh kelompok-kelompok budaya tertentu, seperti masyarakat suku tertentu, kelompok buruh, anak-anak dari kelompok usia tertentu, kelas professional dan sebagainya. identitas dari kelompok tersebut biasanya bergantung pada focus minat atau kepentingan, motivasi dan kode-kode tertentu dan jargon yang tidak terkait dengan bidang matematika akademik. b. anyaman tradisional dan konsep berpikir masyarakat rajapolah menurut hoenigman (wikipedia, 2008) anyaman merupakan wujud kebudayaan, yang termasuk dalam artefak. artefak adalah wujud kebudayaan fisik yang berupa hasil dari aktivitas, perbuatan, dan karya semua manusia dalam masyarakat berupa benda-benda atau hal-hal yang dapat diraba, dilihat, dan didokumentasikan. anyaman pertama kali digunakan manusia, yaitu untuk membantu dalam kehidupannya sehari-hari. anyaman merupakan salah satu bentuk lain dari gerabah yang terbuat dari pengaturan bilahbilah selain dari gerabah yang terbuat dari tanah liat. banyak sekali jenis anyaman tradisional yang terdapat di suku sunda. dimana beda material beda juga nama dan teknik menganyam. di rajapolah sendiri setidaknya ada 3 jenis material yang digunakan yaitu adalah bambu, pandan, dan mendong. tiap bahan memiliki karakteristik dan beberapa diantaranya memiliki filosofi yang sangat kuat. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 28 motif anyaman tradisional sangat beragam hal ini dikarenakan bahan yang digunakan dalam pembuatan anyaman berbeda-beda, namun beberapa motif anyaman meskipun bahan berbeda ada yang diberi nama sama, hal ini melihat dari kesamaan bentuk motifnya. dilihat dari keadaan diatas, masyarakat sunda rajapolah telah memiliki sebuah pemikiran yang sangat logis dan jauh dari sifat mistis dalam pembuatan motif anyaman, sehingga nama yang diberikan merupakan nama anyaman yang diambil dari alam dan kehidupan yang mereka jalani.beberapa bahan anyaman memiliki filosofi yang kuat. bambu adalah salah satu bahan anyaman yang sangat kental dengan makna, apalagi jika kita menghubungkan dengan suku sunda. masyarakat sunda sudah sedemikian lama berhubungan akrab dengan bambu, banyak pengalaman leluhur yang bisa dipetik, sejak lahir hingga mati, orang sunda selalu dipertemukan dengan bambu. menurut pengurus harian yayasan bambu indonesia, jatnika (kompas, 2007), menuturkan "di masa lalu, seluruh rangkaian hidup orang sunda penuh dengan bambu," katanya. pada saat dilahirkan, bayi-bayi sunda dahulu dilepaskan dari ari-arinya menggunakan sembilu dari bambu. lalu bayi tersebut disimpan dalam ayakan atau saringan besar terbuat dari bambu. ketika bayi lelaki disunat, pisau penyunatnya terbuat dari bambu. saat belajar berjalan, orangtuanya membuat tonggak-tonggak dari bamboo di halaman yang bisa dikitari oleh anak tersebut. saat makin besar, ia dibuatkan jajangkungan (mainan dari bambu) untuk berlatih keseimbangan, kakinya akan naik ke bambu yang tinggi dan ia berjalan di atasnya sehingga bisa melihat desa dari atas. makin besar, mereka mengasah keterampilan tangan dan kekompakan dengan teman melalui berbagai permainan, seperti bebedilan atau pistol mainan, mereka juga membuat alat musik untuk hiburan, seperti angklung, calung, dan suling. di kalangan keluarga, mereka menggunakan daun bambu untuk membungkus makanan seperti bacang dan wajit. mereka juga memakan rebung atau anak bambu untuk sayur. seharihari mereka tinggal di rumah bambu dan membuat mebel dari bambu. perkakas rumah tangga seperti pengki (tempat sampah) hingga aseupan (pengukus) terbuat dari anyaman bambu. ketika sudah tua, orang sunda membuat tongkat dari bambu. saat meninggal, ia ditandu dengan keranda bambu dengan penutup jenazah dari anyaman bambu. bambu juga merupakan bahan bangunan yang hingga kini digunakan oleh masyarakat sunda yaitu digunakan dalam pembuatan sekat atau dinding rumah yang tidak lain sering disebut bilik, tentu saja digunakan di rumah-rumah yang terdapat di perkampungan dengan menggunakan 4 hingga 6 buah penyangga dari batu, dan menurut penelitian rumah jenis ini dapat meminimalisir guncangan gempa. selain itu pula bambu digunakan sebagai alat musik, angklung dan suling sudah digunakan orang sunda sejak abad ke-7. selain bambu bahan dasar lain seperti pandan memiliki nilai filosofi dalam kehidupan masyarakat sunda. pandan memiliki karakteristik yang mudah dibentuk, halus, dan lentur. pandan mempunyai nilai filosofi yang cukup tinggi, menurut ali sastramidjaja (2007) nilai filosofi yang terkandung dari pandan dapat kita lihat pada produk anyaman, yaitu adalah tikar pandan atau samak. pada jaman dahulu masyarakat sunda mempunyai kebiasaan bahwa samak merupakan keluarga. hal ini dapat dilihat dari keseharian masyarakat sunda dahulu, mereka lahir diatas tikar, saat ada waktu berkumpul mereka ada diatas tikar dan ketika meninggal ditutup oleh tikar pula. selain itu pandan juga memiliki keunggulan yang mungkin tidak semua suku atau bangsa tahu, yaitu saat bayi suku sunda lahir, darah yang tercecer pada tikar pandan, dapat dibersihkan dengan mudah dan bau dari darah dapat hilang dengan cepat, selain digunakan infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 29 dalam proses kelahiran, samak digunakan pada saat seseorang meninggal, dimana jasadnya akan ditutup oleh kain kafan dan ditutup oleh tikar pandan, menurut warga sekitar dengan tikar itu sendiri maka bau mayat tidak akan tercium, sehingga tidak akan menimbulkan fitnah atau kejadian yang tidak diinginkan. selain dari bahan pembuat anyaman, filosofi kehidupan masyarakat sunda dapat dikaji dari segi bentuk benda anyaman yang mewakili filosofi hidup suku sunda. menurut mamat sasmita (pendiri rumah baca buku sunda) pada boboko (tempat nasi) bentuknya yang unik, bentuk atasnya yang membulat dan bawahnya yang menggunakan alas berbentuk persegi merupakan filosofi hidup masyarakat sunda yaitu “tekad kudu buleud, hidup kudu masagi” yang artinya menurut bahasa tekad harus bulat, dan hidup harus persegi, yang secara garis besar bisa diartikan kita harus mempunyai tekad yang teguh dan tidak goyah dan hidup kita harus teratur. c. ide matematis etnomatematika muncul sebagai konsep baru yang merupakan pengaruh timbal balik antara matematika, pendidikan, budaya, dan politik. etnomatematika dinyatakan sebagai sebuah kajian terhadap ide-ide matematik pada masyarakat primitive. ide-ide matematik terdapat pada setiap budaya, akan tetapi yang diutamakan adalah bagaimana mereka mengungkapkannya dan konteks-konteks khusus yang terdapat pada suatu budaya akan berbeda dengan budaya yang lain. perbedaannya bukan terletak pada kemampuan untuk berpikir abstrak secara logis, tetapi terletak pada pemikiran subjek, anggapan dasar budaya, dan situasi apa yang muncul saat proses berpikir. istilah ide matematis muncul dalam buku yang ditulis maria ascher. dalam buku ini menganalisis mengenai beberapa contoh dari ide-ide matematis, seperti sistem bilangan, sistem kekerabatan, sistem navigasi, karakteristik desain, dan analisis permainan. menurut ascher dan ascher (1997:25), ide-ide matematis meliputi segala sesuatu yang berkaitan dengan bilangan, logika, konfigurasi spasial, dan kombinasi atau susunan dalam sistem dan struktur. ascher mengakui bahwa terdapat dua aspek dalam etnomatematika, yaitu memahami hubungan antara ide-ide matematis dan budaya serta bagaimana memodifikasi pendidikan matematika dengan memasukan ide-ide tersebut. penelitian etnomatematika dalam ranah pendidikan dapat digunakan untuk mengungkap ideide yang terdapat dalam aktivitas budaya tertentu atau kelompok sosial tertentu untuk mengembangkan kurikulum matematika untuk, dengan dan oleh kelompok tersebut. sehingga matematika dapat memiliki bentuk yang berbeda-beda dan berkembang sesuai dengan perkembangan masyarakat pemakainya. sejalan yang diuangkapkan oleh gerdes (1996:930) yaitu “use of ideas embedded in the activities of certain cultural or social (marginalized) group within a society to develop a mathematical curriculum for and with/ bt this group”. d. pemanfaatan etnomatematika kerajinan anyaman rajapolah dalam pembelajaran kajian geometri “a tessellation is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane” (o’daffer, [4]:676). kutipan di atasmenyatakan bahwa teselasi merupakan suatu pola khusus yang terdiri dari bangun-bangungeometri yang disusun tanpa pemisah/jarak untuk menutupi suatu bidang datar.istilah lain yang sering digunakan infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 30 untuk menyebut teselasi adalah pengubinan.“teselasi atau pengubinan merupakan konsep antar cabang ilmu pengetahuan,yaitu matematika dan seni” (rokhmah, [5]:1). ketika teselasi digunakan oleh beberapaseniman dan tukang batu, teselasi mengacu pada konsep artistik. sedangkan dalampembelajaran matematika, teselasi meliputi beberapa konsep-konsep matematika yanglebih dalam seperti segi banyak beraturan, segi banyak tidak beraturan, kekongruenan,sudut dalam, jumlah sudut dalam suatu segi banyak, simetri, translasi, refleksi, danrotasi. prinsip teselasi tersebut banyak diterapkan dalam kehidupan sehari-hari, sepertipada teknik pemasangan ubin, pembuatan motif kain, desain pola wallpaper dan lainlain(depdiknas [3]). bahkan di alam pun bisa ditemukan contoh teselasi yang terjadi secara alami, yaitu pada sarang lebah. bangun-bangun geometri yang bisa menteselasi contohnya persegi, segitiga, segi lima beraturan, segi enam beraturan dan bisa juga berupa kurva. beberapa definisi terkait teselasi diberikan sebagai berikut: 1. regular tesselation “such a tesselation, made up of congruent regular polygons of one type, all meeting edge to edge and vertex to vertex is called a regular tesselation” (o’daffer, [4]:677). hanya ada tiga poligon beraturan yang dapat menteselasi bidang datar yaitusegitiga, persegi, dan segienam beraturan. 2. semiregular tesselation “a tesselation formed by two or more regular polygons with the arrangement of polygons at each vertex the same is called a semiregular tesselation” (o’daffer,[4]:677). dua hal penting yang dimiliki oleh semi regular tesselation adalah teselasi inidibentuk oleh poligon-poligon beraturan dan setiap puncak pada pertemuan polygon-poligonini adalah sama. 3. a demi regular tesselation “a demi regular tesselation is a tessellations of regular polygons that has exactly two or three different polygon arrangements about its vertices” (o’daffer, [4]:688). dari uraian di atas, dapat dikatakan bahwa beberapa kerajinan anyaman rajapolah terkandung unsur matematika salah satunya adalah penggunaan prinsip teselasi atau pengubinan. karena mengandung unsur matematika maka hasil-hasil kerajinan anyaman ini dapat dimanfaatkan dalam pembelajaran dikelas terutama sebagai sumber belajar atau menghasilkan suatu model atau metode pembelajaran berbasis etnomatematika kerajinan anyaman rajapolah. iii. kesimpulan dan saran etnomatematika pada kerajinan anyaman rajapolah dapat dimanfaatkan sebagai sumber belajar dalam pembelajaran matematika, menambah wawasan siswa mengenai keberadaan matematika yang ada pada salah satu unsur budaya yang mereka miliki, meningkatkan motivasi dalam belajar serta memfasilitasi siswa dalam mengaitkan konsep-konsep yang dipelajari dengan situasi dunia nyata. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 31 daftar pustaka ascher&ascher. (1997). ethnomathematics. dalam a.b. powell& m. frankenstein (penyunting), ethnomathematics challenging eurocentrism in mathematics education (hlm 25-50). albany: state university of new york. gerdes, p. (1996). ethnomathematics and mathematics education. dalam international handbook of mathematical education (hlm 909-943). dordrecht: kluwer academic publiser. karnilah, n. (2013). study ethnomathematics: penanggalan sistem bilangan masyarakat adat baduy. (skripsi). universitas pendidikan indonesia. bandung. o’daffer, phares g. 2008. mathematics for elementary school teachers. fourth edition. pearson education. peard, r. (1996). ethnomathematics. dalam review of mathematics in australia 1992-1995 bill atweh, ed. (hlm. 41-49). washington, d.c : eric clearinghouse. rosa, m. & orey d. (2009). ethnomathematics: the cultural aspect of mathematics. revista latinoamericana de etnomatematca, 4(2) hlm32-54. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p229-238 229 the effect of mathematical disposition and learning motivation on problem solving: an analysis masta hutajulu *1 , tommy tanu wijaya 2 , wahyu hidayat 3 1,3 institut keguruan dan ilmu pendidikan siliwangi 2 guangxi normal university article info abstract article history: received july 5, 2019 revised sept 21, 2019 accepted sept 25, 2019 this research was motivated by the low problem solving abilities, mathematical disposition and learning motivation of junior students. this study aims to find and analyze empirically the influence of mathematical dispositions and learning motivation on problem solving abilities. samples were obtained in class vii-2 at smpn 2 cimahi as many as 34 students. study uses correlational quantitative methods. analysis was done by regression method. the data collection was given 2 pieces of study instruments, namely problem solving test instruments and non-test questionnaires in a set of mathematical dispositions and learning motivation. the data was tested for regression and correlation. the results of data analysis that was mathematical disposition and larning motivation were significantly influenced by problem solving ability of junior students, with the regression equation ̂ indicating a positive influence, and the degree of closeness is the pearson correlation coefficient of 0.827 classified in a strong positive interpretation. together, mathematical disposition variables and learning motivation variables can determine the problem solving variable by 68.3%. recommendations from this study, teachers should design learning processes that can improve mathematical disposition and student motivation so that students' problem solving abilities increase. keywords: learning motivation, mathematical disposition, problem solving copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: masta hutajulu, departement of mathematics education, institut keguruan dan ilmu pendidikan siliwangi, jl.terusan jenderal sudirman, cimahi, west java, indonesia. email: masthahutajulu@yahoo.com how to cite: hutajulu, m., wijaya, t. t., & hidayat, w. (2019). the effect of mathematical disposition and learning motivation on problem solving: an analysis. infinity, 8(2), 229-238. 1. introduction problem solving is one of the abilities that need to be owned and developed by every student at the secondary education level. the rationale of the statement includes mathematical problem solving was the ability listed in the curriculum and objectives of the 2006 ktsp mathematics learning and 2013 mathematics curriculum which states that the purpose of mathematics learning was to solve problems that includes the ability to mailto:masthahutajulu@yahoo.com hutajulu, wijaya, & hidayat, the effect of mathematical disposition and learning motivation … 230 understand problems, design mathematical models, solve model, and interpret the solutions obtained (hendriana, rohaeti, & sumarmo, 2017). cooney (hendriana & sumarmo, 2014) suggests that ownership of problem solving ability helps students think analytically in making decisions in daily life and helps improve the ability to think critically in dealing with new situations. thus the mathematical problem solving abilities are very important for students. the importance of ownership of problem solving abilities is reflected in the quote branca (1980) which states that mathematical problem solving is one of the important goals in learning mathematics even the mathematical problem solving process is the heart of mathematics. polya (2004) developed a model, procedure, or heuristic problem solving consisting of steps to solve a problem, namely (1) understanding the problem; (2) devising a plan; (3) carrying out the plan; and (4) looking back. understanding problems refers to identifying facts, concepts, or information needed to solve problems. devising a plan refers to the preparation of mathematical models of known problems. carrying out the plan refers to the completion of the mathematical model that has been compiled. while looking back it relates to examining the suitability or correctness of answers. the stages of problem solving proposed by polya (2004) can be seen as aspects that need to be considered in evaluating problem solving abilities. in other words, mathematical problem solving abilities include the ability to understand problems, make a problem solving plan, implement a problem-solving plan, and examine solutions. based on the description above it is clear that the mathematical problem solving abilities of students need to get attention to be developed. mathematical problem solving ability required to learn and mathematics it self. therefore solving mathematical problems is very important in learning mathematics because it can facilitate students in facing problems in the lives of students today and in the days to come. but the reality in the field shows that the problem solving ability is still low. this is in line with research assesments program for international students (pisa) 2015 (sjøberg, 2015) indicates that student math ability indonesia ranked at 63 out of 71 countries with a score obtained is 386. furthermore, the research results trends in international the 2015 mathematic and science study (timss) shows that ability indonesian mathematics students ranked 44th out of 49 countries with a score obtained is 397. thus, it can be known that mathematical abilities indonesian students are in the low category so that the impact was also wrong one mathematical ability that was a low mathematical problem solving ability (visser, juan, & feza, 2015). many factors can affect solving ability mathematical problems, one of which is the positive attitude of students towards mathematics or mathematical disposition. disposition in a mathematical context relating to how students solve mathematical problems, whether confident, diligent, interested, and thinking flexible to explore various alternative problem solving. disposition mathematically related to how students ask, answer questions, communicate mathematical ideas, work in groups, and solve math problems. disposition mathematically according to nctm (1989) as tendency to think and act positively. in line with nctm, sumarmo (2010) argues that mathematical disposition is desire, awareness, tendency and strong dedication to students to think and do things mathematically in a way that is positive. kilpatrick, swafford, & findell (2001), mathematical disposition is the tendency to view mathematics as something that can be understood, feel mathematics as something useful, believe in diligent effort and resilient in learning mathematics will produce results, do act as an effective learner and the perpetrator of mathematics own. polking (hendriana & sumarmo, 2014), stated that disposition of mathematics shows: (1) confidence in using math, solve problems, give reasons and communicate idea; 2) flexibility in investigating volume 8, no 2, september 2019, pp. 229-238 231 mathematical ideas and trying to find alternative in solving problem; (3) diligently working on the task mathematics; (4) interest, curiosity, and inner meeting power do math assignments; (5) tend to monitor, reflect their performance and reasoning own; (6) assess the application of mathematics to other situations in mathematics and daily experience; (7) appreciation the role of mathematics in culture and value, mathematics as a tool, and as a language. based on explanation of mathematical dispositions at above, it can be concluded that mathematical disposition is a tendency strong for students to get carry out various activities mathematics so that it can complete mathematical problems effectively and efficient. in addition to mathematical dispositions, factors that can influence the ability to solve mathematical problems are learning motivation. this is in line with the opinion of rakhmat (2007) "one of the factors that influence problem solving is motivation". this means that motivation greatly affects the problem solving process. in solving mathematical problems, learning motivation is important elements that must be possessed by students, students who have high learning motivation will be diligent in doing tasks, resilient and never give up in solving various problems and obstacles, interested in the learning process, thinking about solving problems especially those related to mathematical problems. from the opinions above, it can be concluded that learning motivation was thought to improve problem solving abilities. this the authors suspect because of the sliced characteristics (indicators) between learning motivation with the characteristics of problem solving. increasing motivation in solving problems can be through increasing learning motivation and increasing motivation will produce accuracy in problem solving. according to uno (2010) learning motivation indicators can be classified as the following: (a) successful desires and desires; (b) encouragement and needs in study; (c) future hopes and ideals; (d) deep awards learn; (e) interesting activities in learning; and (f) a conducive learning environment, allowing students to learn with well. based on the description above, we can pay attention to the relationship between motivation and mathematical disposition towards problem solving abilities, but this needs to be studied through a study. therefore, the purpose of this research were to find out and examine the correlation of mathematical disposition motivation and learning motivation on problem solving abilities of yunior students. 2. method this study uses correlational quantitative methods. this study includes two independent variables namely mathematical disposition and learning motivation. and one dependent variable is students'problem solving abilities. the independent variable in this study was mathematical disposition and learning motivation, while the dependent variable was problem solving ability. the population of this research all students vii grade at smpn 2 cimahi, to obtain a representative sample the sample been randomly derived class vii-2 at smp 2 cimahi as many as 34 students. the data collection for each student was given 2 pieces of study instruments, namely problem solving test instruments and non-test questionnaires in a set of mathematical dispositions and learning motivation. the problem solving ability instrument in a set of 5 description questions, while the learning motivation questionnaire in a set of 42 statements with 21 positive questions and 21 negative questions and mathematical disposition questionnaire in a set of 44 statements with 22 positive questions and 22 negative questions. data obtained from the questionnaire in the form of ordinal data, is changed by using method successive interval (msi), to become interval data. hutajulu, wijaya, & hidayat, the effect of mathematical disposition and learning motivation … 232 data analysis is quantitative or statistical by using a correlation test that aims to see whether or not there is a relationship between variables. data was collected then processed using multiple linear regression tests and correlation tests, but before the requirements test was carried out as fulfillment of the assumptions needed in multiple regression analysis on things that are very important in practical terms. because the data obtained from tests of problem-solving abilities and questionnaires from disposition and learning motivation have become interval data then test requirements referred to are test (1) normality for the dependent variable, and error, (2) test linearity requirements, (3) multicollinearity requirements test, (4) heteroscedasticity requirements test. based on the method described above, this study uses multiple regression analysis, and various requirements tests including matching tuna tests were also carried out before further analysis. further analysis is carried out by applying multiple regression equations ̂ which is harmonized with research data with various requirements that theoretically have been described in on. the instrument is given in the form of test and non-test, test of problem solving ability, with the indicator identifying the adequacy of the data for trouble shooting, make a mathematical model of situations or everyday problems and complete it, select and implement strategies to solve in mathematic and or outside mathematic, explain or interpret the results according to the origin problem, and the truth of the results or answers. in figure 1, table 1 and table 2 one of the test and non-test instruments (attitude scale) will be presented with mathematical dispositions and learning motivation used in this research: figure 1. one of test instrument for problem solving abilities used in research the following in table 1 presents number of non-test statements mathematical disposition used in research: table 1. instrument lattices of mathematical disposition questionnaire indicator statement responses degree of belief overcomes learning difficulties sa a da sda confidence i feel able to completematerial mathematics assignments around and square area. (+) flexible and try various alternatives in solving problems i feel happy to solve the circumference and square area problems in various different ways. (+) volume 8, no 2, september 2019, pp. 229-238 233 indicator statement responses degree of belief overcomes learning difficulties sa a da sda diligently working on mathematical tasks i study mathematics, when i want to take an exam. (-) interest and curiosity i am afraid to ask the teacher about the material around and the square area that i have not mastered (-) monitor and reflect on performance / learning mathematics during learning i think a lot of other things and don't really listen to what is being discussed in class. (-) assess math applications i study mathematics around and around quadrilateral, useful in solving problems in everyday life. (+) award for the role of mathematics the math lesson is not difficult, provided we diligently study it. (+) description: sa: strongly agree a: agree da: disagree sda: strongly disagree the following in table 2 presents a number of non-test statements motivation to learn are used in research: table 2. instrument lattices of learning motivation questionnaire indicator statement response degree of belief overcomes learning difficulties sa a ds sda the desire and desire succeed i se and completing mathematical tasks circumference and area of quadrilateral material. (+) i always give up when faced with a difficult problem . (-) there is encouragement and need for learning. i always study the material to be learned in class (+) first i never forget the lesson the teacher has conveyed (-) there are hopes or aspirations for the future i feel learning material around and the width of a triangle is useful in everyday life. (+) there is appreciation in learning. i always get low scores on material around the area and square area. (-) there are interesting activities in learning at the time of learning i always keep quiet and copy answers from friends. (-) the existence of a conducive learning environment, allowing students to learn well. i studied in a clean and comfortable classroom. (+) description: sa: strongly agree a: agree da: disagree sda: strongly disagree 3. results and discussion based on the research results, obtained recapitulation of the achievements of students problem solving ability presented in table 3. hutajulu, wijaya, & hidayat, the effect of mathematical disposition and learning motivation … 234 table 3. recapitulation of the average achievement of problem solving ability problem solving indicator average (%) category identify the adequacy of data for problem solving. 65 enough make a mathematical model of everyday situations or problems and solve them 47 less choose and apply strategies to solve mathematical problems and / or outside mathematics 36 less explain or interpret the results according to the origin problem, as well as the truth of the results or answers. 57 enough the problem solving instrument consists of 5 questions with 4 problem solving indicators. based on table 3, the average student can master the completion of the indicator identifying the adequacy of the data for problem solving. 65 % and is the indicator that has the biggest presentation among other indicators. but students still lack the mastery of indicators making mathematical models of everyday situations or problems and completing them also on indicators choosing and applying strategies to solve mathematical problems and / or outside mathematics, with a percentage of 47% and 57%. in addition, 57 % of the indicators explained or interpreted the results according to the original problem and the correct results or answers were sufficiently mastered by the students. the following is presented a table of mathematical disposition results questionnaires with a total of 44 statements consisting of 22 positive statements and 22 negative statements. the results of the learning motivation questionnaire with a total of 42 statements consisting of 21 positive statements and 21 negative statements are as follows (table 4). table 4. results of mathematical disposition and learning motivation questionnaire aspect category total score average (%) mathematical disposition positive statement 1833 59.13 negative statement 1599 51.58 motivation learning positive statement 1795 56.09 negative statement 1561 48.78 table 4 show that mathematical dispositions of sample presentations in answering positive and negative questions, greater presentation of positive questions which is equal to 59.13 % while presentation of the negative questions is 51.58 % from the explanation above, meaning students already have mathematical dispositions although very little difference with students who not yet have a mathematical disposition . furthermore, for the motivation to learn sample presentation in answering positive and negative questions, a greater positive question presentation is equal to 56.09 % while the negative question presentation is 48.78 % from the explanation above, which means that students already have motivation to learn even though there are very few differences with students who do not have learning motivation. volume 8, no 2, september 2019, pp. 229-238 235 after testing various requirements including the match test, further analysis is carried out by applying the multiple regression equation. if the regression will be done by applying multiple regression equations, namely: ̂ . here are presented in table 5, the recapitulation of multiple regression test using spss. table 5. recapitulation of the results of multiple regression tests between mathematical dispositions and learning motivation with problem solving abilities model sum of squares df mean square f sig. 1 regression 51.169 2 25.584 31.298 0.000 b residual 23.706 29 0.817 total 74.875 31 a. dependent variable: score_pm b. predictors: (constant), motivation score, disposing_ score based on table 5, obtained by the sig = 0.000 (<0.05), it can be concluded there are significant mathematical disposition and motivation to learn about problem solving abilities signifikan. a calculation is provided to determine the regression equation (table 6). table 6. multiple regression equation model unstandardized coefficients standardized coefficients t sig. b std. error beta 1 (constant) 1.950 2017 0.967 0.342 disposal_ score 0.121 0.35 0.751 3.481 0.002 motivation score 0.15 0.037 0.86 0.397 0.694 a. dependent variable: score_pm based on table 6, the obtained constant value is 1.95 while the regression coefficient efficient value is 0.121 for mathematical dispositions and 0.015 for learning motivation, so the double regression equation can be made, namely: ̂ , coefficient values are both positive motivation and disposition can be interpreted that the motive a mathematical disposition of the study and a positive influence on solving mathematical ability. to analyze how closely the relationship between mathematical disposition and learning motivation towards problem solving skills , the pearson correlation coefficient values will be determined as shown in table 7. hutajulu, wijaya, & hidayat, the effect of mathematical disposition and learning motivation … 236 table 7. correlation between mathematical dispositions and motivation to learn with problem solving skills motivation score disposal_ score _pm score motivation score pearson correlation 1 0.875 ** 0.742 ** sig. (2-tailed) 0.000 0.000 n 32 32 32 disposal_ score pearson correlation 0.875 ** 1 0.826 ** sig. (2-tailed) 0.000 0.000 n 32 32 32 _pm score pearson correlation 0.742 ** 0.826 ** 1 sig. (2-tailed) 0.000 0.000 n 32 32 32 **. correlation is significant at 0.01 level (2-tailed). based on table 7, pearson correlation coefficients were obtained between mathematical dispositions and problem solving abilities namely 0.875 and pearson correlation coefficients betweenmotivation to learn and problem solving abilities, namely 0.742 this shows that the relationship between mathematical disposition and learning motivation with problem solving abilities is in a very strong classification. positive correlation coefficient shows that between mathematical disposition and learning motivation with problem solving abilities have a positive relationship, meaning that the higher the mathematical disposition and motivation to learn , the greater the ability to solve problems. next to see the correlation between mathematical disposition and motivation to learn together with problem solving skills can be seen in table 8. table 8. correlation between mathematical dispositions and motivation to learn together with problem solving skills model r r square adjusted r square std. error of the estimate 1 0.827 a 0.683 0.662 0.904 a. predictors: (constant), score_disposition, score_motivation based on table 8, the correlation coefficient between mathematical disposition and learning motivation together with problem solving ability is 0.827, meaning that the higher the mathematical disposition and motivation to learn , the greater the ability to solve problems. in table 8 it can also be seen that the determination value of the correlation coefficient is 68.3 %, this can be interpreted that the mathematical disposition and motivation to learn together affect the problem solving ability by 68.3% while the remaining 31.7% is influenced by factors other than mathematical disposition and learning motivation. based on the results of data analysis it was concluded that mathematical disposition and student learning motivation had a significant influence on problem solving abilities, it could see from the double regression equation: ̂ . this is in line with the research of darmawati (2017), fadila, septiana, amelia, & wahyuni (2019), kusmaryono, suyitno, dwijanto, & dwidayati (2019) and taiyeb & mukhlisa (2015), that there were a relationship between mathematical disposition, learning motivation and learning outcomes. likewise, it is also in line with the research of huda (2016) and ningsih & rohana (2016) that mathematical dispositions and student learning motivation are mainly students' activeness in aspects: working together to solve problems, volume 8, no 2, september 2019, pp. 229-238 237 giving opinions when there are group friends who have not understood, resolved dissent, and communicated with friends and teachers during the learning process in the classroom has increased. therefore teachers need to instill students' learning motivation in each lesson (suprihatin, 2015). to embed a mathematical disposition and students' motivation is high, then the teacher needs to create a fun learning environment, enable and develop selfconfidence and always provide good motivation (hidayat & sariningsih, 2018; subaidi, 2016). according to cleopatra (2015) and rosyana, afrilianto, & senjayawati (2018), learning in a structured and meaningful manner can also increase learning motivation. with high confidence and motivation, the students' ability to convey ideas or mathematical ideas will be better. 4. conclusion based on the results and discussion, the conclusion of this study is that problem solving abilities are influenced by mathematical dispositions and learning motivation. as well as mathematical disposition and learning motivation both individually and jointly have a positive effect on students' problem solving abilities, meaning that the higher the mathematical disposition and learning motivationstudents, the higher the problem solving ability of students, the further the correlation coefficient is classified into a very strong classification. based on the results of this study also, the authors recommend that the level of problem solving abilities of students is influenced by mathematical disposition factors and learning motivation, so as to improve mathematical disposition and motivation to learn students also need to pay attention to learning that must be designed as well as possible. acknowledgements place furthermore, we would like to thank the ikip siliwangi who has given full support so that this paper can be realized. also, we thank to smpn 2 cimahi, the place of research. references branca, n. a. 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(2007). psikologi komunikasi. edisi revisi, cetakan ke-24. bandung: pt. remaja rosdakarya offset. rosyana, t., afrilianto, m., & senjayawati, e. (2018). the strategy of formulate-sharelisten-create to improve vocational high school students’mathematical problem posing ability and mathematical disposition on probability concept. infinity journal, 7(1), 1–6. sjøberg, s. (2015). oecd, pisa, and globalization: the influence of the international assessment regime. in education policy perils (pp. 114–145). routledge. subaidi, a. (2016). self-efficacy siswa dalam pemecahan masalah matematika. sigma, 1(2), 64–68. sumarmo, u. (2010). berpikir dan disposisi matematik: apa, mengapa, dan bagaimana dikembangkan pada peserta didik. bandung: fpmipa upi. suprihatin, s. (2015). upaya guru dalam meningkatkan motivasi belajar siswa. jurnal pendidikan ekonomi um metro, 3(1), 73–82. taiyeb, a. m., & mukhlisa, n. 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(2015). home and school resources as predictors of mathematics performance in south africa. south african journal of education, 35(1), 1-10. http://jurnal.unswagati.ac.id/index.php/jnpm/article/view/1027 http://jurnal.unswagati.ac.id/index.php/jnpm/article/view/1027 http://jurnal.unswagati.ac.id/index.php/jnpm/article/view/1027 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/187 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/187 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/187 https://eric.ed.gov/?id=ej1201186 https://eric.ed.gov/?id=ej1201186 https://eric.ed.gov/?id=ej1201186 https://eric.ed.gov/?id=ej1201186 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/214 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/214 https://books.google.co.id/books?hl=id&lr=&id=z_hsbu9kyqqc&oi=fnd&pg=pp2&dq=polya,+g.+(2004).+how+to+solve+it:+a+new+aspect+of+mathematical+method.+princeton+university+press.&ots=oypgpnkyt4&sig=w0zq2_f-ee9_hvospsq6sqiejt0&redir_esc=y#v=onepage&q=polya%2c%20g.%20(2004).%20how%20to%20solve%20it%3a%20a%20new%20aspect%20of%20mathematical%20method.%20princeton%20university%20press.&f=false https://books.google.co.id/books?hl=id&lr=&id=z_hsbu9kyqqc&oi=fnd&pg=pp2&dq=polya,+g.+(2004).+how+to+solve+it:+a+new+aspect+of+mathematical+method.+princeton+university+press.&ots=oypgpnkyt4&sig=w0zq2_f-ee9_hvospsq6sqiejt0&redir_esc=y#v=onepage&q=polya%2c%20g.%20(2004).%20how%20to%20solve%20it%3a%20a%20new%20aspect%20of%20mathematical%20method.%20princeton%20university%20press.&f=false http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/286 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/286 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/286 http://e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/286 https://www.taylorfrancis.com/books/e/9781315708713/chapters/10.4324/9781315708713-14 https://www.taylorfrancis.com/books/e/9781315708713/chapters/10.4324/9781315708713-14 ejournal.unira.ac.id/index.php/jurnal_sigma/article/view/68 ejournal.unira.ac.id/index.php/jurnal_sigma/article/view/68 http://ojs.fkip.ummetro.ac.id/index.php/ekonomi/article/view/144 http://ojs.fkip.ummetro.ac.id/index.php/ekonomi/article/view/144 https://ojs.unm.ac.id/bionature/article/view/1563 https://ojs.unm.ac.id/bionature/article/view/1563 https://ojs.unm.ac.id/bionature/article/view/1563 https://www.ajol.info/index.php/saje/article/view/113801 https://www.ajol.info/index.php/saje/article/view/113801 https://www.ajol.info/index.php/saje/article/view/113801 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p75-86 75 student readiness and challenge in completing higher order thinking skill test type for mathematics kusaeri* 1 , asep saepul hamdani 2 , suprananto 3 1,2 universitas islam negeri sunan ampel surabaya 3 balitbang kementerian pendidikan dan kebudayaan republik indonesia article info abstract article history: received jan 21, 2019 revised feb 13, 2019 accepted feb 28, 2019 mathematics teaching and learning should equip students with the ability to think critically and creatively and problem-solving skill to enable them to compete in the industrial revolution 4.0. this descriptive explorative study explores the readiness and challenges faced by indonesian junior high school (smp/mts) student in solving hots type mathematics test using islamic context. data were collected from a test-based open-ended questionnaire distributed to 164 students in the 8th of five schools in sidoarjo east java comprising 2 junior madrasas (islamic schools), 2 islamic-based junior high schools and 1 public junior high school. descriptive analysis using spss and categorization of responses (referring to response options in the questionnaire such as have ever-never, necessary-unnecessary, answered-not answered, etc) show that considering their experience, the students are “less ready” to face hots type mathematics test. however, students expect that they can be given more practices in doing hots type mathematics questions to help them get accustomed to this type of question. being unaccustomed to nonroutine and lengthy questions, lazy trait and being unfamiliar to islamic terms used as the context of the question becomes the challenges for the students in facing hots type mathematics test. keywords: mathematics hots student readiness student challenge copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: kusaeri, departement of mathematics education, universitas islam negeri sunan ampel surabaya, jl. jl. ahmad yani no.117, surabaya, jawa timur, indonesia. email: kusaeri@uinsby.ac.id how to cite: kusaeri, k., hamdani, a. s., & suprananto, s. (2019). student readiness and challenge in completing higher order thinking skill test type for mathematics. infinity, 8(1), 75-86. 1. introduction implementation of higher order thinking skill (hots) test type for mathematics in the indonesian national exam 2018 has invited various responses from test takers and has become viral in social media. test takers consider that the hots type mathematics test is too difficult. in fact, the questions in the test are not as complex as what the students have complained. it is just a matter of student not being accustomed to such type of question. this phenomenon of student being not accustomed to hots type question may be rooted from the learning trait and culture at school that has not offered students much to be kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 76 accustomed to hots. in consequence, students are not used to dealing and completing questions or problems in mathematics that require hots (kusaeri, sadieda, indayati & faizien, 2018; nursyahidah & albab, 2017). so far, most mathematics problems provided by mathematics teachers are related to the ability to apply mathematics formulas, procedures or algorithms (dewantara, zulkardi & darmawijoyo, 2015; kusaeri & aditomo, 2019; sembiring, hadi & dolk, 2008). there have not been sufficient mathematics problems that lead and motivate students to hone their analytical, evaluative and creative skills. student unfamiliarity in dealing with and completing hots type of test make it difficult for the student to find correct answer to hots type mathematics questions. this condition can bring two possible consequences. first, students with high level of curiosity will find hots type mathematics test as a challenge. once they can solve a hots mathematics problem, they will be excited and motivated to complete other hots questions. in this perspective, hots question can successfully trigger student interest. on the other hand, students having low interest in mathematics may feel desperate when they have to complete hots mathematics questions. they will tend to choose mathematics questions they commonly face; thus, they are not used to solving hots mathematics questions (kamarullah, 2017). student low skill in completing hots type of test does not only happen in indonesia. studies conducted in nigeria (adeyami, 2012), united states (marin & halpern, 2011) and dubai (taleb & chadwick, 2016) have shown similar findings. such low skill seems to be triggered by student deficiency in identifying information provided in question item, in interpreting and sorting information in the question item, implementing strategy to find answer to the question, connecting inter-related concepts, and using reasoning to solve complex hots type test questions (suryapuspitarini, wardono, kartono, 2018). in general, student weakness in completing hots type test is due to student lack of competence to generalize some information in the test item in order to design new strategies to solve the problems. findings from some previous studies on student weaknesses in solving hots type test have been congruous. abdullah, abidin, & ali (2015) found that student difficulties in solving hots type test lie in ability to connect information and find solution to the problem. finding by gais & afriansyah (2017) shows that lack of meticulousness at the process of finishing the task, low mathematics skill, less optimal learning process, and lack of parental monitoring have been indicated to cause student difficulties in solving hots type test. a research by hadi et al. (2018) also finds that student unfamiliarity with hots type test, student low interest in tackling hots problems and student indifference toward lengthy question bring student difficulty in completing hots type test. the aforementioned previous studies have extensively explored student difficulties in completing hots type mathematics test but have not investigated the extent to which student can do hots mathematics test when the question items incorporate context they are familiar with in indonesia, such context can be that of islamic context for students in islamic school or madrasa. this is because these two institutions have more islamic subjects than public school; hence, the students in islamic school or madrasa have been familiar with islamic context. thus, the integration of islamic context to hots type mathematics test may facilitate student familiarity with the context and promote student interest to the questions. the integration referred here is the use of real life context (fiqh and ibadah) in the mathematics problem presented to the students. in the context of indonesian madrasa and islamic school, research by kurniati (2015) and kusaeri, sadieda, indayati & faizien (2018) find that the integration of islamic context to hots type mathematics test has made mathematics more interesting subject volume 8, no 1, february 2019, pp. 75-86 77 because student feels that mathematics has become part of life and its diversity. it can also bring new perspective to the student that it is not hard to relate mathematics to islamic context. the integration of mathematics and islamic context can develop student reasoning in order to build critical awareness about the ultimate truth based on islamic value and learning. as integration of specific context to hots type mathematics test has not yet gained particular attention in the existing theories and previous studies, research that explores student readiness and challenge in solving hots type mathematics test is deemed to be important. in indonesian context, studies about madrasa and islamic school students, particularly those that are related to their ability to complete hots type mathematics questions have not received much attention. this research aims to identify madrasa tsanawiyah (islamic junior high) and islamic school student readiness and challenge in completing hots type mathematics test that integrates islamic context.hence, the questions in this research are: (a) how is the readiness of students (madrasah tsanawiyah, islamic school and general junior high school) to solve hots type omathematics problems that integrate islamic context? and (b) what are the challenges faced by students in solving hots type mathematics problems that integrate islamic context?. 2. method this research is descriptive explorative in that it explores the students readiness and challenges in solving hots type mathematics problems. the results of such an exploration are then described in tables of frequency distribution and percentages. data were collected through test-based open-ended questionnaire, that is a inquiry technique in which participants filled out a questionnaire after taking hots type mathematics test that integrates islamic context. this data inquiry technique has enabled collection of rich and varied data as students were given freedom to respond and express their ideas. so, student readiness and difficulties in completing hots type mathematics test can be better portrayed. 2.1. participants the participants of this research are 164 students in the 8th grade of 5 junior high school (smp/ mts) spread in sidoarjo, east java selected based on the representation of educational institutions’ type, primarily affiliated to islamic school. these five schools /madrasa are smp al-muslim sidoarjo, smp progressive bumi sholawat sidoarjo, mtsn 1 sidoarjo, mtsn tlasih sidoarjo and smpn 5 sidoarjo, each is represented by one class as participants based on their headmaster consideration. the five selected schools are considered to be representative of excellent schools (madrasah, islamic school and general school) in sidoarjo district. the participants are nearly balanced between the male students (46.34%) and female students (53.66%). data were collected in mid of the second semester of academic year 2018/2019. 2.2. instrument test-based open-ended questionnaire was utilized to measure student readiness and challenge in completing hots type mathematics test. two questions aim to explore student readiness are formulated as "how often do you get hots type mathematics test? please elaborate your answer!" and "do you feel the need to be trained in solving hots type mathematics test?" two questions are also asked to explore challenge faced by students: "is it more difficult to complete hots questions that integrate islamic context?" kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 78 and "what challenges you experience in solving hots type mathematics test with islamic context?" to strengthen the result of both data types from the questionnaire, eight (8) twotier multiple choice test using islamic context (fiqh and prayer) was used. the test was also completed with reasoning. level one of the test contains questions with four (4) possible answers, while the second level contains reasons which must be filled by students in reference to their corresponding answer choice. by this method, student readiness, challenge and understanding (either complete or incomplete) to hots type mathematics test can be investigated (treagust, 2006). the piloting of the test items show that the items have distinguishing factors (a) difficulty level of questions (b) and guessing factor (c) as follow: 0.55 <a <1.16; 0.00 <b <1.40; and 0.16 <c<0.29. referring to baker (2001), these eight items are in the category of good item. 2.3. analysis student readiness and challenge collected through questionnaire were analyzed descriptively using spss. similar response pattern were grouped into one category. the two-tier multiple-choice data were analyzed by these following steps: (a) recording/summarizing results of multiple-choice tests with reasoning from all students, and (b) constructing and classifying the reasoning answered by students as a second tier choice. 3. results and discussion this section is organized into 2 (two) sub-sections: (a) student readiness in completing hots type mathematics test, and (b) student chalenge in completing hots type mathematics test. 3.1. student readiness in facing hots type mathematics test data from the questionnaire presented in table 1 indicate that the majority of students (63.1%) stated that they had never received hots type mathematics test at school. this indicates that teachers do not generally give exercises or mathematics tests that require higher-level thinking. in fact, if teachers provide more varieties of this type of test/question, the students will have better ability to explore their reasoning when they face a variety of other questions. they will also used to honing their ability to understand, analyze problems, create a math model, and find solution/answer. table 1. student readiness to solve hots type mathematics test aspects of student’s learning readiness responses number of respondents percentage how often do you get hots type mathematics test? please elaborate your answer. have ever: 61 36.9% once 26 15.8% twice 9 5.3% sometimes 20 12.3% at the mathematics olympics 6 3.5% never 103 63.1% volume 8, no 1, february 2019, pp. 75-86 79 aspects of student’s learning readiness responses number of respondents percentage do you feel the need to be trained in solving hots type mathematics test? necessary/needed: 138 84.2% training high-level thinking skills 43 26.3% requiring particular understanding 37 22.8% preparing for a mathematics test in national exam 12 7% presenting exciting and challenging problem 14 8.8% preparing questions to face the next hots type of test 32 19.3% unnecessary/not needed: 26 15.8% dividing focus on other subjects 9 5.3% adding burden of thought 12 7% being not needed at national exam 5 3.1% in contrast, students who lack experience in dealing with hots type mathematics test will find it difficult because they do not know enough patterns or steps needed to solve the problem/question. evidently, students who answered 'never' in facing hots type mathematics test, based on the answer of multiple choice with reasoning questions, were not yet able to solve questions optimally. however, there is an interesting fact and worth highlighting that students from public junior were in fact gave better results than junior students from islamic school or madrasa (mts). a good indicator can be seen from their more coherent thinking process starting from analyzing problems, making mathematics model and solving problem. this is different from students from mts who mostly have not been able to solve problems in coherent steps and to find the correct answer. a similar case was found from junior high students in islamic school who prefer to answer randomly without calculating or analyzing the question first. this fact is interesting to observe as students from public junior high also feel they have not been familiar with hots mathematics test in islamic context, but in fact they can still perform well. it is certainly influenced by the learning process that occurs in public junior high which is more conducive in giving mathematics problems that demand high reasoning (kusaeri, aditomo, ridho & fuad, 2018). so they are more familiar with tests or questions that require reasoning skills. student familiarity to questions that require higherlevel thinking, according to lee & francis (2018), can make the student ready to solve various forms of mathematics problem. despite the surprising results above, students have expectations to be trained to be able to tackle hots type mathematics test. this is indicated by the larger percentage of respondents who expressed 'necessary/needed' much larger (84.2%) compared to 'unnecessary/not needed' (see table 1). they state that by training of dealing with hots kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 80 type mathematics test will improve their thinking skill. the presumption evoked from their uncertainty in solving hots questions given at the beginning of the research. in addition to the above reason, some of them also acknowledged that solving hots type mathematics test require certain skills, certain comprehension, even a specific strategy (nadapdap & istiyono, 2017). thus, they think it is necessary to have a particular training in order to deal with and solve hots type mathematics test. preparation for the national examination (un) is also another reason considered by students, hence they are ready to be trained in dealing with hots type mathematics test. in this way, they hope that they can successfully get optimal un mathematics score. another factors contributing to the students need to get training is because they really enjoy the questions presented. some students become interested to learn hots mathematics. further exploration of the data also show that students who like challenge and problem complexity are more willing to participate in training. students who assume such training as 'not needed/unnecessary' (table 1) view that mathematics is a difficult subject to understand. not surprisingly, 7% of students expressed that training was not necessary because it will add burden to their minds. regular and routine mathematics question they encounter are considered to be difficult already. the one that integrates the islamic context is considered to give them more burden. this is because such question will involve all levels of thinking from bloom's taxonomy (kusaeri, 2014). it can be concluded from the above data that student readiness in dealing with hots type mathematics test is low when it is referred to student experience. however, they have very good readiness in dealing with hots type mathematics test when viewed from the side of willingness to accept training. 3.2. student challenges in facing hots type mathematics test the results of the two-tier multiple-choice with reasoning test analysis, portray challenge faced by students when solving hots type mathematics test. the analysis was based on the number of students who answered and who did not answer each item. then, the answers were categorized and put into percentages based on the correct and incorrect answers. table 2 presents the result of analysis. table 2. analysis of student answer of multiple choice with reasoning test problem no answer sheet analysis results frequency percentage 1 answered: 73 44.5% true 25 15.2% false 48 29.3% not answered 91 55.5% 2 answered: 113 68.9% true 78 47.6% false 35 21.3% not answered 51 31.1% 3 answered: 130 79.3% true 51 31.1% false 79 48.2% not answered 34 20.7% volume 8, no 1, february 2019, pp. 75-86 81 problem no answer sheet analysis results frequency percentage 4 answered: 47 28.7% true 13 7.9% false 34 20.7% not answered 117 71.3% 5 answered: 128 78.0% true 32 19.5% false 96 58.5% not answered 36 22.7% 6 answered: 108 65.9% true 17 10.4% false 91 55.5% not answered 56 34.1% 7 answered: 46 28.1% true 14 8.5% false 32 19.6% not answered 118 71.9% 8 answered: 51 31.1% true 18 11.0% false 33 20.1% not answered 113 68.9% table 2 informs that number 2, 3, 5 and 6 relatively get more answers from the students although the answers tend to be incorrect. error occurs because students were in a hurry concluding problem given. they assumed those questions were easy; thus, they tended to be sloppy and less focus on what the question actually implies. for theother four question items (number 1, 4, 7 and 8), many students tend notto answer. the context or long narration can be suspected to be their triggers. the students seemed to be not familiar with problems that have long narration. long narration requires in-depth understanding and this can makes students find it difficult to comprehend (laily, 2014). when students face difficulty in understanding questions, consequently the mathematics models they present to explain each sentences or steps can also be wrong (samritin & suryanto, 2016). questions with long narration tend to frustrate students and make them demotivated to complete the questions. table 2 also identifies that students also tend to avoid questions that displays image element, such as numbers 1 and 8. this kind of question, not only requires students to do calculations in order to find a solution, but also to understand the image. if students cannot understand the image presented, they will also find it difficult to answer. this lack of understanding makes it difficult for the students to associate their imagination to what is referred in the question and the image (bokosmaty et al., 2015). some questions were unanswered because of various factors. among others is because the material presented in the question is complex. this finding confirms finding by abdullah et al. (2015) in that students find it difficult when the questions are different from what they normally encounter. it is possible that students have forgotten the material or the various procedures that are commonly used to solve such type of problems. in the end, kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 82 they will ignore this type of questions and choose other questions that are easier to complete. table 3 further shows various difficulties experienced by students when solving hotsmathematics problems. the data show that most students (31.6%) stated that the given hots mathematics problem makes them dizzy and feel confused how to complete. table 3. student challenge based on questionnaire results when solving mathematics routine questions, felt more troubled when doing mathematics hots test. the context of islam used make them think twice, namely thinking about the mathematics models and islamic values contained in the question. in the end, the confusion covers the meaning and purpose hots mathematics problems that are basically used to improve students reasoning skills (dorr, 2017). in addition to creating confusion, research data shows that hots mathematics questions also make it difficult for the students to connect some of information the questions. this type of questions has different characteristics with routine questions that the students usually encounter. no wonder that they feel they need to go through different process. this kind of problem requires the students to be able to analyze problems, explore previous experiences, dig up the materials that have been obtained, to be creative and put a lot of effort (kurniati, harimukti& jamil, 2016). therefore, it is not easy to draw on a wide range of capabilities above to analyze relationship between components in question. despite the above drawbacks of hots type mathematics test with islamic context integration, there is still a positive element of this type of test. the integration of islamic aspect of constraints responses number of respondents percentage is it more difficult to complete hots questions that integrate islamic context? agree: 95 57.9% increasingly create confusion to complete. 52 31.6% spending lengthy time to complete. 11 7.0% more difficult because there is too much information on the questions 14 8.8% not easy to connect some information in the problem 18 10.5% disagree: 69 42.1% mathematics seems to be closer to everyday life and i learn more about islam. 46 28.1% enjoy solving the problem more. 23 14.0% what challenges do you experience in solving hots type mathematics test with islamic context? the test is less familiar 92 56.1% the narration is too long 38 23.2% the time to work on is too tight 12 7.31% i am lazy 7 4.27% the termsusedrelated to islamic context is difficult 15 9.15% volume 8, no 1, february 2019, pp. 75-86 83 values in the mathematics test can bring new perspective to the students. the students who previously thought negatively towards mathematics and perceived that it is hard associate mathematics to everyday life, even in the context of islam, can now get a new insight on the possible relationship between mathematics and islamic values. for the majority of students who come from mts and islamic junior high and do not like mathematics, this can be a breakthrough in the reference about kind of test they get. those who initially did not like mathematics may shifts slightly and can like mathematics more. the impact may be felt indirectly. the students are increasingly able to do mathematics questions because there are links with islamic values. this can change theirparadigm about mathematics and become an appeal forthem to constantly interact with mathematics. these types of problems are not normally met by students and are not monotonous. therefore, this may bring euphoria to the students; they feel the complex thinking in finding solutions to the question. such euphoria makes students enjoy the questions presented and encourage them to continue to complete other question (muis et al., 2015). table 3 also shows that difficulties experienced by most of the students are because they are not accustomed to face this kind of hots mathematics problem. the lack of exercise in non-routine problems duringlearning is suspected to be the trigger. as mentioned earlier by dewantara, zulkardi & darmawijoyo (2015), kusaeri, sadieda, indayati, & faizien (2018), as well as sembiring, hadi, & dolk (2008), problems presented during learning in classroom by mathematics teacher commonly demand more of student's ability to apply formulas, procedures or algorithms. from the descriptionabove, it can be concluded that the stimuli of hots type mathematics test with islamic context make most of the students more confused, because they experience difficulty in connecting information in the problem. lack of exercise during learning and long narration brings difficultiesto the students. 4. conclusion findings from this research show that most of students are 'less ready' to face hots type mathematics test because of lack of experience in dealing with this kind of questions. they expect to be trained and get accustomed to interacting with hots type mathematics test. less readiness in facing non-routine problems, too long narration, laziness and less familiarity with islamic terms are difficulties faced by students in completing hots type mathematics test. these findings suggest the importance of changing the way teachers assign exercises, assignments and exams, to not only take routine questions from the textbooks that require only a low-level thinking. this highlights the need for mathematics teachers to provide students with experience to practice and interact with hots type mathematics problem in their day to day mathematics learning not only in class but also at home as homework and in the formative and summative assessment. for teachers in islamic schools and madrasah tsanawiyah, the hots type of mathematics problem should not be too long in order to avoid student confusion and demotivation to read the questions. this study only collected data on the student challenges from the result of multiple choice test and questionnaire. although these data collection techniques can be considered as appropriate and sufficient, further research using interview as a triangulation is worth doing to provide more robust and comprehensive data. kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 84 acknowledgments this article is part of a research funded by institute for research and community service (lppm) uin sunan ampel surabaya, based on the rector decree no. 187, 2018, april 19, 2018. therefore, a thank note goes to the chair of uin sunan lppm ampel surabaya. the researcher would like to thank principles of the schools and teachers involved in the research (particularly elmita irmanila and m. ikmal faizien). may allah grant them with the best award. references abdullah, a. h., abidin, n. l. z., & ali, m. 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(2006). conceptual change as a viable approach to understanding student learning in science. in teaching and learning science-a handbook (pp. 25-32). praeger publications. kusaeri, hamdani, & suprananto, student readiness and challenge in completing … 86 infinity infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 42 peningkatan kemampuan komunikasi matematik melalui pembelajaran matematika realistik oleh: 1) saleh haji, 2) m. ilham abdullah program studi pendidikan matematika, fkip universitas bengkulu 1 salehhaji25@gmail.com, 2 ilhamabdullah418@gmail.com abstrak tujuan penelitian ini adalah untuk mengetahui pencapaian dan peningkatan kemampuan komunikasi matematik siswa sekolah menengah pertama kota bengkulu melalui pembelajaran matematika realistik. metode penelitian yang digunakan adalah kuasi eksperimen dengan non-equivalent control group design. hasil penelitian sebagai berikut. pembelajaran matematika realistik lebih efektif dalam pencapaian dan peningkatan kemampuan komunikasi matematik siswa dibandingkan pembelajaran konvensional. besarnya pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik adalah 63,96 dan pembelajaran konvensional adalah 47,46. sementara itu, besarnya peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik adalah 0,51 dan pembelajaran konvensional 0,24. kata kunci : komunikasi matematik, pembelajaran matematika realistik abstract the purpose of this study is to determine the achievement and improvement of students’ mathematical communication ability in bengkulu city junior high school through realistic mathematics learning. the method used is a quasi-experimental non-equivalent control group design. the results of this study as follows. the realistic mathematics learning is more effective in the achievement and improvement of students’ mathematical communication ability than conventional learning. the magnitude of the achievement of students’ mathematical communication ability in realistic mathematics learning is 63,96 and conventional learning is 47,46. meanwhile, the magnitude of the improvement of students’ mathematical communication ability is realistic mathematics learning is 0,51 and conventional learning is 0,24. keywords: mathematical communication, realistic mathematics learning i. pendahuluan a. latar belakang kemampuan berkomunikasi dalam pembelajaran matematika di sekolah menengah pertama kurang mendapat perhatian dari para guru. guru cenderung menekankan pada kemampuan berhitung, pemecahan masalah, dan penalaran. sehingga kemampuan komunikasi matematika siswa lemah. siswa kurang dapat mengkomunikasikan ide-ide matematiknya secara jelas dan benar, baik secara lisan maupun tulisan. hasil penelitian rohaeti (2003), wihatma (2004), purniati (2004) menyimpulkan bahwa kemampuan komunikasi matematik siswa sekolah menengah pertama rendah. mailto:salehhaji25@gmail.com infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 43 kemampuan komunikasi matematik merupakan salah satu standar proses dalam pembelajaran matematika di sekolah. van de wall, karp, jennifer, and williams (2000) mengemukakan 5 standar proses dalam pembelajaran matematika yakni: 1. problem solving, 2. penalaran dan bukti, 3. komunikasi, 4. koneksi, dan 5. representasi. standar proses kemampuan komunikasi tersebut memberikan batasan terhadap aspek-aspek komunikasi yang diajarkan pada sekolah. kemampuan komunikasi matematik merupakan kemampuan siswa dalam mengkomunikasikan ide-ide matematik pada orang lain. baroody (1993) menjelaskan bahwa pembelajaran matematik harus dapat membantu siswa mengkomunikasikan ide matematik melalui 5 aspek yaitu: representing, listening, reading, discussing, dan writing. baroody (1993) juga menjelaskan bahwa pentingnya komunikasi dalam pembelajaran matematika, karena mathematics as language dan mathematics learning as social activity. sebagai bahasa, matematika digunakan orang dalam menyampaikan ide dengan menggunkan simbol dan pengertian yang memiliki arti tunggal. matematika yang digunakan di sekolah dalam pembelajaran matematika menghasilkan berbagai aktivitas siswa dan guru dalam mendiskusikan matematik. selain aktivitas di dalam kelas (sekolah), matematik juga digunakan oleh masyarakan dalam aktivitas sosialnya. seperti dalam aktivitas perdagangan,pertanian, pertambangan dan lain-lain. untuk mengatasi kelemahan komunikasi matematik siswa perlu dilakukan perubahan pembelajaran, dari pembelajaran konvensional ke pembelajaran non-konvensional. firdaus (2006) menjelaskan bahwa pembelajaran non-konvensional meningkatkan kemampuan komunikasi matematik siswa. salah satu pembelajaran non-konvensional tersebut adalah pembelajaran matematika realistik (pmr). pembelajaran matematika realistik memandang bahwa matematika sebagai suatu aktivitas manusia. sebagai suatu aktivitas, matematika sebagai suatu sarana yang memungkinkan terjadinya interaksi antar manusia. beberapa hasil penelitian menunjukkan bahwa pmr mampu meningkatkan aktivitas belajar siswa. baik aktivitas fisik maupun mental. karena pmr memuat berbagai kegiatan fisik maupun mental. seperti kegiatan diskusi, refleksi, dan penemuan konsep maupun algoritma. khasanah (2007) menemukan bahwa pendidikan matematika realistik indonesia (pmri) dapat meningkatkan aktivitas siswa dalam pembelajaran keliling dan luas bangun datar. begitu juga, hasil penelitian al muhari (2008) menghasilkan bahwa pmri dapat meningkatkan kreativitas belajar siswa. b. rumusan masalah 1. apakah terdapat peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik? 2. apakah pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik lebih baik daripada pembelajaran konvensional? ii. kajian pustaka a. kemampuan komunikasi matematik secara harfiah, komunikasi memiliki arti pengiriman dan penerimaan berita atau pesan antara dua orang atau lebih (surayin, 2003). ini berarti, dalam komunikasi memuat tiga komponen yaitu isi pesan/berita, proses pengiriman/penerimaan pesan, dan orang yang mengirim/menerima pesan tersebut. dalam konteks pembelajaran matematika, isi pesan/berita infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 44 dalam komunikasi yaitu matematika. suriasumantri, j.s. (2007) mengemukakan bahwa matematika merupakan bahasa yang melambangkan serangkaian makna dari pernyataan yang ingin kita sampaikan. matematika berbentuk simbol yang memiliki makna tertentu. simbol simbol tersebut dapat berupa fakta, konsep, prinsip, maupun algoritma. contoh fakta adalah bilngan π, variabel sebagai suatu konsep, teorema phytagoras sebagai suatu prinsip, dan langkah-langkah penyelesaian operasi hitung dua buah bilangan merupakan algoritma. secara istilah, greenes and schulman (1996) mengemukakan pengertian komunikasi matematik sebagai suatu kemampuan dalam: a. menyatakan ide matematik melalui ucapan, tulisan, demonstrasi, dan melukiskannya secara visual, b. memahami, menafsirkan, dan menilai ide yang disajikan dalam tulisan, lisan, atau visual, c. mengkonstruk, menafsirkan, dan menghubungkan bermacam-macam representasi ide dan hubungannya. berdasarkan pengertian ini, komunikasi matematik lebih detail. isi pesan/berita berbentuk ide matematik. proses pengiriman/penerimaan isi pesan/berita dalam bentuk tulisan, lisan, representasi. national council of teacher of mathematics (2000) menjelaskan bentuk proses komunikasi matematik sebagai: membuat ilustrasi dan interpretasi, berbicara atau berdiskusi, menyimak atau mendengar, menulis, dan membaca. bentuk kegiatan prosesnya adalah menyatakan, memahami, menafsirkan, dan menghubungkan isi pesan (matematika). kemampuan komunikasi matematik merupakan suatu kemampuan dalam mengkomunikasikan matematik. hulukati (2005), menjelaskan komunikasi matematik sebagai suatu kemampuan siswa dalam mengekspresikan, menginterpretasi, mengevaluasi ide-ide dan notasi matematika melalui tulisan, lisan, dan mendemonstrasikannya secara verbal. dari pengertian tersebut, kemampuan komunikasi matematik memiliki karakteristik. sumarmo (2000), menjelaskan karakteristik kemampuan komunikasi sebagai berikut: a. membuat hubungan benda nyata, gambar, dan diagram ke dalam ide matematik. b. menjelaskan ide, situasi dan relasi matematika secara lisan maupun tulisan dengan benda nyata, gambar, grafik, dan gambar. c. menyatakan peristiwa sehari-hari dalam bahasa atau simbol matematika. d. mendengarkan, berdiskusi, dan menulis tentang matematika, membaca dengan pemahaman suatu representasi matematik. e. membuat konjektur, menyusun argumen, merumuskan definisi dan generalisasi. f. menjelaskan dan membuat pertanyaan tentang matematika. van de wall, karp, jennifer, and williams (2000) mengemukakan indikator dari kemampuan komunikasi matematik siswa di sekolah sebagai berikut: a. mengatur dan mengkonsulidasikan pemikiran matematika siswa melalui komunikasi. b. mengkomunikasikan pemikiran matematika siswa secara koheren dan jelas kepada temanteman, guru, dan lain-lain. c. menganalisis dan mengevaluasi pemikiran matematik. d. menggunakan bahasa matematik untuk mengekspresikan ide-ide matematik secara tepat. indikator-indikator tersebut menunjukkan adanya kemampuan komunikasi pada seorang siswa maupun guru. sedangkan national council of teacher of mathematics (2000) menjelaskan indikator kemampuan komunikasi matematik sebagai berikut, yaitu: a. mengungkapkan ide matematik secara tertulis maupun lisan. b. merumuskan definisi dan membuat generalisasi. c. menyajikan matematika dengan pengertian. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 45 d. menjelaskan pertanyaan matematik. e. menghargai daya dan keindahan matematik. menurut hendriana (2009), kegiatan komunikasi tentang matematik berkaitan dengan kegiatan merefleksi proses kognitif, mendeskripsikan prosedur, menggambarkan metakognisi, dan berkomunikasi dengan orang lain tentang matematika. kegiatan komunikasi dalam matematik adalah mencatat secara matematik dan merepresentasikan sesuatu dengan simbol. sedangkan komuniksi dengan matematik sebagai berikut: matematika sebagai alat pemecahan masalah, mencari solusi alternatif, menginterpretasikan argumen, dan menggunakan pemecahan masalah matematik. b. pembelajaran matematika realistik haji dan abdullah (2014) mengemukakan pembelajaran matematika realistik sebagai suatu pola yang sistematis dalam merancang pembelajaran matematika yang efektif untuk mencapai tujuan pembelajaran matematika dengan bertumpu pada kreativitas siswa dalam melakukan doing mathematics yang memandang matematika sebagai suatu aktivitas manusia melalui kegiatan memecahkan masalah kontekstual, merumuskan model, mengkaitkan berbagai topik, berinteraksi dengan berbagai sumber, memanfaatkan berbagai potensi sendiri, berdiskusi, melakukan refleksi, memanfaatkan fenomena pendidikan, mengeksplor, dan akhirnya menemukan (invention) berbagai konsep (prinsip) dan algoritma matematik. dari pengertian tersebut, pembelajaran matematika realistik memiliki karakteristik. treffers (1987) menjelaskan lima karakteristik dari pembelajaran matematika realistik, yaitu: 1. the use of context, 2. the use of models, 3. the use of students’, 4. the interactive character of teaching process, and 5. the intertwinement of varios learning strands. iii. metode penelitian sampel penelitian ini adalah siswa kelas viia smpn 24 kota bengkulu yang terdiri atas 19 siswa. sedangkan populasi penelitian ini adalah siswa kelas vii smpn 24 kota bengkulu yang terdiri atas 85 siswa. instrumen penelitian ini berbentuk tes kemampuan komunikasi matematik sebanyak 3 item. hasil analisis tes kemampuan komunikasi matematik dengan menggunakan software anates sebagai berikut. rata-rata skor = 60,79, standar deviasi = 22,75, korelasi xy = 0,71, reliabilitas = 0,83 (haji dan abdullah, 2014). disain penelitian ini adalah a quasi-experimental design by the non-equivalent control group (cohen, manion and morrison, 2000): experimental o1 x o2 control o3 o4 x adalah perlakuan berupa pembelajaran matematika realistik infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 46 iv. hasil dan pembahasan 1. peningkatan kemampuan komunikasi matematik (kkm) hasil uji hipotesis menunjukkan bahwa terdapat peningkatan kemampun komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik. seperti tampak pada tabel 1. peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realisti sebesar 0,51. sedangkan peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran konvensional sebesar 0,24. peningkatan kemampuan komunikasi matematik siswa lebih besar daripada peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran konvensional. tabel 1. hasil uji – t n-gain kkm siswa kelompok data n ratarata beda rata-rata t df sig. (2-arah) h0 n-gain kkom eksperimen 27 0,51 0,27 3,396 49 0,001 ditolak n-gain kkom kontrol 24 0,24 sumber: haji dan abdullah (2014) peningkatan kemampuan komunikasi matematika siswa yang diajar melalui pembelajaran matematika realistik dikarenakan pembelajaran matematika realistik memotivasi siswa untuk melakukan diskusi terhadap teman-temannya dan guru dalam memecahkan masalah kontekstual maupun dalam menemukan suatu konsep dalam matematika. seperti dalam memecahkan masalah berikut ini, diketahui bilangan-bilangan bulat 786 dan -867. bilangan mana yang lebih besar? mengapa? sebanyak 95% siswa yang diajar melalui pembelajaran matematika realistik menjawab dengan benar permasalahan tersebut. hasil diskusi mereka sebagai berikut: siswa a : 786 lebih besar daripada -867, karena bilangan positif lebih besar daripada bilangan negatif. siswa b : coba kamu tunjukkan bahwa bilangan positif lebih besar daripada bilangan negatif. siswa a : baik, saya akan tunjukkan bahwa bilangan 786 lebih besar daripada -867, dengan menggunakan gambar garis bilangan sebagai berikut: -867 0 786 siswa b : mengapa bilangan -867 letaknya di sebelah kiri bilangan 786 dan diantara ke dua bilangan tersebut terdapat bilangan nol? siswa a : karena bilangan -867 merupakan bilangan negatif, sedangkan bilangan 786 merupakan bilangan positif. bilangan 0 lebih besar daripada bilangan negatif dan lebih kecil daripada bilangan positif. siswa b : oh, kalau begitu saya setuju. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 47 sebanyak 73%, siswa yang diajar melalui pembelajaran konvensional mengalami kesulitan dalam menyelesaikan soal tersebut. sebanyak 62%, jawaban siswa salah. mereka mengatakan bahwa -867 lebih besar daripada 786. karena bilangan 8 pada -867 lebih besar daripada bilangan 7 pada 786. kegiatan diskusi pada pembelajaran konvensional kurang berkembang. umumnya, siswa masing-masing menyelesaikan soal tersebut. 2. pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik hasil uji mann whitney menunjukkan bahwa menerima hipotesis yang menyatakan bahwa terdapat perbedaan pencapaian kemampuan komunikasi matematik siswa yang diajar dengan menggunakan pembelajaran matematika realistik dengan pembelajaran konvensional. pencapain kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik lebih besar daripada siswa yang diajar melalui pembelajaran konvensional. kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik sebesar 63,96, sedangkan siswa yang diajar melalui pembelajaran konvensional sebesar 47,46. hal ini tampak pada tabel 2 berikut ini. tabel 2. hasil uji mann-whitney pretes dan postes kkm siswa kelompok data ratarata u mann whitney z sig. (2-arah) h0 pretes kkmm eksperimen 32,33 221,500 -1,950 0,051 diterima pretes kkmm kontrol 32,00 postes kkmm eksperimen 63,96 149,500 -3,306 0,001 ditolak postes kkmm kontrol 47,46 sumber: haji dan abdullah (2014) kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik lebih cepat daripada siswa yang diajar melalui pembelajaran konvensional. karena pada pembelajaran matematika realistik, siswa melakukan kegiatan refleksi dan penemuan. melalui kegiatan refleksi, siswa memiliki kesempatan untuk memperbaiki kesalahan maupun kekurang tepatan dalam menyelesaikan suatu masalah. kegiatan refleksi tersebut dilakukan siswa dengan meninjau ulang hal yang telah dilakukannya. siswa berkomunikasi dengan dirinya sendiri. begitu pula dengan kegiatan penemuan. siswa melakukan komunikasi dengan teman maupun gurunya. seperti dalam menemukan keliling segitiga. siswa menggunakan seutas tali untuk mengitari benda berbentuk segitiga. sehingga siswa menyimpulkan bahwa keliling suatu segitiga adalah jumlah dari ketiga sisinya. melalui pemahaman seperti ini, sebanyak 77% siswa dapat menjawab dengan benar soal berikut ini. keliling suatu segitiga yang panjang sisinya (a + 6) cm, 28 cm, dan 12 cm adalah 180 cm. berapakan nilai a? jawaban soal tersebut adalah a = 134. salah satu cara dalam menjawab soal tersebut sebagai berikut. ke tiga sisi segitiga digambar dalam satu garis disertai dengan panjangnya. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 48 (a + 6) cm 28 cm 12 cm a + 6 + 28 + 12 180 cm 6 + 28 + 12 = 46 a = 180 – 46 = 134 peningkatan dan pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik maupun melaui pembelajaran konvensional ditunjukkan pada gambar 1 berikut ini. gambar 1. peningkatan dan pencapaian kemampuan komunikasi matematik siswa v. kesimpulan dan saran a. kesimpulan 1. terdapat peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik sebesar 0,51. 2. pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik sebesar 63,96, hal ini lebih besar daripada siswa yang diajar melalui pembelajaran konvensional yakni 47,46. b. saran 1. untuk meningkatkan kemampuan komunikasi matematik siswa, hendaknya guru menerapkan model pembelajaran matematika realistik dengan menekankan pada kegiatan diskusi yang multi arah. 2. untuk meningkatkan pencapaian kemampuan komunikasi matematik siswa, hendaknya guru menerapkan model pembelajaran matematika realistik dengan menekankan pada kegiatan refleksi dan penemuan selama berlangsung pembelajaran. 0 10 20 30 40 50 60 70 pretes postes n-gain 32.26 63.96 0.51 32 47.46 0.24 s k o r k k m s is w a kelas eksperimen kelas kontrol infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 49 daftar pustaka al muhari, r. (2008). penerapan pendekatan pendidikan matematika realistik indonesia (pmri) untuk meningkatkan kreativitas belajar siswa. yogyakarta. baroody, a.j. (1993). problem solving, reasoning, and communicating, k-8. helping children think methematically. new york: macmullan publishing company. cohen, l., manion, l. and morrison, k. (2000). research mathods in education. london: routledge falmer. firdaus (2006). meningkatkan kemampuan komunikasi matematik siswa melalui pembelajaran dalam kelompok kecil tipe team assisted individualization (tai) dengan pendekatan berbasis masalah. tesis. program pascasarjana upi bandung. tidak diterbitkan. greenes, c. and sculman, l. (1996). communication processis in mathematical exploration and investigations. virginia: nctm. haji, s. dan abdullah, m.i. (2014). model pembelajaran matematika realistik untuk menigkatkan kemampuan berpikir matematika tingkat tinggi dan kemandirian belajar siswa. laporan penelitian. universitas bengkulu. hendriana, h. (2009). pembelajaran dengan pendekatan metaphorocal thinking untuk meningkatkan kemampuan pemahaman matematik, komunikasi matematik dan kepercayaan dirisiswa sekolah menengah pertama. disertasi. sekolah pascasarjana upi bandung. tidak diterbitkan. hulukati, e. (2005). mengembangkan kemampuan komunikasi dan pemecahan masalah matematika siswa smp melalui model pembelajaran generatif. disertasi. program pascasarjana upi bandung. tidak diterbitkan. khasanah, f. (2007). aktivitas siswa dalam pembelajaran keliling dan luas bangun datar melalui pendekatan pendidikan matematika realistik indonesia. yogyakarta. national council of teacher of mathematics (2000). principles and standard for school mathematics. reston: v.a. purniati, t. (2004). pembelajaran geometri berdasarkan tahap-tahap teori van hiele dalam upaya meningkatkan kemampuan komunikasi siswa smp. bandung: sekolah pascasarjana upi. tidak diterbitkan. rohaeti, e.e. (2003). pembelajaran dengan metode improve untuk meningktakan pemahman dan kemampun komunikasi matematik siswa sltp. tesis. bandung: sekolah pascasarjana upi. tidak diterbitkan. surayin (2003). kamus umum bahasa indonesia. bandung: yrama widya. suriasumnatri, j.s. (2007). filsafat ilmu sebuah pengantar populer. jakarta: pustaka sinar harapan. sumarmo, u. (2000). alternatif pembelajaran matematika dalam menerapkan kurikulum berbasis kompetensi. makalah seminar nasional fmipa universitas pendidikan indonesia. traffers (1987). three dimentions, a model of goels in theory description in mathematics instruction.dordrecht: reidel publishing company. van de wall, karp, jennifer, and williams (2000). elementary and middle school mathematics teaching developmentally. wihatma, u. (2004). meningkatkan kemampuan komunikasi matematik siswa sltp melalui cooperative learning tipe stad. tesis. bandung: sekolah pascasarjana upi. tidak diterbitkan. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p59-68 59 experience student background and their behavior in problem solving yulyanti harisman* 1 , muchamad subali noto 2 , wahyu hidayat 3 1 universitas negeri padang 2 universitas swadaya gunung djati 3 institut keguruan dan ilmu pendidikan siliwangi article info abstract article history: received jan 14, 2020 revised feb 11, 2020 accepted feb 18, 2020 these students' mathematical problem solving behavior had been presented in the previous paper. four categories of students' mathematical problem solving behavior in junior high schools in indonesia had been obtained. these categories were: naive, routine, semi-sophisticated, and sophisticated. this paper was a continuation of that research. in this session would discuss about external aspects affect student behavior in problem solving. this research used survey method. eighteen students from three junior high schools in indonesia had been interviewed about it. these three aspects were: distance of home from school, family background, contests contests like math olympiads that had been followed. the interview results were coded to get conclusions. research findings were that the external aspects of students did not influence students in behaving to solve problems in mathematics. the implication of this finding is that the main factor influencing student behavior in problem solving is teacher professionalism in learning not from the students themselves, so the teacher must be really prepared in designing all components of learning well. keywords: behaviors, experience student, problem solving copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: yulyanti harisman, department of mathematics education, universitas negeri padang, jl. prof. dr. hamka, air tawar, padang city, west sumatera, indonesia. email: yulyanti_h@fmipa.unp.ac.id how to cite: harisman, y., noto, m. s., & hidayat, w. (2020). experience student background and their behavior in problem solving. infinity, 9(1), 59-68. 1. introduction mathematical problem solving is core of mathematics learning. problem solving occupies a central position and is one of the most important forms of learning in mathematics (napitupulu, 2008). schoenfeld (1980) confirms that the problem solving process is one of the most important aspects of mathematics that must get attention from the teacher. in various curriculums, problem solving is central in mathematics learning. state that ho & hedberg (2005) mathematical problem solving (mps) as the central focus in the primary mathematics syllabus for all singapore schools. furthermore, pratiwi, masfuah, & rondli (2018) state that 21st century skills and competencies according to unesco include critical thinking and problem solving (solving problems), communication and collaboration, social and cross-cultural skills, and information literacy. mailto:yulyanti_h@fmipa.unp.ac.id harisman, noto, & hidayat, experience student background and their behavior … 60 problem solving will make students be creative and critical both in learning and in life. according to andersson (2007) mathematical problem solving will trigger the memory of children for working. such research on problem solving has significant implications. some researchers developed more direct instructional approaches that were designed to help students acquire schemata to solve mathematics problem solving (hidayat & sariningsih, 2018; nunokawa, 2005). goos & galbraith (1996) in her research implement the nature of secondary students' metacognitive strategy use, and how these strategies are applied when students work together on problems. the findings of that research is shed some light on the nature of individual and interactive metacognitive strategy use during collaborative activity. yerushalmy (2000) in his research describes a longitudinal observation of a pair of students that studied algebra for 3 years using a function approach, including intensive use of graphing technology. in his study is an attempt to analyze students construction of function based problem solving methods in introductory algebra. his study also offers terms for describing and explaining what and how do learners appreciate and make out of solving introductory school algebra problems over a three years course. puteh & ibrahim (2010) does research on the self-regulated learning strategies among form four students in the state of perak, malaysia. she did a case study to determine the usage of self-regulated strategies among the students and how it helps the students in solving mathematics’ problems. puteh & ibrahim (2010) research revealed the level of motivation and the existence of self-regulated learning strategies among the students. this research also has shown that there is a strong relationship between the self-regulated learning strategies and the students’ performance of problem solving. furthermore, funke (2013) identified the effects of different strategies on the problem-solving performance of junior high school algebra students. the finding of his research is the efficiency of these strategies depended on the number of steps in the problem solution, the number of possible blind alleys, and memory load requirements. research about how student solve the problem is also important beside the implementation of the strategy by teacher in learning about problem solving. some researchers have done it. harisman, noto, bakar, & amam (2017) and harun et al. (2018) in their research had seen how the gesture of student with different gender and different behavior when they solve the problem. the finding of her research is there is different gesture between male student and female student when they solve problem that given. futhermore, schoenfeld (1982) did the research about how the behavior student in solve problem. the finding of schoenfeld research there was two behaviors of student in solve problem he gave name with expert and novice problem solver. the expert of problem solving tend to recognize the patterns of problems, tend to change strategy when a strategy is not working, and an expert of problem solving can generate its own strategy in solving problems. some of the behavior displayed by novice problem solvers in solving the problem are: only recognize the problem of the surface of it, tend to just manipulating numbers in solving problems, and unable to move strategy if a strategy does not work. this research was continued by muir, beswick, & williamson (2008), the research findings of muir et al. (2008) complemented the research done by schoenfeld. if schoenfeld categorized the behavior of students when solving problems in two categories namely expert and novice, then muir categorized it into three categories: naive, routine, and sophisticated. naive behavior oriented on problem solver behavior associated only with manipulating numbers that exist in the problem. routine behavior oriented on structured behavior and sophisticated problem solvers oriented on problem solver who can generate their own strategies when they were faced with problems. harisman, kusumah, & volume 9, no 1, february 2020, pp. 59-68 61 kusnandi (2018) found four category of problem solving student behaviour that gave named: apathetic, routine, semi-sophisticate, and sophisticated. this research is a continuation of harisman's research (harisman et al., 2018) which focuses on what factors influence students' behavior in problem solving. there are two factors that influence student behavior in problem solving, namely internal factors and external factors. external factors include the teacher who teaches about problem solving and the teks book that used. this has been studied in harisman's research, the finding of this research is teacher beliefs, reflection, didaktik and pedagogi, attitude of teacher and the teks books that used will influent student behavior in problem solving. the research that has been carried out leaves the question of how the factors of the students themselves are like the distance of the homes of students from schools, families and so on. this paper was trying to answer that question. this series of investigations extends earlier work to include the examination of how distance of student home from school, family background of student, contests contests like math olympiads that had been followed might influence problem solving behavior of student (harisman et al., 2018; harisman, kusumah, & kusnandi, 2019a; 2019b; harisman, kusumah, kusnandi, & noto, 2019; kariman, harisman, sovia, & prahmana, 2019). 2. method this research was survey method. eighteen students from three different junior high school with different cluster were chosen as research subject. from each school had chosen six student with different ability (low, middle, and high ability) about mathematic achievement. the students were recommended by school teachers based on mathematical ability. students were labeled t to high-ability students, s for medium students, and r for lower-ability students. furthermore, in labeling schools, s-1 is for the school one, s-2 for the school two, and s-3 is for school three. each student had been classified into four behavior in problem solving at harisman et al. (2018) that categories gave named: apathetic, routine, semi-sophisticate, and sophisticated. the following explanation is in the table 1 names and abilities of students in each school. table 1. name and ability of students at each school no students at the school one students at the school two students at the school three 1 lutfi (r) s-1 dwi nanda (r) s-2 fauzan (r) s-3 2 divy (r) s-1 aulia fauza (r) s-2 rizky (r) s-3 3 fikri (s) s-1 najla (s) s-2 mesya(s) s-3 4 dhea (s) s-1 febrina (s) s-2 dita (s) s-3 5 annisa (t) s-1 zara (t) s-2 yani (t) s-3 6 alvaro (t) s-1 salma (t) s-2 nadhira (t) s-3 each student had been classified into four behavior in problem solving at harisman et al. (2018) that categories gave named: apathetic, routine, semi-sophisticate, and sophisticated. it can be seen in following table 2. harisman, noto, & hidayat, experience student background and their behavior … 62 table 2. recapitulation of each students’ behavior no apathetic routine semi-sophisticate sophisticated 1 lutfi (r) s1 dhea (s) s1 annisa (t) s1 alvaro (t) s1 2 fikri (s) s1 najla (s) s2 zara (t) s2 nadhira (t) s3 3 divy (r) s1 febrina (s) s2 salma (t) s2 4 mesya(s) s3 dwi nanda (r) s2 yani (t) s3 5 dita (s) s3 aulia fauza (r) s2 6 fauzan (r) s3 7 rizky (r) s3 each rubric in the category of mathematical problem solving behavior in harisman et al. (2018) research can be seen as table 3. table 3. a picture of student behavior based on the range of student behavior no factor indicator category of behavior apathetic routine semi sophisticated sophisticated 1 knowledge ownership the application of polya's heuristic steps in mathematical problem solving made a mistake on the four solving steps no effort to verify the solution (make a mistake on some troubleshooting steps) there is an attempt at verifying the solution but still making mistakes on some troubleshooting steps a high score on each troubleshooting step the use of prior knowledge for mathematical problem solving cannot use previously resolved issues can identify a similar problem, but not on the mathematical structure (can identify similar problems only on external features only) can identify a similar problem, but not on the mathematical structure, but can be directed to look at problems in the mathematical structure identify similar problems according to the mathematical structure his many ways are used in mathematical problem solving often use the same way to solve all problems because of the limitations of knowledge possessed focus on one way with the knowledge you have to solve a particular problem, but have no desire to generate the right way to solve the focus on one way with the knowledge you have to solve a particular problem, but have a desire to generate the right way to solve the problem identify other ways of troubleshooting volume 9, no 1, february 2020, pp. 59-68 63 no factor indicator category of behavior apathetic routine semi sophisticated sophisticated problem written and verbal communication in mathematical problem solving written and verbal communication is inadequate written communication is not clear, but can clarify through verbal communication written communication is not clear, but can clarify through verbal communication and can be directed to clarify his written communication written and verbal communication is sufficient (obviously) 2 self control (control) thinking metacognition in mathematical problem solving communication thinking metacognition is invisible, both in written and verbal metacognition does not appear in writing, but metacognition appears verbally metacognition is not without writing, but metacognition appears verbally and can be directed to perform metacognition processes in written answers thinking metacognition appears in written and verbal responses 3 confidence confidence in how to implement strategy in problem solving doing exactly the same strategy implementing the strategy in a systematic way implementing the strategy in a systematic, yet expandable way to generate its own strategy strategy matematis generate your own confidence in the variety of strategies used in mathematical problem solving strategy rely on one strategy relying on more than two strategies, when one strategy does not work does not switch to another relying on more than two strategies, when one strategy does not work, does not switch to another strategy, and can be directed to move strategy desiring to combine several strategies 4 affective often declare lack of confidence in solving problems, but desire to explore further how to solve them with confidence confidence in solving mathematical problems confidence is in line with the quick answer often declare lack of confidence in solving problems appears confident in problem-solving skills harisman, noto, & hidayat, experience student background and their behavior … 64 in this research would focus on factors of the students themselves are like distance of student home from school, family background of student, contests contests like math olympiads that had been followed. each student would give the open questions about it, and all of the answer would be video. the video results will be coded and analyzed using grounded theory which will be clearly specified. 3. results and discussion this section would detail the results of the video code transcript. each type of student behavior would be corresponded with indicator of eksternal factor. the recapitulation of apathetic students’ behavior with indicator eksternal factor can be seen in table 4. table 4. recapitulation of apathetic students’ behavior with eksternal indicator factor no apathetic distance of student home from school family background contests contests like math olympiads that had been followed 1 lutfi (r) s1 far from school rich family never followed 2 fikri (s) s1 far from school rich family never followed 3 divy (r) s1 far from school simple family never followed 4 mesya(s) s3 near from school poor families never followed 5 dita (s) s3 near from school poor families never followed 6 fauzan (r) s3 near from school poor families never followed 7 rizky (r) s3 near from school poor families never followed the recapitulation of routine students’ behavior with indicator eksternal factor can be seen in table 5. table 5. recapitulation of routine students’ behavior with eksternal indicator factor no routine distance of student home from school family background contests contests like math olympiads that had been followed 1 dhea (s) s1 near from school simple family never followed 2 najla (s) s2 far from school simple family never followed 3 febrina (s) s2 far from school simple family never followed 4 dwi nanda (r) s2 near from school simple family never followed 5 aulia fauza (r) s2 near from school simple family never followed the recapitulation of semi sophisticated students’ behavior with indicator eksternal factor can be seen in table 6. table 6. recapitulation of semi sophisticated students’ behavior with eksternal indicator factor no semi sophisticated distance of student home from school family background contests contests like math olympiads that had been followed 1 annisa (t) s1 near from school simple family never followed 2 zara (t) s2 far from school rich family never followed volume 9, no 1, february 2020, pp. 59-68 65 no semi sophisticated distance of student home from school family background contests contests like math olympiads that had been followed 3 salma (t) s2 far from school simple family never followed 4 yani (t) s3 near from school simple family never followed the recapitulation of sophisticated students’ behavior with indicator eksternal factor can be seen in table 7. table 7. recapitulation of sophisticated students’ behavior with eksternal indicator factor no sophisticated distance of student home from school family background contests contests like math olympiads that had been followed 1 alvaro (t) s1 near from school simple family ever followed 2 nadhira (t) s3 near from school simple family never followed based on the result of research, distance of student home from school, family background and contestscontests like math olympiads that had been followed didn’t influent behavior of student in problem solving. the findings of this study contradict with the findings of the study that did by choy (2001). according choy (2001) family background would influent the achievement of the children. in her research choy gave information on how student and family background characteristics and students high school experiences are related to their access to postsecondary education immediately after high school. fuligni (1997) research is also in line with choy (2001). he did his research in latino, east asian, filipino, and european adolescent to determine relative impact of family background, peer suport on academic achievement of student. the finding of his research there was a more significant correlations between family background and academic achievement of student. base on fehrmann, keith, & reimers (1987) parental involvement is considered an important influence on academic progress. time spent on homework and in leisure tv viewing has an important effect on academic learning. many cases about family background had not been covered in this study. we should find the specific thing that could influent student behavior in problem solving. so is the case with social culture, the environment, student activities and so on. 4. conclusion research results show distance of student home from school, family background and contests-contests like math olympiads that had been followed didn’t influent behavior of student in problem solving. further research is needed to examine more in terms of what affects students behaving in problem solving. references andersson, u. 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(2021). development of worksheets based project using the lesson study. infinity, 10(1), 41-52. 1. introduction direct proportion material is the ratio of two or more values of a similar quantity (ben-chaim, keret, & ilany, 2012; zuhra, zubainur, & abidin, 2018). situations that show value comparisons such as quantity, per, value, or price contained in real problems (de la cruz & garney, 2016; lobato et al., 2010). direct proportion can be solved by factors of change, unit rate, and cross multiplication (de la cruz & garney, 2016). however, at present, students still have mathematical modeling difficulties and calculation procedures in solving direct proportion material, and students often lose the meaning of direct proportion because they are accustomed to using cross multiplication in solving the problem (arican, 2018). https://doi.org/10.22460/infinity.v10i1.p41-52 sari & putri, development of worksheets based project using the lesson study … 42 social arithmetic materials are usually found in everyday life such as buying and selling transactions (profit/loss/break-even) in trading activities (as'ari et al., 2016). but in reality in social arithmetic material, students often fail in the principles and calculation techniques in completing social arithmetic (astutik & kurniawan, 2015). another problem in learning mathematics is that students are rarely given problems related to the project because they are often given problems in the form of algorithms (mahendra, 2017). this doesn’t meet the demands of permendikbud concerning assessment standards that to assess competency skills can be done through performance assessments that require students to demonstrate certain competencies using a project (kemdikbud, 2013). the government has also made a program to build the golden generation of 2045, which is equipped with 21st century skills, including 1) the quality of character which generally consists of religion, nationalism, independence, independence, mutual cooperation, and integrity; 2) literacy consisting of language, numeracy, scientific, digital, financial, cultural and citizenship literacy; and 3) competencies consisting of critical thinking, creativity, communication, and collaboration. in this research a project activity will be carried out by making a project plan: the implementation of my school cooperative. some basic mathematical competencies completed in this project include 3.8. analyze direct proportion and indirect proportion using data tables, graphs, and equations; 4.8. resolve problems that occur by using direct proportion and indirect proportion; 3.9. get to know various analyzes related to social arithmetic; 4.9 solve problems related to social arithmetic; and the basic competency of ips is 3.3. understanding the concept of interaction between humans and space produces a variety of economic activities (production, distribution, consumption, demand, and supply) and interaction between spaces for the survival of indonesia's economic, social and cultural life (kemdikbud, 2016). the government plans to build a golden generation of 2045 equipped with 21st century skills requiring three things including 1) quality of character consisting of religiosity, nationalism, independence, mutual cooperation, and integration; 2) literacy consisting of language, numeracy, scientific, digital, financial, cultural and citizenship literacy; and 3) competencies consisting of critical thinking, creativity, communication, and collaboration (kemdikbud, 2016). to complete these efforts, it can be done with lesson study for the learning community (lslc). there are some things that are still lacking in the learning system, among others, some teachers are still unable to manage the class well (bustang, zulkardi, darmawijoyo, dolk, & van erde, 2013); teachers do not pay attention to students who have difficulty in learning because it is difficult for teachers to guide them individually (sato, 2012); and also the education system in schools still pays little attention to the evaluation system and the learning process taking place in the classroom (fauziah et al., 2020). while education discusses how students learn and how to teach teachers, not only on learning outcomes (brooks & brooks, 1999). the application of the lesson study system in learning is by collaborating among colleagues to support, try, discuss and reflect on the learning done, where there will be many improvements that will indirectly improve the quality of learning by redesigning (lewis, 2002; sato, 2012). that is because in learning from lesson study is collaborative learning, where students work in groups with unequal final answers from each student that creates interconnected and mutually supportive relationships between students (sato, 2012). enabling learning can improve the quality of learning or improve educational practices by building learning communities that encourage collaboration and collegatives (octriana, putri & nurjannah, 2019; sato, 2012). this is contrary to vygotsky's theory of the learning process will occur more effectively if with the help of others (vygotsky, 1980). volume 10, no 1, february 2021, pp. 41-52 43 this is supported by several studies which generally explain that lesson study for learning community (lslc) has a positive impact and can improve the quality of learning, among others, according to arini and putri (2019) who use lesson study has a positive impact on students including increasing awareness with friends, reduce the emergence of competition in learning activities, and establish collaboration between students in learning. the same thing according to nuraida and putri (2019) that lesson study also has a positive impact on the learning process because it can help students to understand the concept of a material through their activities in their study groups by collaborating and caring for each other. the purpose of this study is to produce valid and practical problems on worksheets in order to be able to help them with direct proportion material and social arithmetic in school cooperative projects. 2. method this research uses a design research method which is a form of qualitative approach. stages of research development in design research (van den akker, 1999). the development study undertaken is the initial stage of applying the initial idea of a literature study before designing learning activities; and the formative evaluation stage which includes selfevaluation aimed at researchers to evaluate and examine the initial prototype itself, prototyping (expert reviews, one-to-one, and small groups) aimed at validating each prototype in terms of content, structure, and language (tessmer, 1993; zulkardi, 2002) (figure 1). figure 1. stages of research development in design research in this research the development of professionalism that uses lesson study is to plan, do, and see. this research was conducted in seventh grade students of smp n 1 palembang with direct proportions and social arithmetic. this study involved 8 students in small groups. the research lasted for several weeks. 3. results and discussion the stages in this research are development research and lesson study system which include plan; do and reflection. the following is an explanation of each stage carried out by the researcher. sari & putri, development of worksheets based project using the lesson study … 44 3.1. planning stages at this stage, planning is carried out by the researcher and discussed together with participants including teachers and colleagues. the planning included discussing student activity sheets and predicting student answers. figure 2. results of revision of student worksheets and prediction of student answers volume 10, no 1, february 2021, pp. 41-52 45 researchers discuss with the teacher at smp n 1 palembang about the material to be designed. then from the discussion, the researcher obtained information that vii values in the second semester had understood the concept of direct proportion and social arithmetic so the next step was for the researcher to design questions or problems in the form of a school cooperative project. researchers design worksheets that contain problems and then validate worksheets with colleagues and mathematics teachers to obtain information about whether the worksheets are correct and can be understood by students in terms of structure, content, and language (figure 2). the validation results will be considered by the researcher whether to make revisions or not (wahyuni et al., 2020). the results of changes at this stage include increased front-page writing (semester information, work time, and learning objectives) and the difficulty level of the worksheet. researchers along with colleagues and mathematics teachers predict the possibility of students' answers ranging from low ability, medium ability, and high ability that there may be differences in thinking patterns. this is needed so that researchers can plan scaffolding if something happens that was predicted beforehand. some revisions made after the discussion included changing the title of the student’s activity sheet from "come on, entrepreneurship in a school cooperative" to "a direct proportion of economic activity"; figures on the agreed price list to make it easier for students to calculate; on the worksheet the students agree to get, break even, and losses related to selling prices and buying prices; improve more than one answer prediction from students. after that go to the stage one-to-one. questions on the worksheet that have been designed are being tested on three students with various levels of ability including high, medium and low ability students. students are also expected to contribute in validating worksheets in terms of language level and their understanding of the purpose of the questions on the worksheet (figure 3). after that, the results of students' answers are analyzed and students are interviewed with a focus on the work process carried out when completing the worksheets. figure 3. student answers in determining price lists in determining the price list of goods, high-ability and medium-ability students can determine the price list of goods correctly and correctly in different ways (figure 3). h ig h a b il it y s tu d e n t m e d iu m a b il it y s tu d e n t l o w a b il it y s tu d e n t sari & putri, development of worksheets based project using the lesson study … 46 however, in low-ability student, there are still errors in completing the calculation of the direct proportion. the following is an explanation of each students’ abilities, from the results of the answers to high-ability students, she determines the price of goods by making a direct proportion equation, then the ratio on the left which is the selling price of the distributor agent multiplied by the numbers that can produce the same number as the ratio on the right. for example, he wrote to buy as many as five pack pencils and known the price of the distributor agent of rp. 48,000 for two packs. students are correct by making the following direct proportion, 2/48,000=5/(price of 5 packs). after that, students determine the exact number to multiply by 2 to produce the number 5 is 2.5 so that the students make the equation become (2×2.5)/(48,000×2.5)=5/(price of 5 packs). the price for five packs of pencils is rp. 120,000. judging from the procedures that have been carried out by students, he already understands the direct proportion because he tries to equate the proportion on the left with the one on the right to find out one unknown variable from the propotion on the right. then, there are no errors in students in calculating the total purchase, which is rp. 750,000 and enough from the maximum limit to buy the goods specified on the worksheet even though he does not need to process the calculation, but the final results that have been written are correct. in students the ability of being seen from the answer that he first determines the price of goods for one unit and then calculates the number of packs purchased. this can be seen by students writing to buy two packs of pencils. he calculated the price of one unit by dividing the price of goods and the number of packs, namely rp.180,000 ÷ 5 packs = rp. 36,000. after that, multiplied by rp.36,000 × 2 packs = rp.72,000 because he wrote to buy two packs of pencils. students then calculate the total purchase by adding the total price of items purchased. from the procedure that has been done by students is correct and there are no mistakes even though he solved the problem in a different way from high-ability students by calculating the price of a package first and then multiplying it with many items purchased. in low-ability student viewed from the answer that the student has been able to write it in direct proportion. he uses cross multiplication in solving these problems by writing 90,000/5 = (pen a)/20. then multiply the cross to 5 × pen a = 20 × 90,000 after completing it to 5 × pen a = 1,800,000. but there was a mistake he did while looking for the value of pen a, he wrote it into pen a = 1,800,000 × 5 = 5,400,000. he was wrong in calculating what should be the pen a = 1,800,000 ÷ 5 = 360,000. from these answers and the results of interview information that students are still wrong in determining the operation of one of the variables sought in the equation. figure 4. student answers in determining the price of capital per unit h ig h a b il it y s tu d e n t m e d iu m a b il it y s tu d e n t l o w a b il it y s tu d e n t volume 10, no 1, february 2021, pp. 41-52 47 for high ability student, she already understands to calculate the unit price, so the required number of units of the whole pack purchased (figure 4). this can be seen by student writing down a lot of pencil b is 5 packs × 12 pieces = 60 pieces. after that, she wrote down 60/120,000=1/(price of a pencil b) then look for the right arithmetic operation so that the proportion on the left is the same as the proportion on the right by dividing it 60 into (60÷60)/(120,000÷60)=1/(price of a pencil b), then obtained 1/2,000=1/(price of a pencil b) so the price for a pencil b is rp. 2,000 from the whole procedure is correct and there are no errors in the calculation so that it can be said that students have been able to solve the problem of direct proportion. in the medium ability students also already understand to calculate the unit price, it requires the number of units of the whole pack purchased (figure 4). this can be seen by students writing a lot of pencils a is 2 pack × 12 pieces = 24 pieces. after that, he divides it by the total price of pencil a, which is 72,000 ÷ 24 = 3,000 so that the price for a pencil a is rp. 3,000. from the whole procedure it is correct and there are no errors in calculations so it can be said that students have been able to solve the problem. however, it made clear that the students' understanding of the direct proportion was conducted interviews related to the answer and it was found that the students understood the comparative worth of the explanation that is “because the price of one pencil is sought, then the ratio of the value is directly divided between the price of goods with the number of items”. in low ability students, it appears that he only wrote a pen 12 pieces, even though he bought 20 packs of pen a, which should have written 20 packs × 12 pieces = 240 pieces (figure 4). from this it appears that students do not understand the purpose of the problem. it was also strengthened from the results of his interview, he only wrote it according to the guidance table containing 1 pack = 12 pieces, so he wrote 12 pieces. then students calculate the price of capital per unit by writing 5,400,000 ÷ 12 = 441. from the answer the students also showed that students did not understand the division operation. the result of the division should be 5,400,000 ÷ 12 = 450,000. figure 5. student answers in adjusting the selling price to the price of capital from the answers of high-ability student and medium-ability student (figure 5), it appears that they have understood the price of sales must be greater than the purchase price. this can be seen from the answers to the students' high ability to write the capital price of h ig h a b il it y s tu d e n t m e d iu m a b il it y s tu d e n t l o w a b il it y s tu d e n t sari & putri, development of worksheets based project using the lesson study … 48 pencil b for rp. 2,000 and the selling price of rp. 4,000. the same thing is done by mediumability students by writing the capital price from pencil a of rp. 3,000 and the selling price of rp. 3,500. their answers also show that they have been able to determine the ideal price for the selling price by not raising the selling price too much or not raising the selling price by too little. however, for low-ability student (figure 5), from the results of the answers and also the results of the interview he wrote the selling price is not based on the price of capital but based on the selling price he knows in everyday life. this is because the results of determining the wrong capital price will also create confusion in writing the selling price. figure 6. student answers in determining sales conditions from the answers of high-ability student and medium-ability student (figure 6), it is seen that they understand how to calculate the difference between the selling price and the purchase price. this can be seen from the way they calculate the difference between the selling price and the purchase price by subtracting both of them. in high ability student, she wrote the difference in price for pencil b was rp. 4,000-rp. 2,000 = rp. 2,000. the same thing is done by students of medium ability by writing rp3,500-rp3,000 = rp.500. after that, they can also determine the conditions of sale appropriately ie when the selling price is greater than the purchase price, their sales conditions are in a profit. however, in low-ability student (figure 6), it can be seen from the results of answers and interview results that these student have weaknesses in calculating the reduction operations. this can be seen from the way he reduced the price of notebook a which has an error, which is 5,000-151055525 = 45155525 but the results obtained must be 5,000-151055525 = -151050525. in addition, the students were seen from the results of the answers and interview results which he said were confused when determining the conditions of sale due to errors in calculating the difference between the selling price and the purchase price. but after further questioning, he understood that the profit was obtained when the selling price was greater than the purchase price. from the overall results from stage one to one, it was concluded that all students could understand this goal so that making this worksheet would proceed to the small group stage. h ig h a b il it y s tu d e n t m e d iu m a b il it y s tu d e n t l o w a b il it y s tu d e n t volume 10, no 1, february 2021, pp. 41-52 49 3.2. stages of implementation at the implementation stage, the worksheet will be tested on a small group of 8 people and then divided into 2 groups, the distribution of groups is also based on the level of student ability. here the researcher becomes a teacher and conducts a learning process in accordance with the worksheets that are designed and tries out worksheets that have been validated from the previous stages. in filling out the worksheet using the system of lesson study for learning community (lslc). the questions on the worksheet consist of sharing assignments and jump assignments which are packaged in questions in the form of projects. the teacher model has the task to observe the learning process in small groups. the learning process that takes place in small groups will be analyzed whether the learning goes well or not and also the results of student answers will be analyzed to see the extent of students' ability to the worksheets provided. figure 7. student expressions when learning in small groups, the classroom situation seems calm and focused on each worksheet (figure 7). they completed their respective projects by determining the items they wanted to fill in the school cooperatives provided that the total purchase was not more than rp. 750,000. when there are students who have difficulty in solving problems, they ask for help from their friends who can answer it, creating a conducive classroom atmosphere and they do not riot in their groups and their friends teach students who have difficulty until they understand, then students who experience difficulty saying the word "thank you" to friends who have taught them. after that, students who have difficulty in each group are asked to come forward and present the results of their answers to the class. from these, it will create the learning atmosphere desired by researchers, which are students who have a caring nature to their friends who are having difficulties, students who value their friends by saying "please teach me" and the word "thank you" when asking for help from their friends so that they will arise mutual respect for each other, then the nature of the responsibility of students who ask to be taught by their friends to present the results of the answers so that researchers can understand the extent of student understanding in what his friend explained, and also with this atmosphere there are no more students who do not learn during class time because all students can help each other in the study group which makes students no longer get score of 0 when completing their respective worksheet. sari & putri, development of worksheets based project using the lesson study … 50 table 1. recapitulation of student answers student student ability level group student answer average determining price lists determining the price of capital per unit adjusting the selling price to the price of capital determining sales conditions 1 high 1 81 87 75 74 79.25 2 medium 1 68 77 80 80 76.25 3 low 1 80 82 74 84 80.00 4 low 1 73 77 76 56 70.50 5 high 2 86 71 86 74 79.25 6 medium 2 75 73 70 70 72.00 7 medium 2 70 81 86 72 77.25 8 low 2 76 65 63 70 68.50 average 76.13 76.63 76.25 72.50 75.38 table 1 shows that problem 1 and problem 4 are problems of jumping task. this can be seen from the achievement of students' value at the time of problem 1, namely students who determine the purchase price of all items purchased and total purchases with a value of 76.13 and problem 4, namely students determine the choice of sales in profit, profit, or break even with a value of 72.50. then for problem 2 and problem 3 is the sharing task. this can be seen from the achievement of students' value in problem 2, which is determining the price of goods per unit with a value of 76.63 and problem 3, which determines the difference between the selling price and the purchase price with a value of 76.25. from achieving these values, worksheets in the form of projects can already be used because they consist of problems in the form of sharing task and jumping task. in addition, it can be seen in the table that the average value of all students is 75.63. for high-ability students in group 1 with 79.25 and group 2 with 79.25 so the average value for high-ability students is 79.25. these values indicate that high-ability students are above the average value of overall students. for medium-ability students in group 1 of 76.25 and group 2 of 72.00 and 77.25 so that the average value for students of medium ability is at 75.16. these values indicate that students with medium abilities are below the average of the overall value of students. for low ability students in group 1 amounted to 80.00 and 70.50 and group 2 amounted to 68.50 so the average value for low ability students was 73.00. these values indicate that low-ability students are below the average of the overall value of students. but an interesting fact from low ability students is that one of them gets the highest score of 80.00 and also the lowest value of 68.50. collaborative learning from the lslc system has a positive impact on learning because there are no group members who can’t answer at all or get zero marks. 4. conclusion the conclusions of this research are, (1) the process of lslc system, collaborative learning, pmri approach and design research are the right series of learning to be used in learning at smpn 1 palembang; (2) during the learning process, using context in daily life such as school cooperatives can be recognized by students helping them to more easily solve material problems of direct proportion and social arimetika by using school cooperative projects; (3) there are still some students who have obstacles to counting in division and subtraction operations (4) overall, after examining the research, students were greatly helped to solve material direct proportions and social arithmetic in completing school volume 10, no 1, february 2021, pp. 41-52 51 cooperative projects; (5) problems 1 and 4 including jumping task, and problem 2 and 3 including sharing task. acknowledgments the author would like to thank the principal smpn 1 palembang to enable me to do research there. also, to the math teacher at smpn 1 palembang. i will never be able to solve this without your guidance. the research/publication of this article was funded by dipa of public service agency of universitas sriwijaya 2020. sp dipa-023.17.2.677515/2020, revision 01, on march 15, 2020. in accordance with the rector’s decree number: 0687/un9/sk.buk.kp/2020, on july 15, 2020. references arican, m. 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(2012). dialog dan kolaborasi di sekolah menengah pertama: praktek “learning community”. jakarta: penerbit pelita, translation. tessmer, m. (1993). planning and conducting formative evaluations: improving the quality of education and training. london: psychology press. van den akker, j. (1999) principles and methods of development research. in: van den akker j., branch r.m., gustafson k., nieveen n., plomp t. (eds) design approaches and tools in education and training. springer, dordrecht. https://doi.org/10.1007/978-94011-4255-7_1 vygotsky, l. s. (1980). mind in society: the development of higher psychological processes. london: harvard university press. wahyuni, r., efuansyah, e., & sukasno, s. (2020). developing student worksheet based on missouri mathematics project model by using think-talk-write strategy of class viii. infinity journal, 9(1), 81-92. https://doi.org/10.22460/infinity.v9i1.p81-92 zuhra, s. f., zubainur, c. m., & abidin, t. f. (2018). learning direct proportion by using the context of timpan recipes. journal of physics: conference series, 1088(1), 012048. http://doi.org/10.1088/1742-6596/1088/1/012048 zulkardi, z. (2002). developing a learning enviroment on realistics mathematics education for indonesian student teacher. enschede: university of twente. https://doi.org/10.23887/jpi-undiksha.v6i1.9257 https://doi.org/10.22342/jpm.13.2.6714.131-142 https://doi.org/10.1007/978-94-011-4255-7_1 https://doi.org/10.1007/978-94-011-4255-7_1 https://doi.org/10.22460/infinity.v9i1.p81-92 http://doi.org/10.1088/1742-6596/1088/1/012048 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p219-228 219 effect of learning using mathematica software toward mathematical abstraction ability, motivation, and independence of students in analytic geometry yanuar hery murtianto 1 , sutrisno *2 , nizaruddin 3 , muhtarom 4 1,2,3,4 universitas pgri semarang article info abstract article history: received april 30, 2019 revised sept 25, 2019 accepted sept 26, 2019 rapid development of technology for the past two decades has greatly influenced mathematic learning system. mathematica software is one of the most advanced technology that helps learn math especially in geometry. therefore this research aims at investigating the effectiveness of analytic geometry learning by using mathematica software on the mathematical abstraction ability, motivation, and independence of students. this research is a quantitative research with quasi-experimental method. the independent variable is learning media, meanwhile the dependent variables are students’ mathematical abstraction ability, motivation, and independence in learning. the population in this research was the third semester students of mathematics education program and the sample was selected using cluster random sampling. the samples of this research consisted of two distinct classes, with one class as the experimental class was treated using mathematica software and the other is the control class was treated without using it. data analyzed using multivariate, particularly hotelling’s t 2 test. the research findings indicated that learning using mathematica software resulted in better mathematical abstraction ability, motivation, and independence of students, than that conventional learning in analytic geometry subject. keywords: analytic geometry, learning independence, learning motivation, mathematica, mathematical abstraction copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: sutrisno, departement of mathematics education, universitas pgri semarang, jl.sidodadi timur no. 24, semarang, indonesia. email: sutrisnojr@upgris.ac.id how to cite: murtianto, y. h., sutrisno, s., nizaruddin, n., & muhtarom, m. (2019). effect of learning using mathematica software toward mathematical abstraction ability, motivation, and independence of students in analytic geometry. infinity, 8(2), 219-228. 1. introduction rapid development of technology for the past two decades has greatly influenced mathematic learning system. higher order thinking skills ability a person is a requirement the main one is able to compete in this 21 st century and mathematics that has an important role in the development of high-order thinking skills. so that technological and mathematical collaboration is needed in answering this challenge. mathematics pre-teacher students are part of a system which is directly influenced by the development of mailto:sutrisnojr@upgris.ac.id murtianto, sutrisno, nizaruddin, & muhtarom, effect of learning using mathematica software … 220 technology. on the other hand the emphasis was on learning mathematics must be in context since learning should be involve the mental processes of students, should be fun, encourage students, and give students the opportunity to construct their learning experience, so that learning become meaningful (nizaruddin, muhtarom & sugiyanti, 2017). furthermore, motivation provides a very important role (rachvelishvili, 2017), in addition to the use of technology in learning mathematics. as it was delivered some researchers have conducted studies that emphasis the view that the success of students’ problem-solving is an important factor in understanding the beliefs of mathematics; students’ beliefs toward solving the problem also depend on the motivation of the belief in mathematic (muhtarom, juniati & siswono, 2017a; 2017b). one of the technology advances in mathematics learning system is the choice variances of mathematics software. the variances are divided into two important parts; dynamic geometry system and computer algebraic system (malinova, 2010). among the mathematical software are maple, mathematica, geometer sketchpad, matlab, and geogebra. a great number of software that can be used by mathematics pre-service teachers in learning requires them to skilfully utilize software they plan to choose (listyani, 2006). taking account to this phenomenon, it is necessary that the students of mathematics education program are provided with computer software utilization skill in order to meet the demands of the workplace. one of mathematic software that can be applied by the students is mathematica software. mathematica is symbol visualization and algebraic manipulation based computer software that was developed by stephen wolfram. it is one of today’s most up-to-date software and this is confirmed by several previous studies by gocheva (2009), malinova (2010) and vosler (2009) which reveal that mathematica software is effective in mathematics learning at university level which investigates concept and finds its connection to other mathematical concepts or its implementation with technology. a more specific research was carried out by kapustina, popyrin, & savina (2015) indicating that visualization concepts of analytic geometry would be easier when presented using this software. the importance of mathematica software in analytic geometry learning is undeniable. this is obvious from the previous studies, two of which were the research sunandar, murtianto & sutrisno (2015) on the development of analytic geometry teaching materials using mathematica software in developing students’ representation, and the research on improving achievement and differential calculus learning independence of mathematics education program students of universitas negeri yogyakarta using mathematica software (listyani, 2006). a number of studies that have been conducted related to software-assisted mathematics learning indicate that the era of modern mathematics learning in the class continues to grow rapidly. the use of software in mathematics learning in classroom should consider the portion and the conformity level to learning objectives. this is so for it can put an effect on the level of abstraction, motivation, and learning independence of students. this thinkable concept can later be used in more complicated and complex level of thinking. the states can raise first abstraction when an individual focuses on the characteristics of objects observed and then label them through a classification process into several groups on the basis of certain categories. motivation in mathematics education is important for it provides energy in learning which can be adapted to bridge (hannula, 2006). meanwhile, motivation is a natural characteristic of a person that can be explored in line with the needs (pintrich, 2003). indeed, motivation cannot be considered directly, but it can be observed from the manifestation of behavior, attitude, and activity of an individual. the general problem is that the average research that has been done previously only emphasizes the volume 8, no 2, september 2019, pp. 219-228 221 cognitive aspects of the dependent variable. research that takes the dependent variable motivation, independence and abstraction ability is usually also carried out separately and not simultaneously. then the purpose of this study is to find out the impact of learning using mathematica on motivation, abstraction ability and independence simultaneously. thus, mathematics learning using mathematica software demands certain motivation to change behavior and attitude through a process of learning, and this is expected to put an effect on the students’ independence. 2. method 2.1. general background of research this research applied a quantitative approach with a quasi-experimental, because in this research it is not possible for researchers all relevant variables. budiyono (2003) stated that research in education is included in quasi-experimental research because educational research often uses intact groups such as classes as experimental groups and classes as groups comparison. this is so for it is impossible for the researcher to control all the relevant variables (budiyono, 2003). some research experts suggest that researches in the field of education are considered quasi-experimental researches since educational researches often make use of intact groups such as classes as experimental class was treated using mathematica software and control class was treated without using them (conventional learning). in this case, randomization is not applied in determining subject assigned to experimental class and control class. the variables in this research were divided into independent and dependent variables. the independent variables were learning media, while the dependent variables were students’ mathematical abstraction, motivation, and independence. multivariate analysis technique, particularly hotelling's t 2 test statistics, was applied for analyzing data since this research only compared two classes (control class and experimental class) and involved three dependent variables, including students’ mathematical abstraction, learning motivation and learning independence. 2.2. sample of research the population of this research is a group of students attending analytic geometry course on third semester that consist of six class in mathematics education department of universitas pgri semarang in the academic year 2018/2019. the sampling of the research using cluster random sampling. the samples of this research consisted of two distinct classes, with one class as the control class and the other is the experimental class. 2.3. instrument and procedures data collection techniques is used in this research are tests and questionnaires. the test technique is used to measure the ability of mathematical abstraction, the questionnaire is used to measure the students' motivation and student’s independence. the mathematical abstraction ability test is used in the form of description consists of two questions, while the questionnaire motivation and student’s independence in the form of a closed questionnaire with four answer options (strongly agree, agree, disagree, and strongly disagree) consisting of 30 items of statement. all such instruments was used to obtain preliminary and final ability data. the student's preliminary ability data was used for the balance test (i.e. univariate normality test, multivariate normality test, homogeneity of variance test, homogeneity of covariance matrices test, and hotelling’s t 2 test), while the student's final ability data was used as the test of the research hypothesis, as in the balance murtianto, sutrisno, nizaruddin, & muhtarom, effect of learning using mathematica software … 222 test. the research was conducted from august to december 2018, while the preliminary ability data was collected prior to the research. 2.4. data analysis multivariate analysis technique, particularly hotelling's t 2 test statistics, was applied for analyzing data since this research only compared two classes (control class and experimental class) and involved three dependent variables, including students’ mathematical abstraction, learning motivation and learning independence. this data analysis technique was used twice, i.e. in balance test for preliminary research data and in hypothesis test for final research data. balance test was intended to determine that both control and experimental classes were in balance before receiving any treatments. this test was carried out to ensure that any changes after treatment provided were resulted from the treatment. before conducting this test statistics, it was necessary to perform prerequisite tests, including multivariate normality test and covariance matrix homogeneity test. mardia test was used to test the data normality, while box’s m was used to test the covariance matrix homogeneity (rencher, 2002). 3. results and discussion 3.1. results preliminary data of abilities in this research were data of students’ mathematical abstraction ability, motivation and independency in learning analytic geometry subject before experiments were conducted. data of mathematical abstraction ability were obtained using written test, while data of learning motivation and independence were obtained using questionnaires. the test was carried out using multivariate technique, particularly test statistics of hotelling’s t 2 . univariate normality test of population obtained in reference to the results of analysis, it was found that both samples constructed test statistics of d ≤ dα and therefore not rejected h0. this was also supported by the pvalue ≥ α = 0.05 for all samples. thus, and the significance level of 5% for both samples was derived from univariate normally-distributed populations. furthermore multivariate normality test of population have been showed that both samples constructed test statistics of b1,p ≤ b1,p,1-α,n and b2,p,α/2,n ≤ b2,p ≤ b2,p,1-(α/2),n. that b1,p is the slope coefficient if it is less than b1,p,1-α,n then it is said to be symmetrical distribution, b2,p is the coefficient of shaking if it is between b2,p,α/2,n and b2,p,1-(α/2),n then the distribution is normal (mesokurtic). this was supported by p-value ≥ α = 0.05 for all samples and therefore not rejected h0. therefore, the significance levels of 5% of both samples were obtained from multivariate normally-distributed populations. homogeneity test of population variance demonstrated that all dependent variables appeared to construct test statistics of and therefore not rejected h0. this was supported by p-value ≥ α = 0.05 for all variables. hence, with the significance levels of 5%, both populations showed homogenous variances for each dependent variable. moreover homogeneity test of population covariance matrices indicated box’s m = 5.359. the value could eventually be interpreted using critical value table of box’s m test in one condition that every cell in factorial design had equal sample size. however, the condition was not fulfilled in this research, and hence box’s m value could not be directly interpreted. for the purpose of interpretation, f approach was employed. for f ≤ fα and therefore not rejected h0. this was supported by p-value ≥ α = 0.05. in conclusion, with 5% significance level, both populations showed homogenous covariance matrices. volume 8, no 2, september 2019, pp. 219-228 223 on the basis of multivariate test using spss, pillai’s trace test resulted in f = 0.214 with p = 0.886; wilk’s lambda test resulted in f = 0.214 with p = 0.886; hotelling’s trace test constructed f = 0.214 with p = 0.886; and roy’s largest root test produced f = 0.214 with p = 0.886. all the tests resulted in p ≥ α = 0.05 and therefore not rejected h0. this research focused more on hotelling’s t 2 test. the results of analysis revealed that statistical test of t 2 ≤ t 2 α was constructed (t 2 = 0.670 and t 2 α = 6.552). statistical test of f ≤ fα was also derived (f = 0.214 and fα = 2.816). this was also supported by p-value ≥ α = 0.05 and therefore not rejected h0. hence, with 5% significance level, both populations had equal and balanced early abilities. all parts that had changed after receiving treatments in experimental class were influenced by the treatments provided. table 1. univariate normality test of population data source dependent variable n d dα p-value result control class mathematical abstraction 26 0.112 0.171 0.200 * not rejected h0 motivation 26 0.101 0.171 0.200 * not rejected h0 independence 26 0.110 0.171 0.200 * not rejected h0 experimental class mathematical abstraction 22 0.089 0.183 0.200 * not rejected h0 motivation 22 0.127 0.183 0.200 * not rejected h0 independence 22 0.074 0.183 0.200 * not rejected h0 research data were data of students’ mathematical abstraction ability, motivation, and learning independence which they obtained after experiment. the data were demon and served as dependent variables to test research hypothesis. the research hypothesis test was carried out using multivariate technique, particularly hotellings t 2 test statistics. the followings are the statistical discussions of the test analysis results. the results of analysis demonstrated at table 1 showed that both samples constructed test statistics of d ≤ dα and therefore not rejected h0. this was supported by pvalue ≥ α = 0.05 for all samples. hence, with the significance level of 5%, both samples were obtained from univariate normally-distributed populations. table 2. multivariat normality test of population of research data data source n test statistics bα p-value result control class 26 b1,p 0.6769 3.780 0.9830 not rejected h0 b2,p 12.0565 lower = 11.440 upper = 17.420 0.1706 experimental class 22 b1,p 1.1482 4.623 0.9374 not rejected h0 b2,p 11.5547 lower = 11.220 upper = 17.280 0.1402 table 2 clearly as showed that both samples appeared to construct test statistics of b1,p ≤ b1,p,1-α,n and b2,p,α/2,n ≤ b2,p ≤ b2,p,1-(α/2),n,. this was supported by p-value ≥ α = 0.05 for murtianto, sutrisno, nizaruddin, & muhtarom, effect of learning using mathematica software … 224 all samples and therefore not rejected h0. hence, with the significance level of 5%, both samples were obtained from multivariate normally-distributed populations. table 3. homogeneity test of population variance of research data dependent variable data source si 2 f fα p-value result mathematical abstraction control class 64.135 1.0083 lower = 0.4378 upper = 2.3558 0.9942 not rejected h0 experimental class 63.610 motivation control class 63.335 1.0235 lower = 0.4245 upper = 2.2840 0.9465 not rejected h0 experimental class 64.824 independence control class 65.433 1.0245 lower = 0.4378 upper = 2.3558 0.9642 not rejected h0 experimental class 63.870 table 3 indicated that all dependent variables appeared to construct test statistics of , and therefore not rejected h0. this was also supported by p-value ≥ α = 0.05 for all variables. hence, with the significance level of 5%, both populations constructed homogenous variances for each dependent variable. table 4 clearly as showed that the revealed box’s m = 3.160. the value could substantively be interpreted with the assistance of critical value table of box’s m test in one condition that each cell in factorial design had equal sample size. however, the condition was not met in this research, and therefore box’s m value could not be interpreted directly. for the purpose of interpretation, f approach was applied. table 4 indicated that all dependent variables appeared to construct test statistics of f ≤ fα and therefore not rejected h0. this is proven by p-value ≥ α = 0.05. hence with the significance level of 5%, both populations had homogenous covariance matrices. table 4. homogeneity test of population covariance matrices of research data data source si box’s m f fα pvalue result control class 64,135 6,895 11, 532 6,895 63, 335 7,188 11, 532 7,188 65, 434              3.160 0.489 2.101 0.817 not rejected h0 experimental class 63, 610 25, 325 3, 541 25, 325 64,825 16, 589 3, 541 16, 589 63,870            volume 8, no 2, september 2019, pp. 219-228 225 table 5. summary of average research data dependent variable class total average control experimental mathematics abstraction (y1) 63.15 77.09 69.54 motivation (y2) 79.85 89.41 84.23 independence (y3) 76.08 85.18 80.25 table 6. summary of multivariate t 2 test of research data source of variance sscp matrices t 2 t 2 α f fα pvalue result treatments 2314.714 1588.344 1512.171 1588.244 1089.776 1037.579 1512.171 1037.579 987.881           h 91.346 6.552 28.538 2.816 0.000 rejected h0 residual, error 2939.203 704.203 362.671 704.203 2944.703 168.671 362.671 168.671 2977.119             e total 5253.917 884.141 1149.500 884.141 4034.479 1206.250 1149.500 1206.250 3965.000           t on the basis of multivariate test through spss, pillai’s trace test resulted in f = 28.538 with p = 0.000; wilk’s lambda test resulted in f = 28.538 with p = 0.000; hotelling’s trace produced f = 28.538 with p = 0,000; and roy’s largest root test produced f = 28.538 with p = 0.000. all tests constructed p < α = 0.05, and hence, h0 was rejected. this research focused more on hotelling’s t 2 test statistics. the summary of analysis is presented in table 6. in reference to results of analysis, test statistics of t 2 > t 2 α was constructed (t 2 = 91.346 and t 2 α = 6,552). this was supported by p-value < α = 0.05. therefore, with significance level of 5%, h0 was rejected. this indicated different effects of conventional learning and learning using mathematica software on students’ mathematical abstraction, motivation, and independence in learning analytic geometry with α = 0.05. in order to find out the differences, further test was carried out using univariate t test on each dependent variable and the summary of analysis is presented in table 7. table 7. summary of univariate t test of research data dependent variable t df αt f α f p-value result mathematical abstraction (y1) 6.019 46 lower = -2.013 upper = 2.013 36.226 4.052 0.000 rejected h0 motivation (y2) 4.126 46 lower = -2.013 upper = 2.013 17.024 4.052 0.000 rejected h0 independence (y3) 3.977 46 lower = -2.013 upper = 2.013 15.264 4.052 0.000 rejected h0 murtianto, sutrisno, nizaruddin, & muhtarom, effect of learning using mathematica software … 226 based on the results of further test on each dependent variable, test statistics of t < t1-(α/2) was constructed, and this resulted in .dkt in addition, test statistics of f > fα was obtained, and this constructed .dkf  this was also proven by p-value < α = 0.05. in conclusion, h0 was rejected in each dependent variable with the significance level of 5%. to derive conclusion related to the different effects, table 5 needs to be observed, particularly on the average of compared cells. with the significance level of 5%, we can conclude that: 1) learning using mathematica software ( was more effective in improving students’ mathematical abstraction than conventional learning ( was in analytic geometry subject; 2) learning using mathematica software ( was more effective in improving students’ motivation on learning than conventional learning ( was in analytic geometry subject; and 3) learning using mathematica software ( was more effective in improving students’ independence than conventional learning ( was in analytic geometry subject. 3.2. discussion thinking process is an activity occurring in human’s brain (muhtarom, murtianto & sutrisno, 2017). the use of software in learning mathematics in classroom has to be related to portion and conformity level with learning objections. this is for the use of software can influence students’ mathematical abstraction, motivation and independence. one of software well-known for mathematics learning, particularly in analytic geometry subject, is mathematica. no one can deny the benefits of this software. mathematica software is one of today’s latest software and this statement is supported by various researches carried out by gocheva (2009) and malinova (2010). malinova (2010) stated that mathematica software is highly effective for learning mathematics at the level of university which examines concept and the relation with other mathematical concepts or implementations with technology. beside that cognitive regulation variable has contributed a greater influence on cognitive variable, than that of cognitive knowledge variable (kapustina et al., 2015; pantiwati & husamah, 2017), indicating that visualization concepts of analytic geometry would be easier when presented using this software. with the use of mathematical software that displays a variety of visualizations and forms of modeling real geometry it will automatically increase student motivation in learning mathematics. with good learning motivation this will have an impact on student independence. simultaneously that mathematical software has a positive impact on mathematical abstraction abilities, motivation and independence of students' mathematics learning. abstraction ability is defined as a process of depicting a particular situation into a thinkable concept through a process of construction. the thinkable concept can later be used in more complicated and complex level of thinking. motivation in mathematics education is important for it provides energy in learning which can be adapted to bridge. motivation related to anxiety and need for achievement (rachvelishvili, 2017), which reflects the manifestation of the behavior, attitudes, and activities of an individual. thus, mathematics learning using mathematica software requires certain motivation to change behavior and attitude through a process of learning, and this is expected to put an effect on the students’ independence. according to listyani (2006), a great number of software that can be used by mathematics pre-service teachers in learning requires them to skillfully utilize software they plan to choose. mathematical software makes it easy to visualize analytic geometry concepts, with varied visualizations produced by mathematical software that will familiarize students in the process of mathematical abstraction. hence, learning volume 8, no 2, september 2019, pp. 219-228 227 analytic geometry using mathematica software is more effective in improving students’ mathematical abstraction ability, motivation, and learning independence than conventional learning. 4. conclusion learning using mathematica software puts an effect on better mathematic abstraction of students compared to conventional learning in analytic geometry subject, learning using mathematica software gives more motivation to students for studying compared to conventional learning in analytic geometry subject and learning using mathematica software is more effective in improving students’s learning independence compared to conventional learning in 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(2017). self and peer assessments in active learning model to increase metacognitive awareness and cognitive abilities. international journal of instruction, 10(4), 185-202. rachvelishvili, n. (2017). achievement motivation toward learning english language in modern educational context of georgia. problems of education in the 21st century, 75(4), 366-374. rencher, a. c. (2002). methods of multivariate analysis. canada: john willey & sons. inc. publications. sunandar, s., murtianto, y. h., & sutrisno, s. (2015). development of teaching-assisted software mathematica in developing student mathematics representation ability. semarang: universitas pgri semarang. vosler, d. (2009). exploring analytic geometry with mathematica. boston: academic press. https://doi.org/10.12973/iji.2017.10411a https://doi.org/10.12973/iji.2017.10411a https://doi.org/10.12973/iji.2017.10411a http://d.researchbib.com/f/5no2sdnf5hmkdilkw0njafmkzizwnkal80agpgzghjzmp1zqt0al5jmtl.pdf http://d.researchbib.com/f/5no2sdnf5hmkdilkw0njafmkzizwnkal80agpgzghjzmp1zqt0al5jmtl.pdf http://d.researchbib.com/f/5no2sdnf5hmkdilkw0njafmkzizwnkal80agpgzghjzmp1zqt0al5jmtl.pdf sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p247-258 247 hypothetical learning trajectory in realistic mathematics education to improve the mathematical communication of junior high school students ida nuraida 1 , asep amam* 2 1,2 universitas galuh article info abstract article history: received sept 2, 2019 revised sept 26, 2019 accepted sept 27, 2019 this research was motivated by the difficulties of junior high school students in linear equation system material. the focus of this research is to produce the hypothetical learning trajectory (hlt) system of linear equations based on the development of learning trajectory (lt) with the aim of research to improve students' mathematical communication skills. research method used design research with 3 phases: preliminary design, teaching experiment, and retrospective analysis. the subject of study in smp grade vii in tasikmalaya district. this research uses the instrument of communication skills test students. processing of research data using test-t. based on the results of the research obtained: (a) hlt results from the development of lt linear alignment system in rme to improve student mathematical communication skills; and (b) students who acquire rme learning have increased mathematical communication skills greater than those who acquire conventional learning. keywords: design research, hlt, learning trajectory, mathematical communication rme copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: asep amam, departement of mathematics education, universitas galuh, jl. r. e. martadinata no.150, ciamis, west java 46274, indonesia. email: asepamam85@gmail.com how to cite: nuraida, i., & amam, a. (2019). hypothetical learning trajectory in realistic mathematics education to improve the mathematical communication of junior high school students. infinity, 8(2), 247-258. 1. introduction the purpose of learning mathematics in indonesia based on the national education standards agency is that students can develop mathematical abilities. the intended abilities include the ability to understand mathematical concepts, the ability to use reasoning on patterns and traits, the ability to solve problems that include the ability to understand problems, the ability to have an attitude of appreciating the usefulness of mathematics in life, and the ability to communicate ideas with symbols, tables, diagrams, or media another to clarify the situation or problem. this explains that mathematical communication skills are very important possessed by students because by having good mathematical communication skills, students are able to verbally or in writing mailto:asepamam85@gmail.com nuraida & amam, hypothetical learning trajectory in realistic mathematics education … 248 communicate mathematical ideas or ideas with symbols, tables, graphs, diagrams, or other media to clarify the situation or problem. difficulties experienced by students are caused by several factors, including: (1) the learning approach factor, the learning approach used in learning lacks building students' thinking abilities and problem-solving abilities according to the opinion of kansanen (2003). some things that characterize learning practices in indonesia are generally teachercentered learning; (2) the factor of study habits, students are only accustomed to learning by memorizing, this method is not so training in thinking skills and problem-solving abilities, this method is the result of learning that teachers normally use is conventional learning, where students are crammed with concepts, examples of questions, then the matter of practice according to anwar, budayasa, amin, & de haan (2012). related to efforts to improve communication skills, teachers are expected to think about the hypothetical learning trajectory (hlt) through realistic mathematics learning (rme) that will occur in mathematics learning, so that communication skills can be improved, prahmana & suwasti (2014) rme learning is learning that has 4 aspects namely using context, using models, interactivity, utilizing construction results, and interrelationships (nuraida, 2017, 2018). rme is an approach to learning mathematics. learning that uses a realistic mathematical approach does not begin with teaching formal mathematics, but rather to appreciate and understand the importance of mathematics as a human activity. the learning process is carried out in stages through the initial knowledge of mathematics that students already have, presenting problems and the results obtained through vertical and horizontal mathematical processes called mathematical progressives. in the principles of realistic mathematics learning, horizontal mathematical consists of three levels, namely: (1) mathematical world orientation; (2) material models; (3) building stone number relation. rme is a teaching approach that starts from 'real' things for students, emphasizes process skills, discusses and collaborates, argues with classmates so they can find their concepts in solving mathematical problems. one of the main principles of the rme is guided reinvention. according to the principle of reinvention in learning mathematics, it should have endeavored that students have experience in discovering their various concepts, principles, or procedures, with the guidance of the teacher. rme has three main principles formulated by gravemeijer (2004), namely: (1) guided reinvention and progressive mathematizing; (2) didactical phenomenology is a phenomenon; and (3) self-developed model. guided reinvention and progressive mathematizing is rediscovering through progressive mathematization, which states that rme-based learning must provide the broadest opportunity to rediscover concepts or algorithms. students can think from informal mathematics to move towards formal mathematics with the capital of understanding that has been derived from previous student knowledge. the teacher's task in terms of learning is as a companion and guide to go to a mathematical concept that will be found again. didactical phenomenology is a phenomenon that was the character of educating. expected phenomena that are educational in nature that emphasize the importance of contextual problems delivered to students following the level of knowledge students have when learning occurs. learning will be meaningful if the contextual issues of education are following the resolution of contextual problems in learning. self-developed model are models developed by students must be able to bridge between informal knowledge and formal knowledge. students independently develop mathematical models in solving contextual problems. learning rme, which acts as a key or starting point in problem-solving, is contextual. volume 8, no 2, september 2019, pp. 247-258 249 realistic mathematics (rme) is an approach to learning mathematics, which was initially applied by freudhental in the netherlands in 1971. furthermore, treffers (zulkardi & putri, 2010) defines it as follows: 1) mechanistic, this approach is often referred to as the traditional approach based on drill and practice and patterns. this approach regards students as a machine (mechanic), 2) empirically, this approach assumes that the world is realistic, which makes students faced with a situation that requires them to use horizontal mathematical activities, 3) structuristic, this approach is based on set theory and games that can be categorized into horizontal mathematical. but it is determined from the world created according to needs, which have nothing in common with the world of students, 4) realistic, which is an approach that uses real-world situations or a context as a starting point in learning mathematics. at this stage, students do horizontal mathematical activities, which is when students organize a problem and try to identify the mathematical aspects that exist in the issue. then, using vertical mathematical students arrive at the concept formation stage. realistic mathematics combines what mathematics views are, how students learn mathematics, and how mathematics should be taught. to design a learning model based on pmri theory, the model must present pmri characteristics both on objectives, material, methods, and evaluation (armanto, 2002). rme has five characteristics, namely: 1) using context, context is the environment experienced by students in daily life; 2) using a model, the model is taken from everyday life both real and that students can imagine, then directs the model to a more abstract symbol; 3) using student contributions, in principle the constructing material is students themselves, in this case, student contributions are needed, so students are encouraged to be active in learning; 4) interactivity, in this learning cooperation is needed, both between students and students and between students and teachers, so that communication occurs in learning; 5) integrated with other learning topics, discussion of certain material is related to other knowledge, so that learning will be effective and efficient (armanto, 2002). the ability of mathematical communication is the ability to convey mathematical ideas/ideas, both orally and in writing as well as the ability to understand and accept mathematical ideas/ideas of others carefully, analytically, critically, and evaluatively, to sharpen the understanding. indicators of mathematical communication skills include: connecting real objects, pictures, and diagrams into mathematical ideas; explain numerical or verbal concepts, situations, and relationships with real objects, images, graphics, and algebrastates everyday events in mathematical language; listening, discussing, and writing about mathematics; read with an understanding of a written mathematics presentation; compile accurate questions that are relevant to the problem situation; make conjectures, arrange arguments, formulate definitions, and generalizations. also, it also states that mathematical communication skills are abilities that can include and contain various opportunities to communicate in the form of: reflect real objects, pictures, and diagrams into mathematical ideas; model the situation or problem using oral, written, concrete, graphical, and algebraic methods; state everyday events in precise language or symbols; listen, discuss, and write about mathematics; read with an understanding of a written mathematical presentation; make connectors, arrange arguments, formulate definitions, and generalizations; explain and make questions about mathematics that have been learned. the indicators of mathematical communication in this study are (1) stating a situation into a mathematical model, (2) creating a problem situation in its own language, and (3) stating a mathematical idea in writing. nurdin (2011) states that mathematical communication skills are the ability to organize mathematical thoughts, communicate mathematical ideas logically and clearly to others, analyze and evaluate mathematical nuraida & amam, hypothetical learning trajectory in realistic mathematics education … 250 thoughts and strategies used by others, and use mathematical language to express ideas appropriately. indicators of mathematical communication skills are as follows: arrange and consolidate their mathematical thinking through communication; communicate their mathematical thinking logically and clearly with other students or with the teacher; analyze and evaluate the mathematical thinking and strategies of others; use precise language to express mathematical ideas appropriately. the indicators of communication skills to be achieved in this study: states the system statement of linear equations (drawing aspects); explain the strategy for solving a problem of systems of linear equations (writing elements); presenting the solution of problems of linear equation systems in detail and correctly (aspects of mathematical expression). student activities are organized so that they can interact with each other for discussion, negotiation, and collaboration. in this situation, they have the opportunity to work, think, and communicate about mathematics. the role of the teacher is limited to the facilitator or supervisor, moderator, and evaluator. in the evaluation material, it is usually made in the form of open-ended questions that lure students to answer freely and use a variety of strategies or a range of answers or free productions. the evaluation must include formative or summative learning, final unit, or topic. in learning mathematics, by using realistic mathematics, students are expected to be able to come up with models and develop them independently. model development can be developed using informal models and formal models that are known to them. beginning with solving contextual problems from real situations that are known and then found 'models of' (models of) these situations (informal forms), and then followed by the discovery of 'models for' (models for) these forms (formal forms), so found the solution to the problem sought in the way of mathematical knowledge standards. students learn from the real situation stage, the modeling stage (reference), generalization, and end the formal stage (gravemeijer, 2004). the use of a model of and model for explained through the theory put forward by bruner (yağcı, 2010) that children obtain information through three stages, namely: enactive stage, namely the stage of learning a knowledge that is learned effectively by using concrete objects or real situations; the iconic stage, namely the stage of learning a knowledge by presenting it in the form of visual imagery, images, or diagrams which are representations of concrete objects at the enactive stage; symbolic stage, namely the learning stage of a knowledge that is presented in the form of abstract symbols, namely the arbiter symbols used based on the commitment of experts in the field concerned. according to prahmana & suwasti (2014), the development of hlt is formulated in three components, namely: 1) learning objectives; 2) learning instruments to be used; 3) hypothetical learning process that anticipates how students' mathematical thinking processes are developed. based on this, in developing a learning design it is necessary to formulate a hlt (gravemeijer, 2004). the term learning trajectory (lt) is called hlt because the design is still in the form of guesses or hypotheses. lt is used to describe the transformation that results from participating activities in learning mathematics, also used for a series of learning or learning trajectories. hlt is used as part of what is called the mathematics teaching cycle (mathematical learning cycle) for one or two, even more, learning. hlt consists of three components: learning objectives, which define the direction (learning objectives), learning activities, and hypotheses of the learning process to predict how students' minds and understanding will develop in the context of learning activities (bakker & van eerde, 2012, 2015). volume 8, no 2, september 2019, pp. 247-258 251 there is another term for lt, which is mathematical learning trajectory (mlt), because lt has been pursed in learning mathematics so that the name from lt becomes mlt. clements & sarama (2009) said that mathematical learning trajectory has three important parts, namely: (1) mathematics learning goals to be achieved, (2) developmental trajectories that will be developed by students in achieving learning objectives, and (3) a set of learning activities or assignments that are appropriate to the stages of thinking on the developmental path that will assist students in developing their thought processes even to higher-order thinking processes. hlt plays a role in each learning phase, the role and position of the hlt in each stage: (1) preparation and design stage: at this stage, hlt is designed to guide the design process of learning materials to be developed and adapted. the confrontation between general thinking and concrete activities often leads to more specific hlt, (2) experimental design stage: during the learning experiment, hlt serves as a guideline for what teachers will focus on in the learning process, interviews, and observations. the teacher needs to adjust the hlt with learning activities for learning meetings. hlt can change during the teaching experiment phase, (3) restrospective analysis stage: at this stage hlt acts as a clue in determining the focus of the analysis, because predictions are made related to student learning processes, it can be compared between the anticipation of predictions through observation during learning experiments (teaching experiment). this analysis concerning the interplay between hlt and empirical observations can be the basis for forming the theory. hlt was reformulated based on the findings of views and analyses conducted. the new hlt will be a clue to the next design phase (bakker & van eerde, 2015) based on this, hlt or lt is a learning trajectory that must be passed by students to achieve the desired learning goals, conduct learning activities according to the stages of thinking and development of the learning trajectory, obtain learning hypotheses to be made the didactic design in anticipating didactic situations especially learning obstacle. it can be concluded that hlt or lt or mlt is a learning trajectory whose contents are similar to those contained in the learning implementation plan (lip), because in lt or hlt or mlt it contains things that are in the lip, such as learning objectives to be achieved, learning activities or tasks that fit the stage of thinking. the difference is that the lesson plan does not contain hypotheses or alleged students' understanding. in the design of learning activities, the learning trajectory plan contains the allegations made by the teacher and is expected to get a response from students for e ach stage in the learning trajectory. these assumptions are described based on each meeting of an instructional activity plan called the learning trajectory plan (gravemeijer, 2004). a learning trajectory plan includes 1) learning objectives for students, 2) planned learning activities, and 3) an alleged learning process, where the teacher anticipates the development of their mathematical knowledge in class and how students' understanding develops as they engage in learning activities in their groups (cobb, confrey, disessa, lehrer, & schauble, 2003). 2. method this research uses a design research method, a technique that aims to design hypothetical learning trajectory (hlt) cobb et al. (2003). and assessing the improvement of students' communication skills. gravemeijer & cobb (2006) state that there are 3 phases in the implementation of design research, namely: (1) initial design; (2) design of the experiment; and (3) retrospective analysis. nuraida & amam, hypothetical learning trajectory in realistic mathematics education … 252 2.1. phase 1 – initial design (preparing for the experiment/preliminary) the preliminary design functions to implement the initial ideas obtained from the literature review on the subject matter, namely curvature space building, rme approach, curriculum, and design research as the basis for formulating hypotheses of students' initial strategy in learning curvaceous material. next is the hypothetical learning trajectory (hlt) designation, which is a series of learning activities to construct curved side spaces using the rme approach in which the hlt contains three aspects. simon (bakker & van eerde, 2015) mentions elements contained in hlt, namely: 1) learning objectives; 2) learning activities; and 3) prediction of student thinking. this prediction is dynamic so that it can be adjusted to students' reactions during learning and revised in the teaching experiment process. the forecast is used as a guide to anticipate students' thoughts and strategies that emerge and can develop in learning activities. the results of the study from phase 2 are in the form of design of learning activities to achieve learning objectives that have been made at each stage of learning and prediction of the trajectories of student activities in achieving learning objectives. the prototype test was held at the tasikmalaya regency middle school with a total of 12 students (2 low ability students, seven medium ability students, and three high ability students) from november 2018 to january 2019. the material under study was the linear equation system. the first cycle is to test the hlt on the linear equation system material contained in the 2013 curriculum. this stage is a design trial. 2.2. phase 2 – design of the experiment (teaching experiment) simon's (bakker & van eerde, 2015) state that "mathematical teaching cycle," suggests that teachers should try to guess before the occurrence of students 'mental activities (thought experiment), then try to find the students' thought processes that are actually related to those suspected in the teaching process (teaching experiment). therefore, the design of this experiment is divided into two experiments: (1) a pilot experiment, which is a bridge between the initial design phase and the teaching experiment. the objectives of the pilot activity are: (a) tracing the student's fundamental knowledge, (b) collecting data to support the adjustment of the previous learning trajectory plan. (2) the teaching experiment aims as data collection to answer research questions. in the teaching experiment phase, researchers conducted learning in the experimental class with rme learning, while the control class with conventional learning. in contrast to stage 1, in step 2 the researchers took one school to be a place of research. the second cycle is carried out from january to march 2019. based on the findings of a retrospective analysis in the second stage, researchers obtain the final recommendations that produce a prototype. 2.3. phase 3 – retrospective analysis in this phase, the data obtained from the teaching experiment phase are analyzed, and the results of the study are used to plan activities and develop activity plans for further learning. in general, the purpose of retrospective analysis is to build lit. this phase may or may not use the grounded theory method. grounded theory, namely the development of arguments based on data obtained systematically and analyzed in a social framework (mills, bonner, & francis, 2006). grounded theory research there are three sequential stages, namely: 1) open coding; 2) selective coding, and 3) theoretical coding (jones & alony, 2011). researchers at this phase do not use grounded theory because of limited volume 8, no 2, september 2019, pp. 247-258 253 time, funds, and facilities. the phases in this design research are summarized as follows in figure 1. figure 1. stages of research design there are two stages in the study that have been carried out, namely step 1 includes prototype design, hlt trials, and retrospective analysis. phase 2 consists of the provision of realistic mathematics learning to see an increase in students' mathematical communication skills. the following is a chart of research methods (figure 2). figure 2. research chart initial design design of the experiment retrospective analysis hypothetical learning trajectory (hlt) nuraida & amam, hypothetical learning trajectory in realistic mathematics education … 254 in stage 1, there are three main steps that must be carried out, namely the preliminary design stage, the second step is the teaching experiment, and the third step is the retrospective analysis. in phase 2, learning is done in the classroom to see the communication skills of the students obtained. 3. results and discussion 3.1. results in this study the research findings are described which are generally divided into two stages in accordance with the design research stages, namely the first stage begins with the preliminary design, teaching experiment, and ends with a retrospective analysis that has been validated and tested in a limited way (cobb et al., 2003). this research was conducted by researchers to determine the alleged learning trajectory and see differences in the improvement of students' communication skills between rme learning and conventional learning. the study was conducted at rajapolah state junior high school 2 in class viii, namely class viii b as an experimental class and class viii c as a control class using purposive sampling techniques, this aims to determine the presence or absence of differences in the improvement of students' mathematical communication skills after the learning process is carried out. based on observations during the study, it was seen that rme learning is learning that can improve student communication skills. in direct learning, the teacher dominates the class more during the learning process, and students look less active, and learning takes place that is less interesting and boring. some students seem to lack enthusiasm in learning mathematics, it is seen when students are given practice questions, some students are reasonable in order to get out of class to avoid the training given. after the preliminary stage is carried out, subsequent stages of a teaching experiment, thus experiencing some test results. to see the improvement of mathematical communication skills of students obtained as in the following table 1. table 1. test-t improved mathematical communication mean t sig. h0 rme konv 0.52 0.31 4.99 0.000 rejected 0.66 0.37 3.59 0.002 rejected 0.34 0.26 2.31 0.031 rejected the t-test result in table 1 shows the significant value of all less than 0.05, then the zero hypothesis rejects. it is concluded that increased communication skills using rme learning is better than the student's using conventional learning. based on the test results of the average difference and the magnitude of the average value of achieving communication skills between the two learning groups (rme and conventional), it can be concluded: (1) achievement of communication skills of students who get rme learning better than students who get conventional learning; (2) improved communication skills are obtained from the n-gain calculation between pretest and posttest scores. the calculation results of the improvement of communication skills of the two learning groups are then searched for the mean and the standard deviation; (3) a volume 8, no 2, september 2019, pp. 247-258 255 comparison of increased communication skills between the rme and conventional groups were analyzed using the mann-whitney u test. the reason for using the mann-whitney u test was because after the data were tested for normality, it was found that the data were not normally distributed. 3.2. discussion the preliminary design that has been done is the study of literature, which is at the same time a process of discussion with experienced mathematics subjects in mathematics learning the system of linear equations. the learning design study used in the next stage is the teaching experiment stage. hlt has three components, namely: learning objectives, activities, and learning hypotheses (cobb et al., 2003). the following are presented alleged learning trajectories in table 2. table 2. hlt examples in learning linear equation systems learning objectives learning activities learning hypothesis hlt examples in learning linear equation systems learning objectives learning activities learning hypothesis students can find the concept of one variable linear equation system in daily life. find the concept of one variable linear equation. rahma was told by her mother to buy a hairpin with rp. 10,000 and get a change of rp. 4,000 possible questions asked to students. teacher: how many hairpins can rahma buy? student 1: one-piece teacher: how are the similarities? student 1: a + 4,000 = 10,000. teacher: are there other answers? student 2: 2 pieces teacher: write down the equation students 2: 2a + 4,000 = 10,000. furthermore student 3 answers 3 pieces, student 4 answers 4 pieces, student 5 answers 5 pieces and so on. the equation in a row: 3a + 4,000 = 10,000 4a + 4,000 = 10,000 5a + 4,000 = 10,000 6a + 4,000 = 10,000 students are able to recognize the form of the system of equations and are able to improve communication skills. students can recognize the form of a system of equations and can improve their communication skills. based on students' activities in discovering the concept of a single variable linear equation system in daily life can vary in the answer nuraida, kusumah, & kartasasmita (2018), because it is an open matter. teachers should have many allegations that happened nuraida & amam, hypothetical learning trajectory in realistic mathematics education … 256 to the student's activities (wilson, sztajn, edgington, & confrey, 2014). the expected objective of the matter is that students discover the concept of equations so that students can use the "=" sign. 4. conclusion the results of this study can be concluded that the alleged study trajectory that will occur in students' activities is very much so that the teacher, in this case, should prepare a more mature trial. this research is certainly not separated from the constraints experienced by students from the preliminary stage to the retrospective stage. these constraints exist at the preliminary stage because of not ready teachers in designing learning paths. the communication skills of students using rme learning are better than students using conventional learning. acknowledgements we thank to drpm kemenristek dikti that has given funding for research; 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(2021). profile of prospective teachers' mathematical communication ability reviewed from adversity quotient. infinity, 10(1), 93-108. 1. introduction the 21st century is a century marked by the occurrence of a massive transformation from an agrarian society to an industrial society and continues to a knowledgeable society (soh, arsad, & osman, 2010). life in the 21st century requires a variety of skills that must be mastered by someone, education is becoming increasingly important to ensure students have learning and innovation skills, skills to use technology and information media, and can work, and survive using life skills (wijaya, sudjimat, & nyoto, 2016). https://doi.org/10.22460/infinity.v10i1.p93-108 muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 94 scott (2015) states in the international commission on education for the twentyfirst century proposes four visions of learning, namely knowledge, understanding, competence for life, and competence to act. in addition to this vision, four principles known as the four pillars of education are formulated, namely learning to know, lerning to do, learning to be and learning to live together. fridanianti, purwati, and murtianto (2018) state that strengthening character education in schools must be able to foster student character to be able to think critically, creatively, be able to communicate, and collaborate, who are able to compete in the 21st century. this is in accordance with the four competencies that students must have in the 21st century which is called 4c, namely critical thinking and problem solving, creativity, communication skills, and the ability to work together. communication is one of the skills in learning to do, oral and written communication skills contribute to career development in the 21st century. the results of an analytical study conducted by wardhani & rumiati (salam, 2017), the cause of the low mathematics achievement of indonesian students in the 2015 timss results is due to the weakness of indonesian students in working on questions that require several abilities, one of the abilities needed is the ability to communicate in mathematics. this can be caused by student confusion in presenting ideas or ideas in the form of symbols, graphs, tables or other media to clarify math problems. the results of the 2018 pisa assessment (nugrahanto & zuchdi, 2019) show that the mathematical abilities of students in indonesia are still low. one of the low mathematical abilities is mathematical communication skills, this can be caused by student confusion in presenting ideas or ideas in the form of symbols, graphs, tables or other media to clarify mathematical problems. ulfa, buchori, and murtianto (2017) stated that in general the process of learning mathematics in the classroom is teacher-centered. this is in line with hampson, patton, and shanks (2011) who state that high-quality teachers are those who have a strong influence on student achievement. the ability to communicate in learning activities is said to be good if the ability of a teacher and lecturer to create a communicative climate, where between lecturers and students or teachers with students as subjects are actively involved in learning activities, both verbally and nonverbally, in other words this communicative climate as a vehicle for the implementation of learning in accordance with the design and achieving learning objectives (son, 2015). it would be better if the provision of mathematical communication skills is integrated in every lecture. to realize good students’ mathematical communication skills given by the teacher, trained or prepared since becoming prospective teacher. son (2015) also adds, of course it is not effective and efficient if prospective mathematics teacher students only get a theory of mathematical communication in a subject without getting enough opportunities to practice it, it would be better if the provision of mathematical communication skills is integrated in every lecture. hapsari, nizaruddin, and muhtarom (2019) state that teachers play a very important role in improving the quality of learning and learning outcomes that will be achieved by students before going to a higher level. many students still have imperfect mathematical communication skills. paradesa and ningsih (2017) states that the ability of students in the aspect of mathematical communication seen from the ability to provide mathematical evidence in the form of facts and data is still experiencing difficulties. if it is related to the problem of mathematical communication skills, the type of intelligence can be used, namely adversity quotient (aq). aq is often identified with fighting power against adversity. aq is considered to be able to support student success in increasing achievement motivation (hidayat & husnussalam, 2019; hidayat, noto, & sariningsih, 2019; hidayat, wahyudin, & prabawanto, 2018). many studies have been conducted to see the effect of aq, including: hidayat, herdiman, aripin, yuliani, and maya (2018) who try to improve aq and student teacher student mathematical creative reasoning, stating that aq has a positive influence on the volume 10, no 1, february 2021, pp. 93-108 95 development of students' mathematical creative reasoning abilities prospective teacher. kartika and yazidah (2019) also tried to analyze the ability of mathematical proof in real analysis courses based on aq, stating that climbers’ students are more able to compile direct evidence than quitters and campers students. paramita (2017) also conducted research on mathematical communication skills in terms of aq through the application of the scss learning model in students class viii, showing that quitters tend not to be able to meet all indicators of mathematical communication skills, campers subject tends to be able to fulfill two indicators, namely the ability to state a situation to in mathematical language and the ability to visualize mathematical ideas, the climbers subject was able to fulfill all indicators. yuniarti (2015) also conducted research on the analysis of the results of the diagnostic assessment of mathematical communication skills in osborn learning based on aq, and the results showed that the quitter category student subjects had not been able to fulfill almost every mathematical communication indicator, the camper category was quite capable in several indicators of mathematical communication, and the climber category. based on the above explanation that aq has a significant effect in determining the success of students' mathematical communication skills, therefore the mathematical communication skills of students who have high aq or students with climbers level will be different from the mathematical communication skills of students who have aq at the campers and quitters level. thus the purpose of this study is to determine and investigate in depth the aq profile of prospective mathematics teacher students on mathematical communication skills. 2. method the method used in this research is descriptive qualitative research method using written and oral data. because when the research was being carried out, it was during the covid-19 pandemic, social distancing, and work from home, so this research was carried out online, where the aq questionnaire was filled out via google form, and a written test of mathematical communication was carried out via the whatsapp group video call, while interviews were conducted via whatsapp call. the subjects defined in this study were 3 students at 6th semester of the mathematics education study program of the universitas pgri semarang class of 2017 including one student with aq quitters, one student with aq campers, and one student with aq climbers. this study used purposive sampling. sugiyono (2008) states that purposive sampling is a technique of sampling data sources with certain considerations, with the consideration that the person we choose is considered to know best about what we expect, making it easier for researchers to explore the object or social situation under study. the instruments used in this study included the aq questionnaire, the mathematical communication skills test sheet, and the interview guide. the aq questionnaire for sixth semester mathematics education students was given to two classes via google form and obtained 57 respondents. the aq questionnaire instrument was adapted by stoltz (2000) and has been validated by one counseling lecturer and three mathematics lecturers at the universitas pgri semarang. this questionnaire was conducted to select 3 students with the categories quitters (mm), campers (kal), and climbers (ndc). in this study, the climber subject was taken with the highest questionnaire score in the climbers category, the camper subject was taken with the middle questionnaire score in the campers’ category, and the quitter subject was taken with the lowest questionnaire score in the quitters’ category. after determining each subject in the aq category, then an online written test was carried out through the whatsapp video call group for students who had the intelligence of quitters, campers, and climbers. the questions given consist of one story item on calculus muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 96 material, which has been validated by three mathematics lecturers. indicators of mathematical communication skills used include: 1) writing, which is to provide answers using your own language or problems using writing and algebra, listening to, discussing and writing about mathematics, and being able to explain ideas or situations from a picture or graphic with words itself in writing; 2) drawing, namely reflecting real objects, pictures, and diagrams into mathematical ideas and vice versa, and expressing a situation with pictures or graphs; 3) mathematical expressions, namely expressing mathematical concepts by expressing everyday events in mathematical language or symbols, and expressing a situation in the form of a mathematical model. before conducting the interview, the researcher checked back one by one the answers of each subject and checked the location of the truth and error in each indicator of mathematical communication. interviews were conducted online via whatsapp calls to get more in-depth information about the mathematical communication forms possessed by these students. the interview instrument was validated by three mathematics lecturers. the interviews were conducted for approximately 10-20 minutes. to maintain the validity of the data in this study, triangulation was used. the triangulation used was method triangulation. after obtaining the results of the analysis of the written test answers and the interview data analysis, then a comparison is made to determine whether the data obtained is valid or not. and the result states that all data for climbers, campers, and quaitters subject can be said to be valid. 3. results and discussion 3.1. results the first step was to determine the students as categories climbers, campers, and quitters. from the aq questionnaire that has been distributed, it was obtained from 57 respondents that 3.51% of students with aq quitters, 0% of students with aq quitters to aq campers, 31.58% of students with aq campers, 57.89% of students with aq campers to aq climbers, and 7.02% of students with aq climbers as in the following figure 1. figure 1. graph of the number of students for each aq 0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% quitters quitters to campers campers campers to climbers climbers volume 10, no 1, february 2021, pp. 93-108 97 after selecting 3 students with the categories climbers, campers, and quitters, then the three students were given questions on communication skills tests and interviews. instruments used for mathematical communication skills include drawing, writing, and mathematical expression. 3.1.1. ndc subject the ndc subject fulfills following the indicators of mathematical communication drawing skills as shown in figure 2. ndc subject can state the problem in the form of an image correctly and accurately and provide information on the length, width, and height of the problem in the question. figure 2. answers to drawing the ndc subject based on the results of the interviews conducted, the ndc subject is able to meet the indicators of mathematical drawing communication skills and can smoothly explain problems into the form of images correctly and accurately and is able to smoothly explain the length, width, and height of the drawings he has made in his answer sheet. an excerpt from the interview with the ndc subject is presented as follows: researcher : what steps do you take? ndc : this is what is asked for the maximum volume, so the first thing to do is to draw a square first, there are 4 edges, so all of them are cut, so it turns out to be a picture that is 24-2a in length, 9-2a in width and a height. the ndc subject fulfills following the indicators of mathematical communication writing skills as shown in figure 3. ndc subjects can use mathematical language appropriately and correctly, and are able to explain ideas or situations from images that have been made previously in their own words in writing. the subject takes his own side in written form, the subject considers the side of the square which is cut off with the symbol "a", and also write an explanation in determining the interval "a"correctly. muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 98 figure 3. answers to writing the ndc subject based on the results of the interviews conducted, the ndc subject is able to meet the indicators of writing mathematical communication skills and can explain fluently an idea or situation from a previously made image and can also explain the example of the square side used, and be able to state and explain how to determine the interval to meet the volume maximum sought. here are excerpts of interviews with the subject ndc: researcher : yes... then after that? ndc : so after that, suppose the square side is cut identically is “a”, then the length is 24-2a, the width is 9-2a and the height is “a”. researcher : then how to determine the maximum volume how? ndc : to determine the maximum volume with v''= 0 researcher : yes... continue? ndc : to determine the a interval it is 0 < a < 4.5 researcher : the reason? ndc : you see, so there is a value, sis, the height is a, then the “a” is less than 0. researcher : where did you determine the 4.5 from? ndc : that's from the width, the width is 9-2a = 0, we move the segment so 9/2 = a, so 4.5 = a. the ndc subject fulfills the following indicators of mathematical communication mathematical expression skills as shown in figure 4. ndc subjects can state mathematical solutions in writing clearly and precisely, are able to use mathematical symbols and perform calculations or get complete and correct solutions. the subject is able to determine the length of the shape she has previously made with the values 24 2a, and for the width 9 2a, and the height a. then the subject is able to write the volume formula used with v = p.l.t, the subject is also able to apply the first derived properties with v' = 0 and is able to determine the value "a" that meets the maximum volume sought, and performs calculations correctly both in calculating the initial volume, determine the equation v', find the value of a, and determine the maximum volume. volume 10, no 1, february 2021, pp. 93-108 99 figure 4. answers to mathematical expression the ndc subject based on the results of the interviews conducted, the ndc subject is able to meet the indicators of mathematical expression and can explain mathematical solutions clearly and precisely, and is able to explain mathematical calculations and correct answers. an excerpt from the interview with the ndc subject is presented as follows: researcher : what do you do after that? ndc : i determine the volume first, the volume formula is p.l.t, now enter the length is 24-2a, the width is 9-2a, the height is a, after that we operate the volume, the result is 4a3-66a2+216a. researcher : then what is the next step? ndc : so after that, determine the maximum volume with v'= 0, now determining v'= 0, we will derive it from the result of the volume which was 12a2132a+216=0, so continue to divide by 12, now the result is it is equal to 2 or a is equal to 9, now for a = 9 it does not meet. researcher : why not fulfill that for what reason? ndc : because the interval was less than 4.5. researcher : how do you continue to determine the maximum volume? ndc : now the maximum volume uses the formula, which is length times width times height, now we enter the one that is known to be 24-2a in length, 9-2a in width, the height is “a”. that's what v''= 0 has already been obtained which is equal to 2, continue to be added to the formula so the maximum volume is 200 cm3. 3.1.2. kal subject the kal subject fulfills the following indicators of mathematical communication drawing skills as shown in figure 5. kal subjects can state the problem in the form of an image correctly and precisely and are able to provide information on the length, width, and height of the problem in the question. muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 100 figure 5. answers to drawing the kal subject based on the results of the interviews conducted, the subject of kal is able to meet the indicators of mathematical communication skills of drawing and can explain problems into the form of images correctly and accurately and is able to explain the length, width, and height of the drawings he has made. an excerpt from the interview with the ndc subject is presented as follows: researcher : explain the picture that you have made. kal : you draw it, the length is 24, now the width is 9, cut into a square, for example, the square is x, the right and left square is 2, so 24-2x is the length, now the width is the same, it makes 9-2x, keep making the height earlier was the x. it is clearly figure 6 shows that the results of the written work of the kal subject on the indicators of mathematical communication writing skills. kal subject can use mathematical language correctly, and is able to explain ideas or situations from images that have been previously made in their own words in written form but are still incomplete. the kal subject takes the cut side of the square with the symbol “x”. however, the kal subject did not specify the interval of “x”. figure 6. answers to writing the kal subject based on the results of the interviews conducted, the kal subject was able to meet the indicators of writing mathematical communication skills and was able to explain ideas or situations from images that had been previously made but were still incomplete, because the kal subject only explained for example the cut side of the square, namely x, but had not explained the interval from x itself. an excerpt from the interview with the ndc subject is presented as follows: volume 10, no 1, february 2021, pp. 93-108 101 researcher : explain the picture that you have made. kal : you draw it, the length is 24, now the width is 9, cut it into a square, let's say that a square is x. it is clearly figure 7 shows that the results of the written work of the kal subject on the indicators of mathematical communication expression skills. the subject of kal can clearly state mathematical solutions in writing, can use mathematical symbols, and perform calculations but is still incomplete. the subject is able to determine the length of the shape he made previously with the values 24 2x, and for the width 9–2x, and the height a. then the kal subject is able to write the volume formula used with v = p.l.t and its calculations, the subject is also able to apply the first derivative with v'= 0, but the kal subject cannot determine the maximum volume of the given problem. figure 7. answers to mathematical expression the kal subject based on the results of the interviews conducted, the kal subject is able to meet the indicators of mathematical expression communication skills and can explain the solution, but the kal subject cannot determine the maximum volume of the given problem, the subject has tried to calculate the maximum volume but the result is negative, this is because the subject did not previously specify the x interval. an excerpt from the interview with the ndc subject is presented as follows: researcher : after you draw, what steps do you take? kal : finding the volume. researcher : how? kal : use that formula, it means that the length times the width times the height, put in (24-2x) (9-2x) (x). now the result is 4x3-66x2+216x, now it's lowered. researcher : continue kal : the result means12x2-132x+216. researcher : then what is the next step? kal : that can be simplified, so it's x2-11x+18. researcher : continue muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 102 kal : so, you get x = 2 or x = 9. researcher : then after that? kal : so, i just got there, sis. researcher : why is the deck just that way? kal : the problem was that i tried to enter it, but the results were both negative. researcher : that means what you are doing only up to here? kal : yes, miss. it means that the maximum volume has not been obtained. it hasn't reached the final result value. 3.1.3. mm subject the subject of mm did not fulfill the mathematical communication indicators of drawing in solving the problem. subject did not present the data or information from the question in the form of an image. mm subject could not write an explanation of the answer to the problem mathematically and did not use mathematical language or symbols appropriately and correctly. figure 8 showed that mm subject is less able to express mathematical solutions in writing, and perform calculations but is wrong, because the mm subject solves the problem not with the volume block formula but by using the rectangular formula and the determination of the length and width values is still wrong. figure 8. answers to mathematical expression the mm subject based on the results of the interviews conducted, the mm subject is not able to meet the indicators of mathematical communication skills in mathematical expression and the mm subject explains mathematical solutions according to the answer sheet but the answer is still wrong, the mm subject is also still hesitant in answering what shapes roughly correspond to the problem in question. an excerpt from the interview with the ndc subject is presented as follows: researcher : what is the next step after you know what was being asked? mm : i multiplied the length times the width. researcher : what do you think it is up to? mm : square researcher : square? square or shape? mm : square ... rectangle. researcher : then you count the volume of the rectangle how it is? mm : length by width. the maximum volume is 216 cm2. data were also collected through in-depth interviews with the subjects of climbers (ndc), campers (kal), and quitters (mm). written test results data were compared with interview data to obtain valid data. from the research results written tests and interviews conducted by climbers subjects met all indicators of mathematical communication skills used, campers subjects tended to be able to meet all indicators of mathematical volume 10, no 1, february 2021, pp. 93-108 103 communication skills used, while quitters subjects were unable to meet all indicators of mathematical communication skills used. 3.2. discussion from the results of the tests and interviews, the researcher observed that the data obtained was sufficient, so the written test and interview were not continued to the next stage. from the analysis of written tests and interviews of mathematical communication skills, the following results are obtained: 3.2.1. mathematics prospective teacher with aq climbers based on the results of the description and analysis of the written test results, the prospective teacher with the aq climbers category can meet all indicators of mathematical communication skills used by the researcher, including drawing, writing, and mathematical expression. prospective teacher with aq climbers are able to express and describe mathematical ideas in the form of pictures, aq climbers are able to provide answers using their own language or problems using writing and algebra, and are able to explain ideas or situations from an image or graph with own words in written form. prospective teacher with aq climbers is able to state a situation in the form of a mathematical model, and is able to perform mathematical calculations correctly. this is in line with nartani, hidayat, and sumiyati (2015) improving the communication skills of mathematics indicated by students are able to express ideas or ideas with mathematics verbally sentence, students are actively involved in discussions about math, students can formulate definitions and generalizations about the math, students can formulate a definition of mathematics by using its own words. mathematical communication skills are shown by students being able to express ideas or ideas with mathematical sentences verbally, students are actively involved in discussions about mathematics, students can formulate definitions and generalizations about mathematics, students can formulate mathematical definitions using their own words. this is also in line with ansari (2012) who states that drawing communication skills are reflecting real objects, drawings and diagrams into mathematical ideas, writing is stating and explaining a mathematical drawing or model into a mathematical idea form, mathematical expression is express a situation or mathematical idea into a symbol or mathematical model and solve it. it can be concluded that the prospective teacher aq climbers is able to meet all indicators of mathematical communication skills of drawing, writing, and mathematical expression. stoltz (2000) states that the subject of climbers is a group of people who always try to reach the peak of success, are ready to face any obstacles, and always raise themselves to success. this research is in line with the research of paramita (2017), kartika and yazidah (2019), and yuniarti (2015). in paramita's research (2017) which states that the climbers subject is able to meet all indicators of mathematical communication skills including the ability to state a situation in mathematical language, the ability to describe mathematical ideas visually, the ability to explain mathematical ideas in writing, and the ability to evaluate mathematical ideas in writing. in kartika and yazidah's research (2019), which states that climbers students are more able to compile direct evidence than quitters and campers students. in research yuniarti (2015) also states that the climber category is capable of almost all indicators of mathematical communication. muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 104 3.2.2. mathematics prospective teacher with aq campers based on the results of descriptions and analysis of written test results, prospective teacher with the aq campers category tend to be able to meet all indicators of mathematical communication skills used by researchers, including drawing. writing, and mathematical expression. prospective teacher with aq campers are able to state, express and describe mathematical ideas in the form of images. aq campers tend to be able to provide answers in their own language or problems using writing and algebra, and are able to explain ideas or situations from an image or graphic in their own words in written form. prospective teacher with aq campers tend to be able to state a situation in the form of a mathematical model, but have not been able to complete it completely in finding the maximum volume value requested in the problem. this is in line with nartani, hidayat, and sumiyati (2015) improving the communication skills of mathematics indicated by students are able to express ideas or ideas with mathematics verbally sentence, students are actively involved in discussions about math, students can formulate definitions and generalizations about the math, students can formulate a definition of mathematics by using its own words. this is also in line with ansari (2012) who states that drawing communication skills are reflecting real objects, drawings and diagrams into mathematical ideas, writing is stating and explaining a mathematical drawing or model into a mathematical idea form, mathematical expression is express a situation or mathematical idea into a symbol or mathematical model and solve it. it can be concluded that the aq campers tends to be able to meet all indicators of mathematical communication skills of drawing, writing, and mathematical expression. stoltz (2000) stated that campers are a group of people who still have the desire to respond to existing challenges, but do not reach the peak of success and easily give up on what has been achieved. stoltz (2000) also adds that campers do not fully exploit their potential, campers have a limited ability to change, especially major changes, campers live with the belief that after several years or after making a number of efforts, life should be relatively free of difficulties. in this study, new things were found because the subject of aq campers tended to meet all indicators of mathematical communication skills of drawing, writing, and mathematical expression. this is not in line with previous research conducted by paramita (2017) and yuniati (2015). in paramita's (2017) research which states that campers tend to be able to fulfill two indicators, namely the ability to express a situation in mathematical language and the ability to visualize mathematical ideas only. yuniarti's (2015) study which states that the camper category is quite capable in several communication indicators. mathematically and the category of campers make process errors and conclusion errors. 3.2.3. mathematics prospective teacher with aq quitters based on the results of descriptions and analysis of written test results, the prospective teacher with the aq quitters category cannot meet all indicators of mathematical communication skills used by researchers, including drawing. writing, and mathematical expression. aq quitters is not able to meet all indicators of mathematical communication skills of drawing, writing, and mathematical expression. stoltz (2000) states that quitters are a group of people who prefer to avoid and reject opportunities, easily give up, give up easily, tend to be passive, and are not enthusiastic about reaching the peak of success. stoltz (2000) also adds that quitters have limited abilities in facing adversity, quitters tend to resist change and claim its every success, or to avoid it and actively walk away from it. the subject of quitters tends to think that the difficulties that arise will continue to occur, so that they are volume 10, no 1, february 2021, pp. 93-108 105 constantly overshadowed by obstacles that often arise, every difficulty, the cause is also considered something that will continue to appear again in the future (hidayat & husnussalam, 2019; hidayat, noto, & sariningsih, 2019; hidayat, wahyudin, & prabawanto, 2018). it is proven in this study that the quitters subject is not able to meet all the indicators requested by the researcher. this study is in line with the research of paramita (2017), and yuniarti (2015). in paramita's research (2017) which states that quitters are not able to fulfill all indicators of mathematical communication skills, including the ability to express a situation in mathematical language, the ability to visualize mathematical ideas, the ability to explain mathematical ideas in writing, and the ability to evaluate mathematical ideas in writing. yuniarti's research (2015) also states that the quitter category has not been able to meet almost every mathematical communication indicator and almost all types of errors occur in the quitters category. this is consistent with the results of this study where the quitters subject is not able to meet all indicators of mathematical communication skills including drawing, writing, and mathematical expression. the results of this study finally produce a summary of the understanding of mathematical communication skills of prospective mathematics teachers in terms of aq, as shown in the following table 1. table 1. summary of mathematical communication skills no aspect indicator category aq climbers aq campers aq quitters 1. drawing the ability to express, express and describe mathematical ideas in the form of pictures, graphs or visual mathematical models. fulfilled fulfilled not fulfilled 2. writing the ability to provide answers using your own language or problems using writing and algebra, and to explain an idea or situation from a picture or graphic in your own words in written form. fulfilled fulfilled not fulfilled 3. mathematical expression the ability to express mathematical concepts by expressing everyday events in mathematical language or symbols, and expressing a situation in the form of a mathematical model fulfilled almost fulfilled not fulfilled table 1 shows that the results of the study show that the subjects of prospective mathematics teachers who have aq climbers and aq campers are able to meet all indicators of mathematical communication skills, indicators of mathematical communication skills used include drawing, writing, and mathematical expression, while the subject of mathematics prospective teacher. those who have aq quitters are not able to meet all indicators of mathematical communication ability. the indicators of mathematical muhtarom, pratiwi, & murtianto, profile of prospective teachers' mathematical communication … 106 communication abilities used include drawing, writing, and mathematical expression. the results of each individual in communicating the problems obtained are in accordance with their aq. this is in line with syarifah, sujatmiko, and setiawan (2017), mathematical communication is the process of expressing mathematical ideas and understanding verbally, visually, and in writing, using numbers, symbols, pictures, graphs, diagrams, and words. someone. the results of this study are also in line with nopiyani, turmudi, and prabawanto (2016), mathematical communication is the ability to express mathematical ideas or ideas either in writing or in pictures. this is also in line with murtafiah (2016) that mathematical communication is the ability to express mathematical ideas through speech, writing, demonstrations, and visually depicting them in different types for each person. 4. conclusion based on the results of research and discussion that has been done with the subject of climbers, the conclusion is that students are able to solve problems using mathematical communication properly and correctly. the three indicators of communication the subject is able to meet all the indicators of mathematical communication used. the subject of campers is quite capable of solving mathematical communication problems properly and correctly, but there are calculations in resolving incomplete problems. the three indicators of communication the subject tends to be able to meet all the indicators of mathematical communication used. the subject of quitters has not been able to solve problems using mathematical communication properly. the three stages of communication, the subject tends not to be able to meet all the indicators of mathematical communication used. based on the results and conclusions of this study, the following 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(2015). analisis hasil penilaian diagnostik kemampuan komunikasi matematis dalam pembelajaran osborn berdasarkan adversity quotient (tesis). semarang: universitas negeri semarang. https://doi.org/10.30651/must.v2i2.888 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p173-182 173 comparison of algebra learning outcomes using realistic mathematics education (rme), team assisted individualization (tai) and conventional learning models in junior high school 1 masohi anderson leonardo palinussa universitas pattimura, indonesia article info abstract article history: received jul 9, 2020 revised sep 5, 2020 accepted sep 6, 2020 the study aims to examine the application of the realistic mathematics education (rme), the cooperative learning model team assited individualization (tai) type and conventional learning models on algebra. the population in this study were all eighth grade smpn 1 masohi. the type of this study is an experimental research design with quasi experimental research. instruments in this study using the test results, analyzed using anova test and further tests using tukey's hsd test. from the analysis of these study obtained data: (1) based on the normality test results obtained sig. x1 (rme learning model) of 0.976, x2 (cooperative learning model tai type) of 0.889 and x3 (conventional learning model) of 0.906. (2) based on the one-way anova calculation with the spss 20.0 program, obtained a significance value of 0.003. (3) there are significant differences in algebra learning outcomes between classes using the rme learning model, the cooperative learning model tai type and conventional learning model, and (4) the algebra learning outcomes of students used the rme learning model are higher than the students that are used cooperative learning models tai type and the students that used conventional learning model. keywords: learning outcomes, realistic mathematics education, team assitsed individualization, conventional, algebra copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: anderson leonardo palinussa, department of mathematics education, universitas pattimura jl. ir. m. putuhena, poka, tlk. ambon, maluku 97233, indonesia email: apalinussa@yahoo.com how to cite: palinussa, a. l. (2020). comparison of algebra learning outcomes using realistic mathematics education (rme), team assisted individualization (tai) and conventional learning models in junior high school 1 masohi. infinity, 9(2), 173-182. 1. introduction the world of education at the lowest level up to the highest level is basically played by mathematics as a complementary science in all disciplines. in addition, mathematics is very useful to be applied in terms of life, so it becomes a provision that must be required by students. https://doi.org/10.22460/infinity.v9i2.p173-182 palinussa, comparison of algebra learning outcomes using realistic mathematics education … 174 the difficulty of students in working on mathematical problems is inseparable from the teacher's role. according to clarke & roche (2017) teachers have an important role in the learning process. the low number of indonesian students in the 2015 pisa results reflects that learning by teachers still emphasizes the ability to read, write and count. in fact, although mathematics is a subject that plays an important role in education, there are still students who do not master mathematics. because students continue to be used as learning objects in a series of learning activities, while the teacher becomes the center of learning. ratumanan (2015) says that student activities in class are not involved in teaching mathematics. because the role of the teacher is made more dominant to learn material that is not the teacher's job. this causes students to be less active in learning, so understanding concepts about student mathematics is very lacking and poor students learning outcomes. the process of learning mathematics in the classroom found a variety of problems including, the teacher still dominates the learning process and when the teacher explains that only some students pay attention well, while other students are busy telling stories and when given questions students only apply the formula given and student activity is not yet visible in the learning process so the learning becomes meaningless. in addition, teachers only use conventional learning models, so learning in the classroom tends to be monotonous which causes students to be bored and lazy to learn, and result in weak understanding of student material, especially for material that is considered difficult. this resulted in decreased student learning outcomes. to overcome the problems that have been raised, it is necessary to choose an appropriate learning model and can be a solution to improve the quality of learning. the realistic mathematics education (rme) learning model and the team assisted individualization (tai) type of cooperative learning model can be applied to increase understanding and increase student creativity. the realistic mathematics education (rme) learning model is one of a series of classroom learning programs designed with the aim of building students' ability to know things that have not been learned through student activities. mathematical concepts become the basis for students to find them in learning mathematics guided by the teacher. problem solving in the form of contextual problems becomes the principle of students discovering the mathematical concept itself. formal mathematical knowledge is obtained by students by modeling the contextual problems they face while learning. in addition, everyday human activities become mathematics learning materials that are designed with the aim of achieving the goals of mathematics learning (gravemeijer, 1994). one of the cooperative learning models that can be used is the team assisted individualization (tai) type. tai has the meaning that students in groups or teams formed heterogeneously when they have a goal to understand the material provided by the teacher must be assisted by individuals who have good learning abilities. meanwhile, according to (siregar, budiyono, & slamet, 2018) tai has a rationale that is to achieve the ability and achievement of students to adapt to differences in individual abilities. the tai type of cooperative learning model is a learning model designed to solve problems in teaching programs, for example in terms of student learning difficulties individually and can help students be more active in the classroom because students will work together between groups in solving the given problems, students also interact with each other , help each other and complement each other. thus every student who has a low ability when assisted by students who have high abilities have the hope to improve their abilities. with this learning model, the key to success in achieving the mathematics learning goals to be achieved is to adapt students to one another. based on the background above, the problems to be investigated and discussed in this study are (1) is there a difference in student learning outcomes taught with the rme volume 9, no 2, september 2020, pp. 173-182 175 learning model, the tai type of cooperative learning model, and the conventional model on the algebraic arithmetic operations material. (2) which learning model is superior to rme learning model, tai type cooperative learning model and conventional learning models on algebraic arithmetic operations material. 2. method the type of this research used the experimental research. the research design is quasi experimental research because the researcher cannot control the variables outside the research that the researcher did not expect. the variables contained in this study are x1: student learning outcomes in mathematics taught with the rme learning model, x2: student learning outcomes in mathematics taught with cooperative learning model tai type and x3: student learning outcomes in mathematics taught with conventional learning models. the learning tools in this study are in the form of learning implementation plan (rpp), student worksheets (lks), and learning materials. determine the population to be the beginning of this study and choose a sample of the existing population. the population in this study were all students of eighth grade of junior high school 1 masohi, central maluku district. the sample selection is done by purposive sampling technique, which is the technique of determining the sample with certain considerations (bidgood, hunt, & jolliffe, 2010). there are three classes chosen by researchers as research samples and obtained the first class as an experimental class 1 using the rme learning model, the second class as an experimental class 2 using a tai type of cooperative learning model, and the third class as a control class using a conventional learning model. data obtained from the results of research in the form of quantitative data. the quantitative data then tested to answer the hypotheses that have been formulated by researchers in accordance with established test procedures. quantitative data were obtained from even semester test results and post test results. data analysis of the results of even semester tests using microsoft excel 2013 software was carried out to find out that the three experimental classes had almost the same average values. the aim is to ensure that at least there is no difference in the initial capabilities of the three groups. while the post test data analysis uses spss (statistical product and service sulation) version 20. for windows in order to find out accept h0/h1. 3. results and discussion 3.1. results this study began by using the results of the even semester tests to determine the experimental class 1, experimental class 2 and the control class and to determine differences in learning outcomes used the post test. after comparing it turns out that even semester 2018/2019 test results between classes eighth grade-1, eighth grade-6, and eighth grade-7 are relatively the same as shown in figure 1. palinussa, comparison of algebra learning outcomes using realistic mathematics education … 176 figure 1. average daily repeat value from the average semester test scores (figure 1), the experimental class 1 is eighth grade-7 with an average of 71.6154, experimental class 2 is eighth grade-1 with an average of 71.8333, and the control class is eighth grade-6 with an average 72.333. data normality and homogeneity of data were tested before using the anova test. the prerequisite test which includes the normality test uses the chi-square test and homogeneity test using the f test. to find out whether the data is normal or not normal, a chi-square calculation is done for the control class and the experimental classes and the results are as shown in table 1. table 1. normality test results class sig. α experiment 1 (rme) (x1) 0.976 0.05 experiment 2 (tai) (x2) 0.889 control (conventional learning) (x3) 0.906 data decision making is normal if asymp sig. (2-tailed) is greater than the level of significance. based on table 1 obtained sig. x1 = 0.976, x2 = 0.889 and x3 = 0.906 which is greater than the significance level of 5% (0.05), so it can be stated that all data are normal. then the variance homogeneity test is performed. to find out that the ability of students in a homogeneous population, two or more variances were used in common using the levene test (see table 2). table 2. homogeneous variance test class sig. α experiment 1 (rme) (x1) 0.654 0.05 experiment 2 (tai) (x2) control (conventional learning) (x3) average std dev volume 9, no 2, september 2020, pp. 173-182 177 table 2 show that calculation of the similarity of two or more variances using the levene test shows the value of sig. greater than 5% (0.05) is 0.654 > 0.05, from the test criteria for the levene test is accept h0 if fcount < ftable and reject h0 if fcount > ftable, it can be concluded that the variance of the three data groups is experimental 1, experimental 2 and control class is homogeneous because h0 is accepted which is 0.95 < 3.12. from the test scores of learning outcomes and calculations of the mean, standard deviation, one-way anova, for the experimental 1, experimental 2 and control class the following results were obtained (see table 3). table 3. anova calculation source of variation df sum of squares mk fcount ftable decision total 74 18286.207 6.347 3.12 fcount > ftable between groups 2 2740.756 1370.378 in group 72 15545.451 215.909 based on the anova one way calculation used spss 20.0 program, a significance value of (0.003) was obtained, meaning that the value of sig. is less than 5% (0.05), it can be concluded that there are differences in learning outcomes of the three learning models. table 3 show that fcount > ftable is obtained (6.347 > 3.12) then h0 is rejected or there are differences in the learning outcomes of eighth grade students of junior high school 1 masohi who are taught using the realistic mathematics education (rme) learning model, the team assisted individualization (tai) of cooperative learning model and the conventional learning model in the operations of algebra. to determine a better learning model among the three learning models used, the average value of student learning outcomes from the three learning models can be considered which can be presented as follows in figure 2. figure 2. the average value of the three groups after treatment was given figure 2 show that the average value of the highest student learning outcomes is the class taught by the realistic mathematics education (rme) learning model (72.1332). then the cooperative learning model team assisted individualization (tai) type (65.2020) and the lowest is the conventional learning model (57.3356). furthermore, to find out a better learning model among the three learning models used, then using a follow-up test or socalled after anova analysis using tukey's hsd (see table 4). palinussa, comparison of algebra learning outcomes using realistic mathematics education … 178 table 4. average differences between groups x1 x2 x3 x1 6.9312 7.8664 x2 6.9312 7.6904 x3 14.7976 7.8664 interpret hsd values by comparing the average differences between groups with the results of hsd calculations. based on the tuckey's hsd test results obtained the value of hsd = 9.9926. test the difference x1 and x2 = 7.4537. obtained from the difference between the average x1 and x2 ie (72.1332 57.3356) then x1 = x2 because 6.9312 < 9.9926, test the difference x1 and x3 = 14.7976. obtained from the difference between the average x1 and x3 ie (72.1332 57.3356) then x1 ≠ y because 14.7976> 9.9926, test the difference x2 and x3 = 7.8684 obtained from the difference between the average x2 and x3 ie (65202 57.3356) then x2 = x3 because 7.8664 <9.9926. based on tuckey's hsd calculation (see table 4), the average value of the three classes is experimental 1 (x1) which has a higher average number so it can be stated that the learning model that is superior among the three learning models is the realistic mathematics education (rme) learning model. 3.2. discussion in the class taught by the realistic mathematics education (rme) learning model, at the beginning of the learning the teacher provides stimulus in the form of material using contextual examples, so that students can better understand the material provided through these examples. and then by following the steps in the rme learning model the teacher guides students to be able to understand the steps of learning. although initially the students still looked confused, with the teacher's guidance the students were then able to understand the steps in the rme learning model well. after students get an explanation of what rme is and the steps of the rme learning model, students who have been sitting in groups begin to look busy with the material to be completed in the group. rme has superior potential compared to conventional and tai learning models in improving mathematics learning outcomes. in rme students are trained to develop reasoning and logical abilities. mathematics learning through rme is very relevant to students in dealing with daily problems so that students can interact with the teacher continuously to solve problems. various studies with rme in indonesia explore the extent to which rme can be utilized and stimulate improved learning process (sembiring, hadi, & dolk, 2008). during the learning process takes place students are also required to better understand the material by solving questions in the worksheet in groups. this causes a sense of responsibility, mutual respect and mutual assistance in the group during the learning process. the rme learning model itself is a learning model that can structure the level of student understanding so that students can relate information that has just been obtained with existing material with the cognitive structure they have. according to gravemeijer (1994) and afriansyah (2016), there are three main principles in rme, namely: (a) guided reinvention and progressive mathematization); (b) didactical phenomenology; and (c) a selfdeveloped model. the first phase guided reinvention, which students should be given the volume 9, no 2, september 2020, pp. 173-182 179 opportunity as a society to find a process similar to the process in which mathematics is found. during the learning process, students independently have the opportunity to build their own mathematical knowledge. in the second stage of didactic phenomenology, a situation that is relevant to the topic of mathematics is created so that it can be applied to be investigated in learning. therefore, it is also necessary to balance the types of applications or methods that must be anticipated in learning by the teacher. the aim of the phenomenological inquiry is to find problem situations in which a specific approach can be generalized. another aim is to find situations that give rise to a paradigmatic solution procedure. the third stage is a self-developed model. self-developed models play an important role in bridging the gap between limited informal knowledge and formal mathematical knowledge. this model was developed by the students themselves. through mathematical generalization and formalization, this model is developed by students to aid mathematical reasoning (sumirattana, makanong, & thipkong, 2017). the results of this study are relevant to the results of research (batlolona et al., 2019) which found that rme has very good effect on improving the mathematics learning outcomes of junior high school students. rme allows teachers and students to connect the context of abstract learning material to be concrete. it is easier to solve contextual problems that students encounter in their daily lives with rme. therefore, rme assists teachers in designing learning that is relevant to the needs of students in real-life contexts. reality concept is a context of known children's knowledge in their lives, and then becomes components of thinking scheme. the scheme components connect various mathematical contexts and concepts. related to this situation, creative thinking can involve various dimensions of knowledge in every stage of cognitive thinking process. thus, reality and intertwinement as rme principles can be used to encourage someone's learning outcomes. the real learning concept is the context of the child's knowledge that is known in his life, then becomes a component schema of thought. schema components connect various mathematical contexts and concepts. associated with in this situation, thinking process can involve various dimensions of knowledge in every stage of thinking process. thus, reality and linkages as the principles of rme can be used to encourage thought processes so that learning outcomes increase (muhtarom, nizaruddin, nursyahidah, & happy, 2019; nuraida & amam, 2019; sitorus & masrayati, 2016; umbara & nuraeni, 2019). student learning outcomes are much improved with rme compared to stad and conventional learning due to an increase in learning activities as well. learning that is real and in accordance with real-world conditions, encourages students to increase learning activities (arsaythamby & zubainur, 2014). this happens because students become curious about the topic they are studying when it is related to real conditions. students arouse curiosity to reveal how the solution to the problems they face. in addition, students experience learning on their own, so they feel the importance of learning and understand that rme helps them in learning. on the other hand, rme can make learning memorized meaningful because students try to connect information that is already in their minds with information that will be obtained so as to enhance student understanding because it contains a summary of concepts and material relationships in the cognitive structure of students (clarke & roche, 2017). this awareness is very motivating for an increase in learning outcomes compared to tai and conventional learning. in the class taught by the tai type of cooperative learning model the teacher only works as a facilitator who is ready to help groups or individuals who need help. the results of individual work will be brought into their respective groups to be discussed and discussed in groups. all group members are responsible for the entire answer that is done. in this learning model there is no competition between students in groups because students work together to complete the given task and students also respect each other's different ways of palinussa, comparison of algebra learning outcomes using realistic mathematics education … 180 thinking, students not only expect help from the teacher, but also motivated to learn accurately fast on all material. the results of group learning are compared with other groups to get awards in the form of praise from the teacher. this type of tai cooperative learning places more emphasis on group appreciation. of the five groups in this tai class, the fourth group was given credit for their work as the super group or the best group while the first, second, third and fifth groups were given credit for their work as a good group. then each student in the group is given an evaluation in the form of a quiz (fact test). in the other hand, the selection of the tai model as a learning model is felt to be accordance with the existing problems. the use of tai is the use of a very simple learning model and is able to provide understanding concepts to students so easily that it becomes a solution for students in learning difficult material. in tai learning, students are required to actively solve problems given by the teacher individually or in groups. the groups formed in the learning of tai consist of students who have high, medium and low abilities making it easier for students to discuss. students who lack understanding can ask students who understand better, especially the group leader (ikhsanudin, 2014). alimuddin (2017) state that the tai model had a significant influence on the mathematics learning outcomes of seventh grade students at bungap satoro middle school by 13.7%. from the results of the research conducted above it can be concluded that the team assisted individualization model can improve learning outcomes and motivation. in classes taught by conventional learning models, at the beginning of learning the teacher conveys the material to be learned and conveys the learning objectives. after that, during the learning process takes place the teacher dominates the learning process while the students only pay attention and record what is explained by the teacher. the teacher explains the material in stages, then gives examples of questions, after that gives the opportunity for students to ask questions and respond back to what students are asking. conventional learning is more oriented towards achieving curriculum goals so that it ignores efforts to instill concepts that are deep and relevant to student needs (leasa & corebima, 2017). however, if there are no questions from students, the teacher will continue the material. then the teacher provides a summary and assignments to complete. in addition, during the learning process only certain students pay attention, while other students don't pay attention to what the teacher says. this is because the learning model used is more centered on the teacher, so students only accept what is conveyed by the teacher which results in a less active learning process. in line with that aziz & hossain (2010) argues that the conventional learning model or lecture method that focuses students' full attention on the teacher so that only teachers are active here, while students are only subject to listening to the explanation presented by the teacher. 4. conclusion based on research that has been done and discussion of research results, it can be concluded that there is a difference in the learning outcomes of eighth grade students of junior high school 1 masohi who are taught using the rme learning model, the cooperative learning model team assisted individualization (tai) type and the conventional learning model. other than that, the superior model used to teach algebraic operations is the rme learning model. rme has a positive impact on student learning outcomes when compared team assisted individualization (tai) and the conventional learning model. volume 9, no 2, september 2020, pp. 173-182 181 references afriansyah, e. a. 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(2019). implementation of realistic mathematics education based on adobe flash professional cs6 to improve mathematical literacy. infinity journal, 8(2), 167-178. https://doi.org/10.22460/infinity.v8i2.p167-178 https://doi.org/10.1007/s11858-008-0125-9 https://doi.org/10.1088/1742-6596/1108/1/012073 https://doi.org/10.1088/1742-6596/1108/1/012073 https://doi.org/10.1016/j.tsc.2016.09.007 https://doi.org/10.1016/j.kjss.2016.06.001 https://doi.org/10.22460/infinity.v8i2.p167-178 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p57-74 57 incomprehension of the indonesian elementary school students on fraction division problem yoppy wahyu purnomo* 1 , chairunnisa widowati 2 , syafika ulfah 3 1,3 universitas muhammadiyah prof. dr. hamka 2 universitas negeri jakarta article info abstract article history: received des 7, 2018 revised jan 17, 2019 accepted feb 1, 2019 the purpose of the study is to investigate the indonesian students’ performance in solving fraction division case including the difficulties, relations, and implications for classroom instruction. this study employed a descriptive case study to achieve it. the procedures of data collecting were initiated by giving a context-based problem to 40 elementary school students and it then according to the test result was selected three students for semistructure interviewed. the findings of the study showed that the tendency of students’ procedural knowledge dominated to their conceptual knowledge in solving the fraction division problem. furthermore, it was found several mistakes. first, the students were not accurate when solving the problem and unsuccessful to figure out the problem. second, students’ conceptual knowledge was incomplete. the last was is to apply the laws and strategies of fraction division irrelevant. these findings emphasized other sub-construct of fractions instead of part-to-whole in the teaching and learning process. teaching and learning of fraction in the mathematics classroom should take both conceptual and procedural knowledge into account as an attempt to prevent faults and misconceptions. in conclusion, it was substantial to present context-based problems at the beginning of the lesson in order for students to be able to learn fraction division meaningfully. keywords: fraction division fractional parts conceptual knowledge procedural knowledge elementary school students indonesia copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: yoppy wahyu purnomo, departement of mathematics education, universitas muhammadiyah prof. dr. hamka, jl. tanah merdeka, jakarta, indonesia email: yoppy.w.purnomo@uhamka.ac.id how to cite: purnomo, y. w., widowati, c., & ulfah, s. (2019). incomprehension of the indonesian elementary school students on fraction division problem. infinity, 8(1), 57-74. 1. introduction fractions are one of the topics studied in elementary mathematics which serves as a cornerstone for comprehending further mathematics topics such as arithmetics, algebra, probability, data analysis, geometry, and measurements. they are also capitalized on for communicating and solving daily life problems. nevertheless, fractions and the operation are the most challenging elementary mathematics topics as they are difficult to understand. in addition, most elementary school students are presented with meaningless instruction mailto:yoppy.w.purnomo@uhamka.ac.id purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 58 (geller, son, & stigler, 2017; lin, becker, ko, & byun, 2013; wang, chen, & lin, 2014), especially in fractions division (kribs-zaleta, 2006; sidney, hattikudur, & alibali, 2015; tirosh, 2000). fractions division learning emphasizing heavily on algorithms acquisition inevitably tends to be accepted and dominated teaching and learning process. the most common procedures used in coming to grips with fractions division problems are the invert and multiply (or keep-flip-change) strategies. these procedures are perceived as the most mysterious rules in elementary mathematics topics (van de walle, karp, & bay-william, 2010) as they are not frequently understood by the teachers and the students. skemp (1987) name these procedures as rules without reason. the ability to manipulate symbols and implement rules without understanding may create trouble in making sense and students may make mistakes when encountering problems which have to be solved using unfamiliar procedures (purnomo, kowiyah, alyani, & assiti, 2014). 1.1. procedural and conceptual knowledge of fraction procedural knowledge has been described as knowledge about how to do something (hallett, nunes, & bryant, 2010). it refers to students’ ability to implement, calculate, and execute symbols representation system and algorithms to solve problem accurately, efficiently, and appropriately (lauritzen, 2012; rittle-johnson, siegler, & alibali, 2001). skemp has identified this knowledge as an instrumental understanding which is described as rules without reason. procedural knowledge deals with symbols, rules, formulas, and algorithms in a discrete manner while conceptual knowledge refers to a knowledge that is rich in connections (rittle-johnson et al., 2001). skemp (1976) label it as a relational understanding. connection or relation between mathematical concepts and integration to a contextual situation is the heart of conceptual knowledge. it is able to assist students in making sense of the fraction concept. conceptual knowledge and procedural knowledge may support each other. concept-first and procedures-first were acknowledged by kinds of literature (rittlejohnson et al., 2001). development of students’ conceptual knowledge at the outset of a lesson may contribute to the ability to address varieties of mathematical tasks successfully. moreover, students are able to build conceptual knowledge by establishing various types of algorithm initially. focusing heavily on procedures in fraction learning may impede students to engage with the real-world context used as a bridge for developing conceptual knowledge. without context, students may encounter puzzlement in coming to grips with the concept of fraction and its application in various situations (sharp & adams, 2002; yim, 2010). traditionally, in the case of fraction division, students mostly are presented by procedure-oriented, memory-based, and meaningless instructional approaches. the common algorithm used for fraction division is by multiplying the dividend by the reciprocal of the divisor. this algorithm is straightforward. hence, students are able to use it easily in dealing with routine problems that have been exemplified by their teachers. however, they get trouble in an attempt to solve unfamiliar problems such as word problems and non-routine problems. the difficulties can be caused by students’ lack to understand the concept of fractions. therefore, procedural knowledge and conceptual knowledge should be interwoven and integrated with each other (kilpatrick, swafford, & findell, 2001). volume 8, no 1, february 2019, pp. 57-74 59 1.2. interpretation of fraction division traditionally, fraction division is able to be explained by the extension of the division interpretation of whole numbers, namely, partitive and measurement concepts (alenazi, 2016; purnomo, 2015a). furthermore, recent studies have discussed others interpretation of fraction division, namely, the determination of a unit rate, the inverse of multiplication, and the inverse of a cartesian product (alenazi, 2016; sinicrope, mick, & kolb, 2002). in our study, we focus on the traditional concepts of fraction division that are partitive and measurement concepts because these concepts are typically introduced to learn the fraction division in first because of related concept whole number division. in addition, these concepts are considered to be relevant and appropriate for developing the understanding of elementary school students on fraction division problem. 1.2.1. fractions division as measurement (repeated subtraction) this model explains fraction as the number of times we can subtract the denominator from the numerator before we attain 0 (zero). this meaningful interpretation can be applied in case of fraction division. for instance, in the case of division of 1/2 by 1/4, students may reason it as a quarter goes into 1/2 two times. it can be interpreted contextually, for instance, if someone has half of a cake and she/he wants to divide it into quarters, then you have two pieces of 1/4. 1.2.2. fractions division as partitive (equal share) this model is mostly known as partitioning or equal sharing. it represents to share activity which distributes a collection or quantity equally among some number of people. in the case of fraction division, for example, division 1/2 by 1/4, students may reason it contextually as a process of distributing half of a cake to several numbers of people in order that each person gets 1/4 of cake equally. 1.3. the present study in indonesia, the elementary school consists of classes from 1st grade up to 6th grade. it is commonly classified into a lower elementary (1st – 3rd grade) and an upper elementary (4th – 6th grade). indonesia has two simultaneously applied curricula, namely the school-based curriculum (known as kurikulum tingkat satuan pendidikan; in abbreviated as ktsp) that has been implemented since 2006 and the curriculum of 2013. one of the fundamental differences of both curricula is a pattern of material organizing. at the ktsp, a relationship among subjects is more mutually exclusive, while for the curriculum of 2013, it is integrative thematic. nevertheless, the implementation of the 2013 curriculum is still a limited trial phase applied in selected schools. at the elementary school level, this curriculum is only applied in grade 2 to represent the lower elementary and grade 4 to represent the upper elementary. in indonesia and most of the international curriculum, fractions are firstly introduced in the third grade of the elementary school (wijaya, 2017), while the divisions are introduced in the fifth grade of elementary school (see also purnomo, widowati, aziz, & pramudiani, 2017). further, wijaya (2017) states that the introduction of the fractional concept as parts of the whole becomes the only construct having a space to learn from both mathematical textbooks and the teacher's teaching method, while the fractional operations are dominated by rigid rules to solve problems. purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 60 in indonesia, some previous studies have mentioned that the fractions topic becomes a difficult topic for the students and a serious concern because of the weakness of the students' performance on this topic (purnomo et al., 2014; trivena, ningsih, & jupri, 2017; wijaya, 2017). in his study, wijaya (2017) has analyzed students' difficulties on the fractions topic from timss results in 2015 and attributed it to the students' opportunities to study the fractional at school. based on the analysis of timss results, his research has found that indonesian students have a weakness on the understanding of fractions, particularly in story problem cases. students do not have space to explore their ideas because teachers resist getting out of the content and sequence of material in the book. similarly, trivena, ningsih, and jupri (2017) also have found that elementary school students are oftentimes misconception the concept of addition and subtraction of fractions. these studies show the fraction is one of the materials requiring attention and handling. however, those studies and literature related to it have not focused on more specific content that is fractions division. focusing on more specific issues helps to handle the problems more precise at hand. in addition, it is also substantial to know how the students' strategy in dealing with the case of division and what the difficulties are. based on the above description, this study aims to explore the indonesian elementary school students’ performance in solving fractions division cases including the difficulties, relations, and implications for the classroom instruction. research questions may arise i.e. how is indonesian elementary school students’ understanding of fraction division? 2. method 2.1. context and participants the method of this research employed two phases. the first one was a descriptive study to gain insight into students’ performance in coming to grips with fraction division problems. the participant of this phase was 40 fifth grade elementary school students in jakarta. it was collected fifth-grade students as the participant because they had learned a fraction from the definition up to the operation of fractions division. the second one was a case study to investigate the students’ knowledge further about fraction division and its underlying epistemological factors. several participants were selected based on their achievement in the test and teacher’s suggestions such as their ability of verbal expression and confidence. according to these considerations, three students were selected and pseudonyms were used to address ethical issues. the first student was ummu, an 11-yearsold, javanese girl, she was an outstanding student being a top three in her class every academic year. she comes from a middle-income family. the second one was nunu, an 11-years-old, javanese boy, he was an average student. he comes from a low-income family. his salient characteristics are that he is an active student selected as a leader in his class. the last was cici, a 12-years-old, sundanese girl, she was a student categorized as a low-achiever. however, she is involved actively in several school activities such as flag hoisting troop. she comes from a low-income family. a similarity among them is that they have settled in jakarta city since they were born. 2.2. data colletion the data collection processes were done by using a written test and an interview. the written test was administered to obtain data from the participants’ performance in dealing with fraction division problems. meanwhile, the semi-structured interview was volume 8, no 1, february 2019, pp. 57-74 61 conducted to explore epistemological factors related to the understanding of fraction and its difficulty. the written test composed of three question items about fractions division with different indicators. table 1 below demonstrates the question items. table 1. indicators of the written test case descriptions 1. , with (this case relates to a measurement concept of the fraction division) you have 2 birthday cakes given to your friends, each of them is 1/2 parts. how do you know the number of your friends who will get the cakes? 2. , with and (this case relates to partitive concept of the fraction division) father has 1/2 pizzas given to 2 of his children named mila and damar equally. how do you know how big parts will damar get? 3. , with and (this case relates to measurement concept of fraction division) mrs. vivi has 3/4 kgs flavor. to make 1 donut, mrs. vivi spent 1/4 kgs flavor. how do you know that how many donuts can be made? the interviewing questions based on the items of the question above and responses of the students on each question. first, we re-questioned “how do they solve the problem”. then, we asked, “why do they choose those strategies”. the third was “where do they know the knowledge”. 2.3. data analysis we used a rubric to analyze participants’ written answers. the rubric (see table 2) took indicators into consideration that were, understanding the problems, planning, and the answer’s accuracy. each indicator had 2 for the maximum score and 0 for the minimum score, so the highest score possible for each item is 6. in total, the highest score possible is 18 and the lowest score possible is 0. table 2. assessment rubric for written test assessment criteria assessed indicators score problem understanding comprehensive and organized understanding 2 there is an effort to organize but some problems could not be figured out 1 do not understand the problems, are not organized and systematic 0 strategic planning the strategy used is relevant and well explained (if it is implemented, it will be valid) 2 some strategies are relevant but are not well explained 1 the strategies are irrelevant, unclear and difficult to get to the point 0 accuracy of calculation using the right strategy leading to the right answer 2 some algorithms applied are correct but there are errors found. as the result, the answer is not valid 1 there are no answer 0 purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 62 the percentage of students’ correct answers on every item is counted and classified into the whole right answer, partly right answer, and wrong answer. the written test data also were analyzed descriptively including mean and standard deviation. the mean and standard deviation are used to determine selected students for a case study. the written test data is grouped into high, medium, and low category. the high category had a score more than ̅ . the low category had a score less than ̅ . the scores between both criteria are categorized as a medium category. one student of each category would be chosen for the case study. on the other hand, the interview result data were transcribed and coded based on pattern responses to explore further. triangulation was done by confirmation the written test data and interview result. we also used students’ worksheets as additional data. the students’ worksheets were taken during the interview when students explained their understanding to the interviewer. 3. results and discussion 3.1. results 3.1.1. profile of the fraction division performance the students’ written answers represented their understanding of each case given. descriptive statistic of students’ answers is shown in table 3. table 3. percentage of student responses for each case of fraction division no. item correct partially correct incorrect 1 10% 50% 40% 2 8% 25% 68% 3 15% 28% 58% based on table 3, most participants encountered difficulties when dealing with the presented items. item number 2 obtained the lowest response among others. this item was only whole right answered by 8% of students and 68% of them responded wrong. table 3 also shows that at most only 15% of respondents answered a whole right answer for each given item. overall, the number of participants who correctly answered the items was much less than that of those who obtained partially correct and incorrect answers. it might indicate that fractions division is a problematic and challenging elementary mathematics topic for students. the results of the data analysis on the written test have obtained a mean at 4.525 and a standard deviation at 4.391 with the highest score was 14 and the lowest score was 0. based on the criteria for each category that we had previously set, there were 7 students (18%) as the high category, 21 students (53%) as the medium category, and 13 students (33%) in the low category. 3.1.2. a case on the natural numbers divided by fractions ummu’s responses in accordance with her written response addressing the first case, ummu sliced each cake into two pieces equally, then four half-pieces were obtained from two cakes. in the interview, she said, “this cake is sliced into two similar parts, so does this (another volume 8, no 1, february 2019, pp. 57-74 63 cake) one. after slicing the two, we have four similar pieces. this one (by referring to the shaded part) is for one person. so, there will be four”. ummu tried to address the problem by considering the number of half piece of cakes. therefore, a strategy implemented by ummu is likely to be in line with the concept of division as repeated subtraction. nevertheless, we discern that ummu’s explanation in the interview session tends to be different from her written answer in the test. the same approach also she did when answering subsequent questions we gave. ummu was asked to give explanations of how to share twelve doughnuts with her friends in which each of them would get two doughnuts. she explained, “it will be two doughnuts for one person, two for the other, so do this, this, this (putting marks to every two objects)”. ummu’s strategy was that she tried to distribute two for each person so that it would end at twelve for the number of doughnuts and six for the number of the person. based on this, ummu’s knowledge of the concept of division tends to converge on the idea of division as an inverse of repeated addition. nunu’s responses when dealing with the first case, nunu employed invert-and-multiply rule. figure 1 describes nunu’s efforts to address the question given. figure 1. nunu’s response in written test through the interview, we tried to gain deep information related to nunu’s written response during the test. the following are excerpts of the interview with nunu conducted after the test. interviewer : nunu, could you explain your answer? nunu : this was division, i mean that 2/2 is divided by 1/2. interviewer : which 2/2 did you mean? nunu : as there were two cakes, so 2/2. because we divided this, the second one was reversed and then we multiplied.the result was 4/2. interviewer : so, you thought that 2/2 was two cakes, how about 5/5? nunu : there were five cakes. based on the excerpt above, it is obvious that invert-and-multiply rule is capitalized on by her to solve the question. however, even though she was able to solve it in a correct manner, she did not seem to have an accurate comprehension concerning a concept of fractional parts. this inaccuracy occurs as she tends to assume that the value of a fraction, whose numerator and denominator are equal numbers, is the same as its numerator and denominator. for instance, nunu presumed that 2/2 could be represented as two units. by this method, she also regarded that 5/5 could be represented as five units. based on this interesting fact, further queries were posed to find out about nunu’s comprehension of the concept of fractional parts. purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 64 figure 2. nunu’s response in interview session fig. 2 shows that nunu’s understanding of the fractional concept parts is weak. nunu focused on how many parts are divided and then she presumes that each part b and part c are 1/3. she said, “because we have three parts from one cake”. however, this statement is in contradiction with nunu’s answer that is 1/2 for part a. nunu realized that she added up the three parts and turned out to produce more than 1. cici’s responses cici’s responses to the first case were similar to what ummu undertook. in the interview, she said, “this cake is cut into two parts. then, i get two 1/2s. the other is also divided by two, so i get 1/2 and 1/2. thus, 1/2 + 1/2 = 2/4. since there are two cakes, thus 2/4 × 2/4 = 4/4. so, the result is 4/4 person. 4/4 refers to four persons”. when we elaborated on an aforementioned explanation by posing a further question, unfortunately, she was not able to reveal her argumentation concerning the reason why she used addition and multiplication. “i don’t know why?” she replied. figure 3. cici’s response in written test figure 4. cici’s response in interview session based on the aforementioned description, we disclosed several mistakes and misconceptions made by cici’s work. firstly, cici did mistake when adding two fractions with like denominator, for instance, 1/2 and 1/2. secondly, cici was likely to have an inaccurate understanding of addition and multiplication conception of numbers. thirdly, volume 8, no 1, february 2019, pp. 57-74 65 similar to what nunu did, cici claimed that the value of a fraction with similar numerator and denominator were equal to its numerator and denominator. 3.1.3. a case on the fractions divided by the natural numbers ummu’s responses in the second case, ummu tried to respond by splitting pizza out into two equal parts and wrote 1/4 in each part. therefore, according to ummu’s responses, the result of a division of 1/2 by 4 is 1/4. the following figure presents ummu’s written response to this case. figure 5. ummu’s response in written test figure 6. ummu’s response in interview session ummu’s written response in the test was likely to reflect her understanding of division as equal sharing. however, its conception was not demonstrated when she was interviewed. she tended to present her procedural knowledge. it was obvious when she tried to solve the question presented. even though her procedures and obtained answer were correct, she seemed to express her puzzlement concerning her answer. the following was an excerpt from the interview with ummu. interviewer : could you explain the way how you get 1/4 as your final answer? ummu : by dividing. 1/2 is divided by 2 equals ¼. wait, it is divided by 2… it is 1/2, isn’t? interviewer : what do you mean by 1/2? ummu : this pizza is cut into two. it is 1/2, isn’t? damar has one part, and …… interviewer : your previous obtained answer was 1/4. why do you have the different answer? ummu : because a half pizza is cut into two. interviewer : could you show me, which part of the figure indicates 1/4? ummu : this one (she refers to damar’s part), but i am a bit confused because it is divided by two. but i am sure that 1/4 is the correct answer (by showing her written response) based on the above excerption, ummu was not able to convince herself that 1/4 was the result of division 1/2 by 2. ummu tended to rely heavily on her procedural knowledge and got confused when there was a contradiction between her work showing that 1/2 divided 2 equals to 1/4 and her mental image presuming that something divided by 2 equals to 1/2. the reason might lay in the fact that ummu’s primary focus was the result purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 66 of division instead of paying attention to its dividend and divisor. in addition, a similar response was presented by ummu when she was asked further questions during the interview. ummu : each person gets one. the rest is cut into four like this. let me sign these, 1, 2, 3, and 4. then, i add one cake with this. so, each person will get . ummu : it’s the same. i divide this by three. as each person gets one, then we divide the rest by 3. so the answer is . figure 7. ummu’s response in interview session figure 8. ummu’s response in interview session it is apparent from fig. 7 that ummu was able to address a case in which the dividend of the fraction is whole numbers. however, she was likely to have difficulty in coming to grips with the division problem in which its dividend was a rational number. from fig. 8, ummu presumed that division of a half circle by three results in 1/3. in addition, ummu seemed to have a weak understanding of fractional parts concept. it was obviously observed when she was asked about the fraction representing each sector in fig. 2 and her answers were that sector b and c were 1/3 and sector a was 2/3. fractional parts concept is a fundamental aspect in comprehending division of fraction and other fraction operation. nunu’s responses nunu’s response to the second case indicated that nunu capitalized on procedural knowledge, yet her works were difficult to interpret. based on the interview response, it seemed that nunu’s concept of fraction and division of fraction were still weak. volume 8, no 1, february 2019, pp. 57-74 67 figure 9. nunu’s response in written test figure 10. nunu’s response in written test interviewer : nunu, could you explain this answer you have obtained during the test? nunu : 1/2 is divided by 2/2 interviewer : what does 2/2 mean? nunu : two children interviewer : was 3/4 the answer to the question? nunu : yes, it was. 3/4 for damar. but, wait. it is wrong. it should be 2, not 3. so, the correct answer is 2/4. the dominance of procedural knowledge over conceptual knowledge in fraction concept might lead nunu to make mistake as she was not able to catch on what the presented problem was. for instance, she presumed that 2/2 stood for two units. figure 11. nunu’s response in interview session figure 12. nunu’s response in interview session in the further interview, nunu said i got 1/4 as we divide a cake by four. then, 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5/4. therefore, each child will have 5/4 cake (see fig. 11). in this case, nunu was able to deal with division case whose dividend was whole numbers, yet from fig. 12 we knew that she encountered difficulty as the dividend was not whole numbers. purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 68 cici’s responses on the third case, cici tried to solve the question by cutting the cake into two equal parts. according to cici’s statement during the interview, we found that cici was likely to focus her attention on the result of the fraction instead of noticing the dividend and divisor. figure 13. cici’s response in interview session figure 14. cici’s response in interview session based on the cici’s work and statements, one issue was paramount when it came to her misconceptions about the fraction, that was, she presumed that if all things were cut into three equal parts, then the result would be 1/3. she passed over the form of the thing being divided. another cici’s weaknesses were found when she assumed that a region c and d were greater than region a and b as illustrated in fig. 14. however, surprisingly this response was at odds with her statement when she was asked about the fraction unit that named each part of the divided whole. she claimed that each part represented 1/4. this state of an affair might be attributed to cici’s lack of understanding of equal sharing concept at the fractional parts. 3.1.4. a case on fractions divide by fractions ummu’s responses ummu tried to address the third case using the subtraction method. the following figures illustrate ummu’s written responses during the test and the interview session. figure 15. ummu’s response in written test figure 16. ummu’s response in interview session volume 8, no 1, february 2019, pp. 57-74 69 we were not able to identify ummu’s argumentation, as she viewed this problem as a case of subtracting 1/4 from 3/4. we tried to have ummu read the question meticulously, yet she was likely to stick with its stance in her opinion. nunu’s responses nunu’s responses tended to be similar to that of ummu in which the third case could be addressed using subtraction. it could be discerned clearly when interviewing nunu. her answer was 2/4 as a result of subtraction of 1/4 from 3/4. she accounted for it as 1/4 floor used for making a doughnut. she performed single subtraction in lieu of multiple subtractions. figure 17. nunu’s response in written test figure 18. nunu’s response in interview test cici’s responses cici’s response to the third case indicated that she made the use of her procedural knowledge to address the question. fig. 19 shows cici’s written response in the test. she multiplied 3/4 and 1/4 and during the interview, she was not able to uncover the reason behind her strategy. therefore, it seemed that cici encountered difficulty in grasping the problem presented thoroughly. figure 19. cici’s response in written test purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 70 3.2. discussion tirosh (2000) has summarized that there are at least three main categories of mistakes made by children when solving fractions division problems, to wit: algorithmbased mistakes, intuition-based mistakes, and mistakes derived from formal knowledge. in our study, we explored how children knowledge solves fraction division regarding tirosh’s work and any possibility beyond his work. the findings of the study indicated that most of the participants encounter difficulties in solving problems of fraction division. they face difficulties in solving problems related to the case of dividing fractions by whole numbers. in other words, their equal sharing concept is still weak. this finding is not in accordance with our predictions who reveal that this case is the easiest case compared with other cases. nevertheless, the other cases also need an attention because 58% of the students provided an incorrect answer for the case of dividing fractions by fractions and only 10% answered correctly for the case of dividing of whole numbers by fractions. in addition to being weak in equal sharing’s conceptual knowledge, the majority of the students in this sample still depend on procedural knowledge which is not accompanied by a strong conceptual knowledge. this finding was also supported by purnomo et al. (2014) when examining primary school students number sense. in their study, purnomo found that the students encountered some difficulties in understanding the meaning and the concept of numbers, especially the fractions and decimals. students have a misconception about the fractional concepts and make some errors when performing calculations as they pay more attention to its rules and algorithms. the findings were clarified and reinforced by the response of the three case study samples, namely ummu, cici, and nunu. all three participants are weak in equal sharing concept in the division of fractions. the equal sharing concept is used to interpret divisions which involving whole numbers as divisors. more precisely, this difficulty occurs when they encounter the case of an incomplete part of an object and they asked to determine how much each part divides the incomplete part. they focus on how many parts have been divided regardless of the shared part. this case can be exemplified by ummu when responding to what part was received by three children when they shared the 3 1/2 cakes equally. ummu assumed that the 1/2 part divided by 3 is 1/3 (fig.8). this is also similar to cici's response to the problem (fig 13). we connect this equal sharing conceptual problem to an intuition-based mistake from tirosh (2000). the intuition-based mistake encounters stem from students’ tendency to generalize the concept of equal sharing overly. the students in the sample of this study think that "everything shared by a certain number of a part is one per part of a dividing part". we also recognize that the conceptual problem for equal sharing of these fractions is related to students' misconceptions on the concept of fractions part. the students often focus only on how many parts are shared but they do not notice whether a value of fractions is equal (see fig. 2; fig. 14). the concept of the fraction part is a foundation for children to learn a fraction meaning, fractions operations, and advanced concepts of fractions. therefore, when these fundamentals are not robust, it will affect the understanding of fraction operations including fraction divisions. in an attempt to reduce this problem, it is critical not only to focus on the meaning of the fractions as part-towhole, but also to emphasize other sub-fraction constructs in the learning process of fraction concepts (clarke, clarke, & roche, 2011; purnomo, 2015a; siebert & gaskin, 2006), among others fraction as division, fraction as ratio, fraction as operator, fraction as measure. this is also alluded to by wijaya (2017) in which most mathematics textbooks in indonesia only introduce the concept of fractions with the concept of fractions as part-to volume 8, no 1, february 2019, pp. 57-74 71 whole. the intuition-based mistakes and misconceptions about the concept of fractional parts also discourage students from using the correct terminology. we encounter these things when the child considers 2 as 2/2, 5 as 5/5, and so on (see fig. 1; fig. 3; fig. 4). concept issues for equal sharing are not stand-alone. there are other obstacles related to student difficulties when students face fraction divisions. moreover, we have found that most students still depend on procedural knowledge without being aligned with conceptual knowledge. some researchers agree that focusing only on procedural knowledge may block a development of intuitive sense and the conceptual knowledge itself (forrester & chinnappan, 2010; purnomo et al., 2014, 2017). these problems can be verified by participants' work on the fractions division case which most of the students did it using invert and multiple rules. these rules are not based on a comprehensive explanation, students employ these rules to obey and apply procedures properly. however, applying the rules by ignoring the conceptual knowledge often causes errors in calculations. one example of the errors in employing this strategy can be seen in nunu (see fig.1 and fig. 9) and cici work (see fig. 3 and fig. 19). this has been mentioned by tirosh's study (2000) that he has categorized it as an algorithm-based mistake. generally, obedience to the rules and how to perform procedures properly require the students to memorize them. when they forget a few steps then it will certainly lead them to make mistake. the last problem encountered is an inability of the children to comprehend the fraction division case, particularly the fractions divided by fractions. this makes sense because they are not accustomed to confronting-context related to sources in both teaching and learning process. in indonesia, context-based teaching is still unique because education systems still focus on performance and result (purnomo, 2015b, 2016; purnomo, suryadi, & darwis, 2016; wijaya, 2017). in addition, mathematics textbooks in indonesia tend to consist of a set of rules and the use of performance-oriented algorithms (purnomo et al., 2014; wijaya, 2017). consequently, students are more likely to cope with regular problems and they encounter hardness in dealing with context-based problems. 4. conclusion the results of this study indicate that most of the participants of this study still tend to grape with difficulties in working on the fractions division. there are at least three crucial problems creating students difficulties in working on the case of fractions division. first, students' struggles are based on a shortage of conceptual understanding about equal sharing and fraction parts. second, the difficulty is based on an overemphasis on procedural understanding but not guided by a solid conceptual understanding. the last is students' unfamiliarity on the context-based problems leading to difficulties in interpreting the problem. references alenazi, a. 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(2010). children’s strategies for division by fractions in the context of the area of a rectangle. educational studies in mathematics, 73(2), 105–120. https://doi.org/10.1007/s10649-009-9206-0 purnomo, widowati, & ulfah, incomprehension of the indonesian elementary school … 74 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p189-198 189 the influence of concept attainment model in mathematical communication ability at the university students lilis marina angraini* universitas islam riau article info abstract article history: received march 23, 2019 revised sept 23, 2019 accepted sept 25, 2019 based on data from observation and interview with lecturer who teaches mathematics course on the subject under study, it was found that students' mathematical communication ability is still low. this study aims the students’ mathematical communication ability through concept attainment model learning. the method in this study is two groups randomized subject post-test only. the number of subjects in this study were 82 students. the kolmogorov-smirnov test, levene test, t test, anova one and two-way were used to analyse the data. the results of this study showed that (1) there is difference grade on the student’s mathematical communication ability between experimental group and conventional group as a whole, (2) there is no difference on the students’ mathematical communication ability of experimental classes based on their prior mathematical knowledge (pmk); (3) there is no interaction between the learning that is used with the students’ prior mathematical knowledge on the students’ mathematical communication ability. the concept attainment model provides a better influence on students’ mathematical communication ability, it is also necessary to see the influence of the concept attainment model on students' high-level mathematical thinking ability. keywords: concept attainment model, mathematical communication, prior mathematical knowledge copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: lilis marina angraini, departement of mathematics education, universitas islam riau, jl. simpang tiga, pekanbaru, riau 28288, indonesia. email: lilismarina@edu.uir.ac.id how to cite: angraini, l. m. (2019). the influence of concept attainment model in mathematical communication ability at the university students. infinity, 8(2), 189-198. 1. introduction committee of the undergraduate program on mathematics (cupm) provided six of basic recommendations for the department, programs, and also courses on mathematics (barker et al., 2004). one of the recommendation stated that each course on mathematics should be an activity that will assist students on the development of analytical, critical reasoning, problem solving, and communication skills. the cupm recommendation that is explained, the task of institution in charge of the educating prospective teachers who will teach mathematics, including preparing students for having mathematical communication ability. institutions of higher education personnel (ihep), departement of primary school mailto:lilismarina@edu.uir.ac.id angraini, the influence of concept attainment model in mathematical communication … 190 education which is on charge of childbirth prospective teachers who will teach mathematics partially take responsibility for preparing their students for having mathematical communication ability. mathematical communication is the ability that could be cultivated, the lecturer has a role in the development efforts of students’ mathematical communication ability. mathematical communication is an ability to communicate the mathematical knowledge properly and influenceively (wood, 2012). communication is the essential process in learning mathematics. through communication, students are able for organizing, reflecting upon and clarifying ideas, relationships, mathematical thinking and the mathematical arguments. during mathematics learning, students communicate for various purposes (for presenting or justifying a solution, for expressing mathematical arguments or to put a question) and to different audiences (teacher, colleague, group of students, whole class) (vale & barbosa, 2017). based on data from observation and interview with lecturer who teaches mathematics course on the subject under study, it was found that students' mathematical communication ability is still low. mathematical communication ability is important for students to have, because by having good mathematical communication ability students will find it easier to communicate mathematical ideas they have. concept attainment is a learning model that can bridge students 'mathematical communication ability, because in concept attainment learning there are steps that can train students in expressing mathematical ideas that they have, one of the steps that supports students' mathematical ability is the data presentation stage. bruner (joyce & weil, 2011) said learning process will be going well and creatively, if lecturer give students the opportunity for finding a rule (including concepts, theories, definitions, and so on) through examples that describes/ represents rules that became a source, in other words, students are guided inductively for understanding a common truth. process of study was presented by bruner is in line with theory of concept attainment model according by joyce & weil (2011), concept attainment model is more focused on ways to strengthen the internal human impulses on understanding science, by digging and organizing, as well as developing language for expressing it. bruner, goodnow, and austin reveal that concept attainment model deliberately designed for helping students to learn the concepts that can be used for organizing information, so it makes easy for students to learn concepts in more influenceive way (anjum, 2014; jones & hilaire, 2014). concept attainment model has several stages of learning, the stages in concept attainment models help to train students on mathematical communication ability. the first stage, namely the presentation of the data and the identification of the concept, in this stage, students are asked to compare the characteristic features on example and nonexample, students were asked to create and test the hypothesis, then the student making the definition of the concept on the essential characteristic features, here the students are trained to use critical thinking and also mathematical communication, because the students requested for comparing a characteristic features on examples and non-examples, and they find a definition of concept on characteristic of concept essential features (bhargava, 2016; jain & upadhyay, 2016). the next stage is testing an achievement a concept, the students were requested for identifying examples a concepts and creating additional instances, the students are exercising for using critical thinking and mathematical communication, students who think the characteristics are represented in a concept and kind of example that fulfills a criteria of a concept (kaur, 2014; ostad & soleymanpour, 2014). volume 8, no 2, september 2019, pp. 189-198 191 the last stage is the analytical thinking, the students are requested for communicating mathematics, the students are asked for expressing a concept by their own words, explains the reasons relating for creating additional instances, and writing down a steps for resolving a task on concept being learned, and formulating a mathematical concept, and this is learning model proposed by joyce & weil (2011) in theoretically sure that students’ mathematical communication ability will more better (kumar & mathur, 2013; widiastuti, 2014). kauchak & eggen (2012) suggestion that concept attainment model is an inductive learning model, designed by lecturer for helping students to learn the concepts and trained the students in practicing high-level thinking skills. concept attainment model is very relevant for teaching mathematics (mondal, 2013), because this model can foster understanding and appreciation the students to understand the concepts, principles so grows the power of reason, think logically, critically, systematically and others. concept attainment model is a learning model that aims for helping students understanding a particular concept, it is more appropriate when the emphasize of learning is more focused in the introduction of a new concept, so as trained high-level thinking skills (aningsih & asih, 2017; sharma & pachauri, 2016). the purposes of the study are: (1) to see the difference on students’ mathematical communication who got the concept attainment model and students who got the conventional teaching; (2) to see the difference on students’ mathematical communication who got the concept attainment model seen from the prior mathematical knowledge; (3) to see the interaction a learning and the prior mathematical knowledge on students' mathematical communication. 2. method the research was contained at one of islamic state universities. the population is all students in departement of primary school education 5th semester. all population chosed as sample the control and experiment class. the number of subjects in this study were 82 students.the researcher chooses the fifth semester in departement of primary school education student, because mathematics courses in odd semester can be used as objects, to examine students' in-depth communication ability about mathematical concepts only in semester v, in addition, departement of primary school education students semester v are also a group of students who are deemed ready to accept the treatment of this research, with the aim of becoming their own experience in teaching mathematical concepts before they go into the field, to carry out teaching practices in schools. the population was chosen randomly from two classes as research samples, from these two classes then randomly selected one class that would be the experimental group, and one class that would be the control group with sampling techniques adapted to population conditions. the test is two-dimentional figure that contains in a subject of mathematics education ii. the instrument test mathematical communication composed three questions. an example of the problem. angraini, the influence of concept attainment model in mathematical communication … 192 you are required to call a grade 2 elementary school students, who is not familiar with cartesian coordinates, but they are familiar with geometric shapes. you must explain the pqrs trapezoid image below (by telephone), so that the students can draw it exactly like the picture. write what you explained by telephone. the data about mathematical communication ability of students is obtained through tests of mathematical communication ability. the kolmogorov-smirnov test, levene test, t test, anova one and two-way were used to analyse the data. the t test was used to determine differences students’ mathematical communication who got the concept attainment model and students who got the conventional teaching. the one-way anova test was used to determine differences students’ mathematical communication who got the concept attainment model seen from the prior mathematical knowledge. two-way anova test was used to determine the interaction a learning and the prior mathematical knowledge on students' mathematical communication. before using the t test, the one-way and two-way anova tests, the prerequisite tests are the normality of the data distribution test using the kolmogorov-smirnov test and the homogeneity variance test of the data groups using the levene test. 3. results and discussion 3.1. results the data of prior mathematical knowledge is got and analysed for determining the students’ prior mathematical knowledge before the research was worked. prior mathematical knowledge was got from grades a subject of mathematics education, it acquired in a fourth semester. the grade is clustered by 3 categories of high, medium and low. the results are prior mathematical knowledge data (table 1). table 1. data equality test of prior mathematical knowledge t-test data criteria n 82 h0 accepted sig. (2-tailed) 0.23 table 1 explains there is no difference of prior mathematical knowledge between students who got concept attainment model and students who got conventional model. the data equivalence of students’ mathematical communication based on models shows in the table 2. table 2. equality of matematical communication t-test data criteria n 82 h0 rejected sig 0.00 volume 8, no 2, september 2019, pp. 189-198 193 table 2 explains there is difference between students’ mathematical communication who got concept attainment model and students who got conventional model. a results of students’ mathematical communication who got concept attainment model shows in the table 3. table 3. test of one-way anova the sum of squares df average f sig. inter-group 666.97 2 333.48 2.44 0.10 in a group 5602.66 41 136.65 total 6269.63 43 table 3 explains there is no difference students’ mathematical communication who got concept attainment model based on prior mathematical knowledge. furthermore, a results of an influence of interaction between learning model and prior mathematical knowledge shows in the table 4. table 4. test of two-way anova the sum of squares df average f sig. h0 learning 3242.61 1 3242.61 26.95 0.00 rejected prior mathemetical knowledge 1727.85 2 863.92 7.18 0.00 rejected interaction 72.03 2 36.01 0.29 0.74 accepted error 9142.45 76 120.29 total 132264.00 82 table 4 explains a learning factor that used in each group has impact on students’ mathematical communication. prior mathematical knowledge factor had significant impact on students’ mathematical communication. there is no influence caused by an interaction between learning model and prior mathematical knowledge on the students’ mathematical communication. 3.2. discussion based on overall there is difference in mathematical communication ability of students who are taught with the concept attainment model with students who are taught with conventional learning. from the previous data it can be seen that the mathematical communication ability of students who get the concept attainment learning model is better than students who get conventional learning. angraini, the influence of concept attainment model in mathematical communication … 194 there is a better improvement in the mathematical communication ability of students who are taught with the concept attainment model, than students who are taught with conventional learning, theoretically due to the concept attainment model, there are steps that can facilitate an increase in students' mathematical communication ability, these steps are: the first learning stage (joyce & weil, 2011), namely the presentation of data and identification of concepts, at this stage students are asked to compare the characteristics in the examples and non examples, students are asked to make and test hypotheses, then students make definitions of concepts for essential features, here students are trained to communicate because students express their ideas about the essential features of the concept. the second stage of learning (joyce & weil, 2011), namely the stages of testing the achievement of concepts, at this stage students are asked to identify examples of concepts and make additional examples, here students are re-trained to communicate their ideas about what characteristics represent a concept and examples of what meets the criteria of the concept. in the third learning phase (joyce & weil, 2011), students are asked to express concepts in their own words, express reasons related to making additional examples, and write steps for problem solving of the concepts being studied, as well as formulate the mathematical concept. during the learning process there are several things that concern the researcher regarding the mathematical communication skills of students at each stage of learning using cam. at the data presentation stage, the student as a whole looks carefully at it. the data presentation stage is the introduction stage of the general description of the concepts learned and an explanation of the work steps of the concept in the problem solving process. the role of students in this stage is to examine it, capture the intent and meaning, analyze the characteristics possessed by the concept. at the stage of testing the concept achievement, student activities take place by discussing with friends beside him. students are asked to discuss to identify the characteristics of the concepts learned then find concepts based on the characteristics they have found. the role of the lecturer here provides stimuli so that they are able to find concepts through their own discoveries. furthermore, at the stage of thinking strategy analysis, students are asked to reveal the reasons relating for formulating the concept by their own words and writing the concept through writing. the following will be presented the answers to the results of the discussion they did on the high line material, dividing lines, and the weight lines on the triangle. figure 1. answers of experimental class discussions figure 1 shows the answers of students when conducting discussions during learning. group 1's answer states that the high line is a perpendicular line forming a right angle. bisector is a line that forms the same angle. a weight line is a line that divides two volume 8, no 2, september 2019, pp. 189-198 195 sides of equal length. while the other group states that the height of the triangle is the perpendicular line showing the height of the triangle forming a right angle. bisector is a line that can show several new triangles (a line that forms two equal angles). a weight line is a line that divides two sides of equal length. the results above illustrate that the thinking process they are doing is right and there is something that is not right. after the discussion process took place, the lecturer gave confirmation about the accuracy of the concepts learned, so that students who were not right in the process of finding the concept got directions about the mistakes they did. based on the results of tests on students' mathematical communication skills tests, several errors were found by students who received learning concept attainment models, namely errors in algebraic completion, for example: 5a + 105 = 0, then 5a = 105, then a = 21. then there are students who solve questions like the following 13 + 13 + √50 + √50 = 13 + 13 + 10 = 36. although during learning, lecturers always straighten out the mistakes made by students in algebraic operations, but when the variables change, these errors still occur. this is evident when students complete questions about mathematical communication skills in the final test. learning by using concept attainment models (cam) provides a fairly good influence on the mathematical communication skills of students, this is evident from the acquisition of the average overall mathematical communication skills obtained by the experimental class is 6.9. this average is taken from the results of normalizing the data on the mathematical communication skills of students. although the average mathematical communication ability of the control class is higher, which is 8.5, this does not mean that the learning of the cam model is not better than conventional learning. the average communication ability of the experimental class is lower than the control class caused by various factors. the first factor is there are still many students in the experimental group who did not yet know ta basic concepts of algebraic, during a learning the researcher is more concern on correcting concepts, consequently learning in concept attainment model is less impact because the student's unbalanced abilities. concept attainment model emphasizes finding a concepts by their own selves. a students are still incorrect on their mathematical concepts, the learning is increasingly constrained, lecturer must correct concept that are incorrect, then can only continue a learning material. the second factor is the big number on students in one group (experimental group), which is 44 students, the researcher is somewhat inundate on providing impulse and they will reach a concept. from 44 students, only 15 students have a good point basic concept. during a learning lecturers came to students (whose basic mathematical concept were not well enough) step by step to provide impulse about teaching material so that students still get their own concepts. the third factor, the number of students who are late in class, so the influenceiveness of learning time is not optimal, it should be 2.5 hours of learning time but because many students are late the average learning only runs for 2 hours each time meeting. lecturers are constrained by students who arrive late, because they cannot balance the material with their friends. especially if there are students who are not present at the previous meeting, this makes the lecturer have to explain to the student about the previous material. angraini, the influence of concept attainment model in mathematical communication … 196 4. conclusion the conclusions of the problems in this researchs are: (1) there is difference on students’ mathematical communication who got the concept attainment model and students who got the conventional teaching; (2) there is no difference on students’ mathematical communication who got the concept attainment model seen from the prior mathematical knowledge; (3) there is no the interaction a learning and the prior mathematical knowledge on students' mathematical communication. the concept attainment model provides a better influence on students’ mathematical communication ability, it is also necessary to see the influence of the concept attainment model on students' high-level mathematical thinking ability. references aningsih, a., & asih, t. s. n. 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(2020). primary students‟ mathematical literacy: a case study. infinity, 9(1), 49-58. 1. introduction there has been increased attention to students‟ mathematical literacy nowadays and have widely admitted as one of interesting issues in mathematics education by many scholars both theoretically and practically. it is due to the need for everyone to have such ability to encounter todays complex problem mathematics literacy becomes one focus on the programme for international student assessment (pisa). the organization for economic co-operation and development‟s (oecd) programme for pisa and timss have heightened international awareness of the value and significance of mathematical literacy. the document of pisa 2003 explained that mathematical literacy plays important roles in solving pisa problem as described by hayat (2010). mathematical literacy is an individual‟s capacity to formulate, employ, and interpret mathematics in a variety of contexts. it includes reasoning mathematically and using mathematical concepts, procedures, fact, and tools to describe, explain, and predict phenomena. it assists individuals to recognize the role that mathematics plays in the world and to make the wellmailto:rooselynaekawati@unesa.ac.id ekawati, susanti, & chen, primary students’ mathematical literacy … 50 founded judgments and decisions needed by constructive, engaged and reflective citizens, oecd (2014). there is a growing concern on literacy program in the national level such as surabaya indonesia. it starts to develop reading culture at school by lead students to read book 10 to 15 minutes before the lesson start. in addition to it, mathematical literacy is as important as proficiency in reading and writing. mathematics is so entwined with today‟s way of life that we cannot fully comprehend the information that surrounds us without a basic understanding of mathematical ideas (ojose, 2011). these phenomena can influence the positive awareness of the needs of learners‟ mathematical literacy skills. report of international assessment of the pisa mathematics surveys showed that indonesian students‟ performance has not shown satisfactory results which posited below the average score. in the period 2003-2009 almost 80% of 15-years-olds students were only able to reach below level 2 out of 6 levels of questions examined (widjaja, 2011). moreover, on pisa survey 2009 almost all indonesian students only reached level 3, whereas only 0.1 % of indonesian students reaching level 5 and 6 (stacey, 2011). the report by widjaja (2011) shared 15-years-olds students‟ performance. therefore, indonesia is admitted to be able to give chance to students to learn mathematics especially in solving pisa problems. however, there is a limited study of exploring primary students‟ mathematics literacy in more details. numerous studies have reported the assessment of the domain of mathematics literacy, for example stacey (2011) reported the pisa view of mathematical literacy in indonesia which consider the pisa score. furthermore, wijaya, van den heuvelpanhuizen, doorman, & robitzsch (2014) pointed that indonesian secondary students mostly experienced difficulties in the early stages of solving pisa tasks, namely understanding the meaning of a problem and transforming a word problem into an appropriate mathematical problem. in terms of in european countries, höfer & beckmann (2009) discussed the supporting mathematical literacy which were applied in several schools in denmark, finland, slovenia and germany. the supporting model that delivered to those several european countries was believed to suit to the local situation. in order to have similar opportunity in supporting early learners‟ mathematical literacy in indonesia, it needs a clearer picture of the current indonesian students especially primary students‟ mathematical literacy. in this study, we aimed at comprehensively describing the profile of indonesian primary students‟ mathematical literacy in solving mathematics problem. in order to put forward the operationally reliable measure, we started with reviewing several literatures about mathematical literacy and afterwards developing framework for instrument used as well as undertaking detailed item analysis. 1.1. mathematical literacy there are several definition of mathematical literacy from pisa 2003 to pisa 2015. based on definition of pisa 2003, pisa 2012 elaborated mathematical literacy as „individual‟s capacity to recognize, do and use mathematics in a variety of contexts, and to identify the role that mathematics plays in the world. more mathematically literate individuals are better able to use mathematics and mathematical tools to describe, model, explain and predict phenomena, in order to make the well-founded judgements and decision required by constructive, engaged and reflective citizens (oecd, 2010). based on that definition, mathematical literacy is close related to problem in daily life that can be solved. the important aspect of mathematical literacy is considering problem solving in variety of context. the defining feature of mathematical literacy is that it has to do with thinking mathematically while simultaneously thinking about the context in which the mathematics is being used (stacey & turner, 2015). volume 9, no 1, february 2020, pp. 49-58 51 there are three connected aspects in mathematical literacy in oecd (2013) namely process, content and context. the categories of mathematical content are quantity, uncertainty & data, change & relationship; space & shape. furthermore, real world context categories consist of personal, societal, occupational and scientific. dealing with pisa problem, it necessarily involves mathematical thought and action that has three components such as mathematical concepts, knowledge and skills. the process of mathematical literacy begins with identifying problem in context and formulating the problem mathematically based on the concepts and relationship inherent in the problem. afterwards, the process of employing, interpreting as well as evaluating are considered. concepts and procedures, which are the backbone of traditional mathematical curriculum, are not the main focus of the pisa curricular framework (sáenz, 2009). therefore, through pisa problem, students need knowledge that can be productively used to problem situations. figure 1 showed the mathematical literacy in practice framework that described in oecd (2010). figure 1. mathematical literacy in practice (oecd, 2010) 2. method the participants in this study were sixth grade primary students. we delivered the instrument to 254 sixth graders from five region of surabaya, east java indonesia with various social and economic background. students‟ participants were asked to complete their identitiy and worked on the mathematics literacy tasks (mlt) phase 1: developing mathematics literacy tasks to document primary students‟ mathematical literacy, we developed paper pencil test instrument called mathematics literacy tasks (mlt). the characteristics of mlt satisfies the pisa framework. the developing of mlt item test involves two phases such as (1) defining conceptions of item components and (2) developing and validating test instrument with pilot study. the first phase: defining conceptions of item components. in this phase, we consider the pisa framework and considering context, content, process as well as level of problem. for the validating test instrument, we explore the reliability of instrument. problem in context formulate mathematical problem employ mathematical result interpretation results in context evaluating ekawati, susanti, & chen, primary students’ mathematical literacy … 52 phase 2: developing and validating the test instrument. the mlt went to multiple stages of revision before applied to larger number of participants. discussion with mathematics education experts and piloting test among 53 sixth graders in 90 minutes were executed and considered as the process of validating process of the test instrument. the comprehensive coding scheme for mlt were coded as „correct=score 1” and “incorrect=score 0”. cluster analysis in this study, cluster analysis was applied to the data mainly concerning students‟ responses to the mlt items. ”cluster analysis is required to divide a set of observations into groups or clusters in such a way that most pairs of observations that are placed in the same group are more similar to each other than are pairs of observations that are placed into different clusters (vidal, ma, & sastry, 2016)”. cluster analysis based on factor score was done to get the pattern description from students‟ responses. by the result of cluster analysis, primary students‟ mathematics literacy can be determined. in addition, we explored students‟ challenges in solving mlt with regard context, content, process as well as level of problem. besides, the interview were done to explore more on the picture of students challenges and difficulties. 3. results and discussion 3.1. results the mathematics literacy task (mlt) consist of 15 developed items, and the reliability result of cronbach‟s alpha of the piloted mlt was 0.55 for 53 students‟ respondents. significance level 5% is 0.2706. with criteria of decision of cronbach‟s alpha= 0.550 which is greater than r table then it considered reliable items. table 1 showed the description of mlt after some revision by considering pilot study result. table 1. description of mlt no context content process level description 1 personal quantity employ 1 sorting decimal in contextual problem 2 personal quantity interpret 2 using proportional reasoning for estimating the length of a thing. 3 personal uncertainty and data interpret 3 interpreting a statement related to average to explore its truth. 4 the work space and shape interpret 2 interpreting the form from above of a pile of cubes. 5 personal quantity employ 3 determining quantity of an object within contextual problem. 6 personal uncertainty and data interpret 2 interpreting data that showed in bar diagram volume 9, no 1, february 2020, pp. 49-58 53 no context content process level description 7 social space and shape interpret 4 interpreting visual form of object with given its volume 8 social quantity interpret 5 examining the truth of statement that related to percentage concept 9 personal space and shape formulate 5 formulate a simple mathematics model that related to concept of pattern 10 scientific quantity employ 3 using mathematics‟ rule that presented in contextual problem. 11 personal space and shape interpret 4 drawing cubes‟ net based on the cube‟s side 12 the work space and shape intepret 6 interpret the plane based on given perimeter. there are 6 levels of mathematical literacy problems given to students in this study. three problems were categorized in „high‟ level problems due to those are in level 5 and 6. five problems were in the level 3 and 4 that categorized in „medium‟ level problem. furthermore, four problems are categorized in „easy‟ level problem. furthermore, the detailed means and standard deviations of mlt items obtained from the factor analysis process are shown in table 2. table 2. description of statistics of the data mathematics literacy tasks number of samples mean standard deviation mlt1 254 0.3386 0.47416 mlt2 254 0.5827 0.49409 mlt3 254 0.0630 0.24343 mlt4 254 0.6260 0.48482 mlt5 254 0.4173 0.49409 mlt6 254 0.7717 0.42060 mlt7 254 0.5248 0.46792 mlt8 254 0.1992 0.35397 mlt9 254 0.0409 0.15647 mlt10 254 0.4551 0.48742 mlt11 254 0.0157 0.12474 mlt12 254 0.2087 0.40715 ekawati, susanti, & chen, primary students’ mathematical literacy … 54 as shown in table 2, the maximum standard deviation of the mlt items were around 0.4 that indicated the dispersion of the data point tended to be close to the mean and considered as normal data. cluster analysis in this study, cluster analysis was applied to the data mainly concerning students‟ responses to the items. cluster analysis based on factor score was done to get pattern of students‟ responses. there are three clusters categories to group the students in each school from five region of surabaya who responded to the items in similar ways so that the cluster could be obtained. the three groups g1, g2 and g3 could be interpreted as students with “good”, “middle”, or “low” students‟ mathematical literacy, respectively. table 3 showed students‟ mathematical literacy from five region of surabaya, east java, indonesia. table 3. the level of primary students‟ mathematical literacy no. schools’ region cluster good middle low 1. centre region 21.6% 43.14% 35.26% 2. east region 9.5% 40.5% 50% 3. south region 17% 28.8% 54.2% 4. west region 36% 62% 2% 5. north region 35.6% 47.4% 17% figure 2. students‟ mathematical literacy level based on data above (figure 2), in general, most students were in the middle categories of students in solving mathematical literacy problem. but in two regions, it volume 9, no 1, february 2020, pp. 49-58 55 shows students were in low category. in addition, with regard to the level of mlt problem, students‟ performance (mean score of item in each problem‟s categories) can be drawn in figure 3. figure 3. students‟ performance based on level of mlt problem the data shows that the average score for the easy level problem is less than 0.7 and for the high level, the average score is less than 0.22. it can be determined that in general, students need more opportunity to learn and get used to solve literacy problem. although students could attain high average score for easy problem, most students need more opportunity to learn about decimal fraction within contextual problem. they have difficulties in determining place value in decimal as well as in understanding the contextual problem given. furthermore, within all medium level problems, problem of statistics and geometry are more demanding than other. most students consider have challenge in solving problem related to ”mean, median and modus”. figure 4. mlt problem & a student‟ response on uncertainty & data figure 4 shows a problem on content uncertainty and data in which students are asked to determine the meaning of the given statement. student response that mean can be described as the average of students‟ weight (the total of weight divided by the number of students). however, he also considered that the mean can be interpreted as ‟most students has weight 42 kg‟ which can be determined as mode. in addition, some challenges were found by students in space and shape content problem. figure 2 shows a mlt problem on space and shape. 0 0.2 0.4 0.6 0.8 high medium easy students' mathematical literacy with regard level problem pay attention on the statement below in a classroom, there are 33 students with the average weight is 42 kg circle “yes” or “no” for every statement below based on the statement above statements is the statement below correct? most students has weight 42 kg yes/no if we make an order from the yes/no if we sum up all the students‟ weight and the result is divided by 33, the result is 42 kg yes/no ekawati, susanti, & chen, primary students’ mathematical literacy … 56 figure 5. space and shape content problem and most student‟s response there were only 1.6% students could answer the problem of figure 5 properly. students mention that they found difficulty in following the cutting direction and draw the net of cube that they familiar with. by this, it can be considered that they have difficulty in making translation between 3d objects and 2d nets. 3.2. discussion the result of quantitavive analysis of 254 students‟ responses on paper and pencil mathematics literacy test showed that most students were in middle and low clusters. in terms of mathematics literacy test problems, students performed best in determining graphical representation of content uncertainty and data that considered as easy level problem. this was due to they experienced about this lesson at school based on the document of curriculum. however, students faced challenges in higher level problem on content uncertainty and data. they had misunderstanding the concept of mean, median and mode. they were asked to determine the statistical statements given with the context of weight. the problem regards the students‟ mathematical literacy that also influenced by logical reasoning as also considered by ni‟mah, junaedi, & mariani (2017). furthermore, most students fail in level 3 of statistical literacy at school by watson (2003) namely inconsistent understanding. they could not engage with the context given and interpret the statistics incorrectly. for instance, students interpreted the average of students‟ weight as mode and median of the data of weight. this might happen since the texbook is only cover the data representation and not yet pointed to central tendency. besides, the most challenging problems faced by students are the shape and space content‟s problem. students need more opportunity to understand properties of objects and their relative positions in space and shapes. in other words, students are required to get more understanding the relationship between space and shape (visual representation) and includes understanding how three-dimensional objects can be represented in two dimensions (ojose, 2011). several research result showed that students have great difficulties in conceptualising the translation of two different representation modes (3d and 2d) (gutiérrez, 1992; ben-chaim, lappan, & houang, 1989). this phenomena happen a net of a cube box can be constructed by opening the edge of the box. the direction arrow shows the direction of the edge cutting of a box. draw a net of a cube box based on the picture above volume 9, no 1, february 2020, pp. 49-58 57 since the contruction of 2d nets need mental processes that students might not have. it needs an activity for students to utilize drawing 3d objects convention to avoid the misread drawing. overall, primary students need more opportunity to learn and get used to the mathematics literacy problems. to be more specific, the space and shape content and uncertainty and data contents were two demanding contents for primary students. 4. conclusion this study finds that indonesian students were able to work on content uncertainty and data problem. however, the most challenging problem they faced were shape and space contents‟ problem. in addition, a comprehensive overview of students‟ mathematics literacy is essentially needed in future studies to develop effective hypothetical mathematics learning trajectory that support students‟ mathematics literacy. references ben-chaim, d., lappan, g., & houang, r. t. 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(2014). difficulties in solving context-based pisa mathematics tasks: an analysis of students' errors. the mathematics enthusiast, 11(3), 555-584. https://doi.org/10.1007/s10649-008-9167-8 https://doi.org/10.1007/s10649-008-9167-8 https://doi.org/10.1007/s10649-008-9167-8 https://doi.org/10.1007/978-0-387-87811-9_2 https://doi.org/10.1007/978-0-387-87811-9_2 https://doi.org/10.1007/978-0-387-87811-9_2 http://iase-web.org/documents/papers/isi54/3516.pdf http://iase-web.org/documents/papers/isi54/3516.pdf http://iase-web.org/documents/papers/isi54/3516.pdf http://iase-web.org/documents/papers/isi54/3516.pdf http://dro.deakin.edu.au/eserv/du:30050761/widjaja-towardsmathematical-2011.pdf http://dro.deakin.edu.au/eserv/du:30050761/widjaja-towardsmathematical-2011.pdf sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p179-188 179 mathematical anxiety among engineering students rully charitas indra prahmana* 1 , tri sutanti 2 , aji prasetya wibawa 3 , ahmad muhammad diponegoro 4 1,2,4 universitas ahmad dahlan 3 universitas negeri malang article info abstract article history: received sept 2, 2019 revised sept 21, 2019 accepted sept 30, 2019 mathematical anxiety has a negative relationship with mathematics performance and achievement. further explained, mathematics anxiety has an indirect effect on mathematics performance. this research explores sources or factors related to mathematics anxiety among engineering students at a private university in indonesia. a total of 47 engineering students participated in this survey that randomly chosen based on gender, major, and age. two main factors are affecting the mathematics anxiety of engineering students, namely internal and external factors. the results show that mathematics anxiety among engineering students is manifested into three aspects. firstly, the home aspects are talking about the influence of parents and sibling. secondly, society's issues are discussing self-efficacy, social reinforcement to hate mathematics, and social stereotypes. lastly, the classroom aspects are talking about the traditional mathematics learning process and classroom culture, namely the experience of learning mathematics in classrooms and relationships between friends during learning. the details of the statements under the aspects also highlight unique problems and are not covered by previous research in mathematical anxiety. next, differences in mathematics anxiety by gender and faculty were examined. keywords: descriptive quantitative, engineering student, gender, higher education, math anxiety copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: rully charitas indra prahmana, master program on mathematics education, universitas ahmad dahlan, jl. pramuka 42, pandeyan, umbulharjo, yogyakarta, indonesia. email: rully.indra@mpmat.uad.ac.id how to cite: prahmana, r. c. i., sutanti, t., wibawa, a. p., & diponegoro, a. m. (2019). mathematical anxiety among engineering students. infinity, 8(2), 179-188. 1. introduction nowadays, mathematics and its ability is an essential and frequent phenomenon in education (hannula, 2012; sundayana, herman, dahlan, & prahmana, 2017). mathematical anxiety determines more than mathematics that overcomes the manipulation of numbers and the ability to solve mathematical problems both in everyday life and in the academic world (gresham, 2010). mathematical anxiety is the adequacy of intelligence for intelligent people to overcome quantification, faced with mathematical problems. in his research, blazer (2011) challenged mathematics universally as a non-intellectual factor that mailto:rully.indra@mpmat.uad.ac.id prahmana, sutanti, wibawa, & diponegoro, mathematical anxiety among engineering … 180 inhibited mathematical achievement. mathematical anxiety raises negative attitudes towards subjects, and results in poor and reasonable academic performance, increasing student performance in the mathematics learning process (gresham, 2010). therefore, it is necessary to have a discussion that discusses mathematics. mathematics underlies the universal development of technology (stoet & geary, 2018). mathematical knowledge is directly related to the ability to do logic, analytic, systematic, critical, and creative thinking (hoover, mosvold, ball, & lai, 2016; widodo, istiqomah, leonard, nayazik, & prahmana, 2019). engineers have to study math during lectures to support their work on technology innovation. in fact, prospective engineers still have difficulty in learning mathematics (vitasari, herawan, wahab, othman, & sinnadurai, 2010). engineering students have low comprehension and negative attitudes toward mathematics (kargar, tarmizi, & bayat, 2010; vitasari et al., 2010). they tend to avoid mathematics since their beliefs cannot solve math problems (bates, latham, & kim, 2011; charalambous & philippou, 2010). students with mathematical difficulties will look confusing, helpless, shy, nervous, and feel weak in concentration (charalambous & philippou, 2010; cranfield, 2013; hersh & john-steiner, 2010). engineering students still have a problem in learning mathematics because of their beliefs that affect their physical and emotional condition. engineering students have an affiliation with mathematics, where mathematics is essential for engineering as a language for describing physical, chemical, and other formulations in terms of mathematical inquiry (vitasari et al., 2010). furthermore, erden & akgül (2010) stated that high mathematical anxiety correlated with poor mathematical performance proposed at the university. students need a high level of concentration compared to other subjects so that it is possible to create anxiety about mathematics among engineering students (vitasari et al., 2010). mathematics anxiety feelings arise from the consideration of having symptoms such as fear, loss of interest, lack of concentration, impatience, confusion, and tension (gresham, 2010). on the other hands, female students more anxious than male in learning mathematics (goetz, bieg, lüdtke, pekrun, & hall, 2013; taylor & fraser, 2013). therefore, engineering students need a high level of concentration and low mathematics anxiety to learn mathematics, and female students don't have both of them yet. in this paper, we present a survey to explore mathematical anxiety among engineering students. this survey aims to explore sources or factors related to mathematics anxiety among engineering students at a private university in yogyakarta, indonesia. furthermore, the differences in mathematics anxiety base on gender and majors were examined. this survey involves extracting quantitative data from voluntary engineering student questionnaire groups, grouping variables under the theme, gender, and describe several aspects of the engineering students' mathematics anxiety based on their majors. 2. method the respondents were 47 students, consisting of 29 males and 18 females. they were informed to fill in the questionnaires based on what they experienced and learned during the lecture. these undergraduate students are from four engineering department: informatics engineering department (ifd), industrial engineering department (ied), electrical engineering department (eed), and chemical engineering department (ced). the previous research on mathematics anxiety used to develop the mathematics anxiety questionnaire. this survey contains ninety-two items with four scales: strongly disagree (1), disagree (2), agree (3), and strongly agree (4). students have to answer questions based on their experiences, feelings, and thoughts about mathematical anxiety volume 8, no 2, september 2019, pp. 179-188 181 felt while studying on campus. reliability and validity tests have been carried out. the result of the reliability test is 0.940, more than 0.70 as recommended by raykov & marcoulides (2011), accessing construct validity was interpreted by inter-correlation items (drost, 2011). the instrument used to measure the mathematics anxiety of engineering students consisted of 92 items, developed from shields (2005), whyte & anthony (2012), and also zakariya (2018) instruments, as shown in table 1. table 1. the rubric of mathematics anxiety questionnaires aspect indicator item number total items home 1. parents and siblings give a low status to the students’ mathematical ability and judgment that mathematics is complicated 2. parents let the child stop trying when the child has a mathematical fracture 3. parents demand excessive math success in children. 42, 65, 67, 68, 69, 70, 71, 72, 73, 74, and 75. 11 society 1. self-efficacy (men are better than women in mathematics). 2. social reinforcement to hate mathematics. 3. social stereotype (language skills are more critical and socially acceptable than mathematical abilities). 11, 15, 24, 25, 26, 27, 28, 34, 36, 37, 39, 43, 48, 49, 51, 52, 53, 54, 59, 60, 63, 64, 66, 77, 78, 79, 81, and 82. 28 classroom 1. the classroom aspects whose talk about the traditional mathematics learning process and classroom culture 2. the experience of learning mathematics in classrooms 3. relationships between friends during learning 5, 8, 9, 14, 16, 17, 18, 19, 22, 31, 32, 33, 38, 44, 45, 46, 47, 55, 56, 62, 76, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, and 92. 32 personal 1. physical and behavioral symptoms 2. perception of difficulty 3. low motivation 1, 2, 3, 4, 6, 7, 10, 12, 13, 20, 21, 23, 29, 30, 35, 40, 41, 50, 57, 58, and 61. 21 the survey was conducted during the holiday period after students finished the final examination. the inspector consulted several lecturers in each department to select some students to fill the questionnaire. next, the inspector gives a questionnaire link to be filled by voluntary respondents, who assumed had no awareness in learning mathematics anxiety. the time required for respondents to complete the survey was less than 60 minutes. students must read and answer questions as guided by the inspector. afterward, they have to answer the poll based on their learning experience. 3. results and discussion table 2 presents the demographic data of respondents. the results showed differences found in mathematics anxiety between engineering students on aspects, gender expectations, and variations based on majors. data analysis uses statistics descriptive data analysis. prahmana, sutanti, wibawa, & diponegoro, mathematical anxiety among engineering … 182 table 2. the demographic data of respondents demographic information frequency percentage gender male female 29 18 61.7% 38.3% department eed ied ifd ced 12 17 11 7 25.5% 36.2% 23.4% 14.9% age 18-20 21-23 24 above 2 29 12 4.3% 65.9% 29.8% total 47 100% 3.1. the home aspects: the influence of parents and sibling in engineering students' mathematics anxiety the home aspects were explained such as improving math scores when students increase their study time at home, parents' attention to mathematics learning outcomes, parents and siblings assessment of mathematics learning outcomes, and parents and siblings judgment about mathematics. figure 1 shows that informatics engineering students get the highest while industrial engineering students get the lowest home aspects score in mathematics anxiety. it means that the home aspect has the weakest contribution to the factors affecting mathematics anxiety for informatics engineering students. figure 1. the influence of parents and sibling in engineering students' mathematics anxiety the result was supported by shields (2005) that parents who suffer from math anxiety can accidentally transfer it to their children. parents, especially mother, are a consistent example for their children because their children pay close attention to the attitude of the mother. the position of mathematics attitude that shown to children in the way mothers sees mathematics as a valuable and understandable lesson (makur, prahmana, & gunur, 2019). however, parents can inadvertently increase mathematical anxiety in their children by giving them reasons to stop trying when they are frustrated or upset because of 28 29 30 31 32 33 34 electrical engineering industrial engineering informatics engineering chemical engineering volume 8, no 2, september 2019, pp. 179-188 183 difficulties with math assignments (stolpa, sloan, daane, & giesen, 2004). therefore, parents who suffer from math anxiety can inadvertently transfer this anxiety to their children. 3.2. the society issues: self-efficacy, social reinforcement to hate mathematics, and social stereotypes the society issues consist of several aspects affecting the engineering students’ math anxiety. the first aspect is self-efficacy, which stated that boys are better than girls in mathematics. the next aspect is social reinforcement to hate mathematics. the last aspect is the social stereotype that language skills are more critical and socially acceptable than mathematical abilities. figure 2 showed that electrical engineering students get the highest and informatics engineering students to get the lowest society issues to score in mathematics anxiety questionnaire. it means that social problems contribute the weakest to the factors affecting mathematics anxiety for electrical engineering students. social factors such as mathematical myths can also induce or strengthen mathematics anxiety for some students, i.e., the myth that boys are better than girls in mathematics and that only a few people have a 'mathematical mind' can damage positive self-confidence (whyte & anthony, 2012). a study confirmed that failure in mathematics was socially acceptable participants were less embarrassed about the lack of mathematical skills compared to language skills (latterell, 2005). figure 2. the society issues score 3.3. the classroom aspects: the traditional mathematics learning process and relationships between friends during learning the classroom aspects discuss the traditional mathematics learning process and classroom culture, the experience of learning mathematics in classrooms, and the relationships between friends during learning. here, most engineering students agree that mathematics is boring. however, although they also have difficulty in learning mathematics, they still want to learn mathematics successfully. simple observation found that some the students feel boring because of lack of calculation activities, the students more interest in reading than count (vitasari et al., 2010). mathematics requires higher levels of concentration compared than other subjects (rattan, good, & dweck, 2012). figure 3 describes that electrical engineering students reach the top of the lowest classroom aspects to score in mathematics anxiety, which means that this factor does not give significant influence to them. 68 70 72 74 76 78 80 electrical engineering industrial engineering informatics engineering chemical engineering prahmana, sutanti, wibawa, & diponegoro, mathematical anxiety among engineering … 184 figure 3. the classroom aspects score personal characteristics and academic variables have a rare influence on mathematics anxiety (karimi & venkatesan, 2009). traditional teaching can contribute to mathematics anxiety and also class culture defined as the behaviors and norms that guide class interaction (whyte & anthony, 2012). traditional teaching means teacher center learning with direct instruction without discussion. the experience of learning mathematics in a structured and rigid classroom includes several opportunities to debate or discuss, focus on finding the right answers, offer limited encouragement to reflect on thinking, expect quick responses, and emphasize the test of time (shields, 2005). in such classrooms, the possibility of teacher’s behavior implicitly grows students' mathematical anxiety (mensah, okyere, & kuranchie, 2013). 3.4. the personal aspects: physical and behavioral symptoms, perception of difficulty, and low motivation the personal aspects consist of three issues, namely physical and behavioral symptoms, perception of difficulty, and low motivation. this aspect represented in 21 of 92 questions in the mathematics anxiety questionnaire. the problem started about the student condition during math class, such as my limbs trembled, sweated a lot, had difficulty breathing, my heartbeat fast, felt weak, and cold and hot contribute the significant score for engineering students mathematics anxiety. students who are nervous, bored, afraid, or believe that mathematics is not essential; tend to avoid learning mathematics (furner & berman, 2003), yet want to get a satisfactory grade in mathematics. overcoming this anxiety of the students for mathematics is the real challenge of every lecturer and also made a good mathematics instruction to solve them (mensah et al., 2013; tanujaya, prahmana, & mumu, 2017). figure 4 describes that electrical engineering students get the highest while informatics engineering students get the lowest of personal aspects to score in mathematics anxiety questionnaire. it means that the personal aspects contribute the weakest to the factors affecting mathematics anxiety for electrical engineering students. 80 82 84 86 88 90 92 94 electrical engineering industrial engineering informatics engineering chemical engineering volume 8, no 2, september 2019, pp. 179-188 185 figure 4. the personal aspects score 3.5. the differences effects of gender and engineering major towards mathematic anxiety the differences in gender towards engineering students’ mathematic anxiety were investigated. the results show that female engineering students more anxious than male engineering students as stated by karimi & venkatesan (2009). female students may perform the same level as the male students when they are given the right educational tools and have visible excelling in mathematics (else-quest, hyde, & linn, 2010). figure 5 shows that the male electrical engineering students get the highest and male informatics engineering students to get the lowest mathematics anxiety score in mathematics anxiety. figure 5. the differences effects of gender towards mathematic anxiety the differences effects of engineering major towards mathematics anxiety were examined. electrical engineering students get the highest score in 3 of 4 aspects that affecting engineering students’ mathematics anxiety. it means that these aspects contribute the lowest to the factors affecting mathematics anxiety for electrical engineering students as showed in figure 6. 46 47 48 49 50 51 52 53 54 electrical engineering industrial engineering informatics engineering chemical engineering 0 50 100 150 200 250 300 electrical engineering industrial engineering informatics engineering chemical engineering female male prahmana, sutanti, wibawa, & diponegoro, mathematical anxiety among engineering … 186 figure 6. the differences effects of engineering major towards mathematic anxiety 4. conclusion mathematics may cause learning anxiety among engineering students. mathematical anxiety among engineering students is manifested in four aspects: personal, classroom, community, and home aspects. the difference in scores on mathematics anxiety found that females were more anxious than male students. furthermore, informatics engineering has the highest level of mathematics anxiety compared to other department. therefore, investigations to reduce mathematics anxiety must be sought to improve student academic performance. it can help engineering students overcome their fear and improve of the quality of learning mathematics. for better and more comprehensive research, further discussion of a mathematics anxiety should be focused on other majors, educational levels, and diversity of sources. acknowledgements the authors would like to thank the director general of strengthening research and development, ministry of research technology and higher education of the republic of indonesia that supported and funded this research under the research grant namely penelitian dasar unggulan perguruan tinggi based on decree number 6/e/kpt/2019 and 7/e/kpt/2019. the researcher also thanks to universitas ahmad dahlan for giving the opportunity and facilities to complete this research. lastly, the authors thank all respondents and their lecturer for their participations in this research. references bates, a. b., latham, n., & kim, j. a. 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https://iopscience.iop.org/article/10.1088/1742-6596/1188/1/012087/meta https://iopscience.iop.org/article/10.1088/1742-6596/1188/1/012087/meta http://ijopr.com/index.php/ijopr/article/view/43 http://ijopr.com/index.php/ijopr/article/view/43 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p239-246 239 teacher’s and student’s needs for mathematical problems in disaster context nuzulidar 1 , rahmah johar *2 , sulastri 3 1,2,3 universitas syiah kuala article info abstract article history: received sept 5, 2019 revised sept 27, 2019 accepted sept 30, 2019 indonesia is an archipelagic country lies on the pacific ring of fire, resulting in the country being vulnerable to disaster. teachers need to accustom students to manage natural disaster situation in a more logical approach. therefore, it is necessary to develop mathematical problems in disaster contexts. this research is an early stage of developmental research. the purpose of this study was to analyze the needs of teachers and students of mathematical problems in disaster contexts. the participants in this study were a mathematics teacher and 53 year 7 and 8 students at one of the public junior high school, located in a tsunami affected area, in banda aceh. data collection involved an open questionnaire, and data analysis was carried out descriptively. the results showed that the teacher often provided mathematical problems in learning but had never read mathematical problems in disaster contexts. also, only three students had ever read such problems. the results also revealed that nearly half of the students (41.5%) liked to solve mathematical problems. besides, both the teacher and 71.7% of the students agreed and were willing to participate in the learning process involving mathematical problems in disaster contexts. the results of the study also showed that the mathematical questions in disaster contexts were limited. thus, it is necessary to develop mathematical problems in disaster contexts. keywords: disaster context, mathematical problems copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: rahmah johar, departement of mathematics education, universitas syiah kuala, jl. tgk. hasan krueng kalee, kopelma darussalam, banda aceh, aceh 24415, indonesia email: rahmah.johar@unsyiah.ac.id how to cite: nuzulidar, n., johar, r., sulastri, s. (2019). teacher’s and student’s needs for mathematical problems in disaster context. infinity, 8(2), 239-246. 1. introduction indonesia, geographically, is a disaster-prone country due to the circum-pacific belt, the earthquake zone surrounding the pacific ocean. this circumstance requires resilience in every element of society toward disaster risk. disasters are events that occur due to natural or non-natural factors that lead to casualties and damage to infrastructure (mcdonald, 2003). disaster events can be triggered by natural events, human actions or a combination of both (amri, 2017). an important part of disaster management is the mitigation, the measures to reduce disaster risk, by improving the physical development, the awareness and the ability to mailto:rahmah.johar@unsyiah.ac.id nuzulidar, johar, sulastri, teacher’s and student’s needs for mathematical problems … 240 confront disaster threats. disaster mitigation efforts are efforts to save one self from natural disasters that occur suddenly and threaten one’s life (setiawan, 2016). communities threatened by disasters are so diverse that the most strategic way to educate people about disaster mitigation is through formal and informal education. one of the efforts that the school can do as a formal educational institution in reducing disaster risk is through the integration of disaster risk education in extracurricular activities and various subjects (amri, 2017). one of which through the regular lesson, such as mathematics (shadiq, 2016) because mathematics is a clear and logical means of thinking to solve contextual problems (apec, 2012). aceh is one of 34 provinces in indonesia, with a total of 2,000 schools in a highrisk disaster area (sinambela & nugrahini, 2016). since ten years after the tsunami in aceh, disaster risk reduction has been integrated into the school curriculum, and schoolbased disaster preparation programs have also been carried out (tdmrc unsyiah, 2014). so, disaster is used as one of the contexts in learning mathematics. the contexts provided in learning can be gained from nature, social life, culture, economy, and religion. therefore, the content of learning materials is inseparable from the student’s learning environment and the daily life (zakiyah & rusdiana, 2014). contexts play an important role in achieving the goals of learning mathematics because they can help students understand the materials meaningfully (johnson, 2002). they also benefit the students in solving mathematical problems, making it easier for students to choose the way to solve problems and to provide students with strategic solutions in solving problems (van den heuvel-panhuizen, 2005). provision of contexts in the learning environment can provide answers to students' problems "why should i study this?" and can bring meaningful learning (johar & hanum, 2016). schools have a critical role in developing knowledge to build the community resilience to disasters (oktari, shiwaku, munadi, & shaw, 2015). previous researchers, such as fatmawati (2016), have developed the mathematical problems in disaster contexts but the availability of such questions remains lacking. besides, research that examines the needs of students and teachers for mathematical problems in the context of disaster is limited. therefore, the purpose of this study was to analyze the needs of teachers and students of mathematical problems in disaster contexts. 2. method this study is an early stage of developmental research, particularly the preliminary stage of the tessmer model (tessmer, 2013). the participants were a mathematics teacher and 53 year 7 and 8 students in one of the public junior high school, located in a tsunami affected area, in banda aceh. the participants could define disaster well, provide some examples of disasters, and explain what needs to do in the event of an emergencies. they mentioned some places to use as an early evacuation site, including the evacuation building, hill or mountain, floating ship of a diesel power plant, tsunami museum, tall buildings, and mosques. they admitted that they obtain information about disaster through electronic media, books, schools, and parents. the data collection involved an open questionnaire and interview with several participants to analyze the unique cases of student answers deeply. both the questionnaire and the interview were used to investigate the teacher’s and student’s needs for mathematical problems in disaster contexts. data analysis was conducted descriptively. the example of mathematical problems in disaster contexts provided in the questionnaire were adapted from khalid & ali (2016) and presented as follows. volume 8, no 2, september 2019, pp. 239-246 241 abu was standing on a beach when he heard the tsunami warning siren. he immediately decided to run to a safe place. he had two choices: going to a small hill or a shelter built for tsunami, which can be reached via two ways. the hill is 500 metres and the shelter is 800 metres away from the beach. however, to reach the top of the hill, abu need to climb 300 steps of staircases. abu can run at an average rate of 5m/s and he can climb the stairs at the rate of 3 steps/s. he may also use the curved road (specially made for the shelter) which is 900 metres long, where he can run at 6m/s. in your opinion, which way should abu choose? please explain! the finding of research conducted by khalid & ali (2016) revealed that students were motivated to solve the problem and he suggested teachers to guide students in solving contextual problem in disaster context. in this study, students were asked to respond to the problem in a disaster context, whether they had read/solved such a problem before, and whether they were interested in and willing to solve such a problem? 3. results and discussion the results of teacher questionnaire showed that the mathematics teacher, who taught year 7 and 8, had often provided mathematical problems in mathematics learning but she had never read/solved mathematical problems in disaster contexts. the teacher agreed that mathematical problems in disaster contexts should be given in mathematics learning because it can improve students’ mathematical literacy. the questionnaire results of year 7 students about the student’s needs for mathematical problems in disaster contexts are presented in figure 1. figure 1. year 7 students’ needs for mathematical problems in disaster contexts figure 1 shows all students’ preferences, meaning that 20 students were happy with mathematical problems, but only 17 students were interested in the problems in disaster contexts. only three students had ever read/solved mathematical problems in disaster contexts. in addition, 26 students agreed if the teacher gives mathematical problems in disaster contexts in learning and they were willing to solve mathematical problems in disaster contexts. the questionnaire results of year 8 students concerning the students’ needs for mathematical problems in disaster contexts displayed in figure 2. 20 3 26 17 26 0 10 20 30 preference participation approval interest willingness nuzulidar, johar, sulastri, teacher’s and student’s needs for mathematical problems … 242 figure 2. year 8 students’ needs for mathematical problems in disaster contexts only two of year 8 students felt happy with mathematical problems. however, 12 students said that they were interested in mathematical problems in disaster contexts. all students had never read/solved mathematical problems in disaster contexts. also, 12 students asserted that they agreed if the teacher gives mathematical problems in disaster contexts in mathematics learning, and they were willing to solve such mathematical problems. seven students admitted that they were not happy with the mathematical problems, but they attracted to mathematical problems in disaster contexts. providing contextual problems can increase student motivation in learning mathematics (khalid & ali, 2016). that’s why the participants who were not pleased with the mathematical problem showed positive attitudes towards mathematical problems in disaster contexts. the questionnaire results of year 7 and 8 students regarding the students’ needs for mathematical problems in disaster contexts are presented in figure 3. figure 3. the student’s needs of mathematical problems in disaster contexts figure 3 illustrated the preferences of all students. twenty-two students (41.5%) were happy with mathematical problems, and nearly half of the students (49.1%) were interested in the problems in disaster contexts. only three out of 53 students (5.6%) had ever read/solved mathematical problems in disaster contexts. in addition, more than 70% of the students agreed if the teacher gives mathematical problems in disaster contexts, and they were willing to solve mathematical problems in disaster contexts. 2 0 12 9 12 0 5 10 15 preference participation approval interest willingness 22 3 38 26 38 0 10 20 30 40 preference participation approval interest willingness volume 8, no 2, september 2019, pp. 239-246 243 table 1 presents the questionnaire results of year 7 students about the students’ needs for mathematical problems in disaster contexts. table 1. the year 7 students’ needs for mathematical problems in disaster contexts subject question number 1. do you like to solve mathematical word problems? 2. have you ever read/solved mathematical problems in disaster problems? 3. do you agree if the teacher gives a mathematical word problems in disaster context as the example? 4. are you interested of mathematical problems in disaster context as the example? 5. are you willing to solve mathematical problems in disaster context (see an example)? a not happy never disagree not interested willing b happy never agree not interested willing c happy never agree interested willing d not happy never agree not interested willing e happy never disagree not interested not willing f happy never agree not interested willing g not happy never agree not interested willing h not happy never agree interested willing i not happy never agree interested willing j not happy never agree interested willing k happy ever agree interested willing l happy ever agree not interested willing m happy ever agree interested willing n not happy never agree interested willing o happy never agree interested willing p happy never agree interested willing q happy never agree interested willing r not happy never agree not interested not willing s happy never agree not interested willing t not happy never agree interested not willing u happy never agree interested willing v happy never agree not interested willing w happy never disagree not interested willing x happy never disagree not interested not willing y happy never agree interested willing z not happy never disagree not interested not willing aa not happy never agree not interested willing ab happy never agree interested willing ac happy never agree interested willing ad happy never agree interested willing ae happy never agree interested willing here are the interview excerpts with one of the participants (k). k was one out of three year 7 students who had ever read/solved a mathematical problems in disaster context: q : do you like mathematics? a : yes miss. q : which one do you prefer, problems on number or word problems? a : word problems miss q : you answered question number 2. what is the problem and where do you read it? a : yes mis, the problem was about how much medicine needs for disaster victim. i read it in the elementary school, "thematics" book. both the questionnaire and the interview results of k showed that she liked mathematics and word problems. she had read/completed mathematical problems in disaster contexts from the book "thematics" when she was in elementary school. nuzulidar, johar, sulastri, teacher’s and student’s needs for mathematical problems … 244 "thematics" book consists of themes to link the contents of several subjects and to develop materials based on the environment to provide a meaningful experience for students. five out if six students interviewed mentioned similar answer concerning their reasons for the low interest in mathematical problems in disaster contexts. here is an interview excerpt with student b. q : do you like mathematics? a : not really miss q : which one do you prefer, problems on number or word problems? a : both of them miss q : why did you said happy for number 1 but your response was not interested for number 4? a : because of the length of the problem miss. both the results of the interview and questionnaire of b represented five other people who mentioned a similar reason. thus, it can be concluded that they liked mathematics. they agreed and would solve the problem if the teacher gave it, but they had a low interest in mathematical word problems in disaster contexts. the negative response was due to the length of the problem, indicating that they tried to abstain from more complex problems. the responses to student questionnaires and the interviews with several participants concluded that there were some students who were happy with mathematical problems but were not interested in mathematical problems in disaster contexts because the sample form presented was too long. however, students should accustom to solving contextual problems because those without this experience will face difficulties in the future, even leading to the refusal to solve them. contextual problems alone cannot directly help students in understanding the concept or motivate them (boaler, 1993; carraher & schliemann, 2002). moreover, the problems do not necessarily guarantee students to learn meaningfully. so, teachers need to engage students to interpret the contexts and to explore ideas in solving mathematical problems (widjaja, 2013). students who are not familiar with solving contextual problems will experience difficulties in learning because they need to connect their knowledge and real-life applications. therefore, mathematics contextual problems should also be developed by using a sentence that easily understood by relatively average student. in regard to the textbooks in indonesia, wijaya, van den heuvel-panhuizen, & doorman (2015) found that only 10% of the tasks in the textbooks are context-based. the use of contextual problems support students to develop mathematical understandings (dolk, widjaja, zonneveld, & fauzan, 2010). therefore further research needs to develop mathematics problem or mathematics textbooks in the context of disasters and analyze their impacts on students' problem solving skills. 4. conclusion this study on the importance of mathematical problems in disaster contexts resulted in several findings. the teacher often provided mathematical problems in learning but had never read mathematical problems in disaster contexts. besides, nearly half of the students (41.5%) liked to solve mathematical problems, but only three students had ever read the problems in disaster contexts. both the teacher and 71.7% of the students agreed and were willing to participate in the learning process involving mathematical problems in volume 8, no 2, september 2019, pp. 239-246 245 disaster contexts. further research developing mathematical problems in disaster contexts is necessary to develop literacy skills and raise students' awareness of disasters. acknowledgements we would like to thank direktorat riset dan pengabdian masyarakat (drpm) from kementerian ristek dikti republik indonesia for funding this research through the graduate thesis grant, the year 2019, no: 099/sp2h/lt/drpm/2019, date: 8 march 2019. references amri, a. (2017). pendidikan tangguh bencana: mewujudkan satuan pendidikan aman bencana di indonesia. jakarta: kementerian pendidikan dan kebudayaan. apec. (2012). the role of education for natural disasters. an extraction from the proceedings of the criced 10th anniversary symposium. japan: university of tsukuba. boaler, j. (1993). the role of contexts in the mathematics classroom: do they make mathematics more" real"?. for the learning of mathematics, 13(2), 12-17. carraher, d.w. & schliemann, a.d. (2002). is everyday mathematics truly relevant to mathematics education. journal for research in mathematics education monograph, 11, 131-153. dolk, m., widjaja, w., zonneveld, e., & fauzan, a. (2010). examining teachers’ role in relation to their beliefs and expectations about students’ thinking in design research. a decade of pmri in indonesia, 175-187. fatmawati, d. (2016). pengembangan soal matematika pisa like pada konten change and relationship untuk siswa sekolah menengah pertama. mathedunesa, 5(2). johar, r., & hanum, l. (2016). strategi belajar mengajar. yogyakarta: deepublish. johnson, e. b. (2002). contextual teaching and learning: what it is and why it's here to stay. corwin press. khalid, m., & ali, d. h. p. h. (2016). inculcating tsunami awareness in a mathematics lesson: improving students’ collabirative problem solving via lesson study. southeast asian mathematics education journal. 6(1), 19-31. mcdonald, r. (2007). introduction to natural and man-made disasters and their effects on buildings. routledge. oktari, r. s., shiwaku, k., munadi, k., & shaw, r. (2015). a conceptual model of a school–community collaborative network in enhancing coastal community resilience in banda aceh, indonesia. international journal of disaster risk reduction, 12, 300-310. setiawan, b. (2016). agenda pendidikan nasional. yogyakarta: ar-ruzz media. shadiq, f. (2016). how can seameo witep in mathematical help indonesian mathematics teachers to help their students to be independent learners in the case of disaster risk reduction (drr)?. southeast asian mathematics education journal, 6(1), 3-17. nuzulidar, johar, sulastri, teacher’s and student’s needs for mathematical problems … 246 sinambela, a., & nugrahini, e. (2016). media komunikasi dan inspirasi: jendela pendidikan dan kebudayaan iv/agustus-2016. media komunikasi dan inspirasi: jendela pendidikan dan kebudayaan, 4, 04-20. tdmrc unsyiah. (2014). recovery assessment after 10th years earthquake and tsunami aceh 2004. final report. banda aceh (in press). tessmer, m. (2013). planning and conducting formative evaluations. routledge. van den heuvel-panhuizen, m. (2005). the role of contexts in assessment problems in mathematics. for the learning of mathematics, 25(2), 2-23. widjaja, w. (2013). the use of contextual problems to support mathematical learning. indonesian mathematical society journal on mathematics education, 4(2), 157-168. wijaya, a., van den heuvel-panhuizen, m., & doorman, m. (2015). opportunity-to-learn context-based tasks provided by mathematics textbooks. educational studies in mathematics, 89(1), 41-65. zakiyah, q. y., & rusdiana. (2014). pendidikan nilai: kajian teori dan praktik di sekolah. bandung: cv pustaka setia. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p43-56 43 development and comparison of mathematic mobile learning by using exelearning 2.0 program and mit inventor 2 tri nova hasti yunianta* 1 , anissa putri 2 , dani kusuma 3 1,2,3 universitas kristen satya wacana article info abstract article history: received jan 3, 2019 revised jan 29, 2019 accepted feb 2, 2019 mathematics mobile learning influences student learning outcomes. making this application need skills and also the appropriate programs. the first goal in this study is to develop valid and effective mathematics mobile learning. the second goal in this study if using two different programs it would get the same or different results, is it different between the two applications made with different programs? this type of research is research and development (r & d). the first application research subjects were mathematics teachers and 8th grade students of junior high school in salatiga consisted of 28 students and the subjects in this study were divided into two, namely students in the 8g class who used the mobile learning application and who did not use it and this study was conducted in the second semester academic year 2014/2015. the second application research subject was 10th-grade students of senior high school in salatiga, consisted of 24 students. this research was conducted in the second semester of the academic year 2015/2016. the first application development model was using assure model and developed with exelearning program, while the second application used the addie model and developed with mit inventor 2 program. data on learning outcomes were obtained by giving initial tests and final tests. data were analyzed using n-gain enhancement test. the mathematics mobile learning application used the exelearning 2.0 program to obtain validation results with very good criteria for the display and material sections while for the improvement of student learning outcomes in the high category amounting to 0.7. the second mathematics mobile learning application obtained very good display validation results and for the material section in the good category, while the increase in learning outcomes obtained an increase in the high category which was 0.71. both of these applications possessed differences and the characteristics of mobile learning applications also depend on which software is used. it has a unique impression of using the exelearning 2.0 application and mit inventor 2. keywords: unique impression mathematics mobile learning exelearning 2.0 mit inventor 2 copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: tri nova hastiyunianta, departement of mathematics education, universitas kristen satya wacana, jl. diponegoro no 52-60, salatiga, jawa tengah email: trinova.yunianta@uksw.edu how to cite: yunianta, t. n. h., putri, a., & kusuma, d. (2019). development and comparison of mathematic mobile learning by using exelearning 2.0 program and mit inventor 2. infinity, 8(1), 43-56. mailto:trinova.yunianta@uksw.edu yunianta, putri, & kusuma, development and comparison of mathematic mobile … 44 1. introduction everyone needs to keep learning to deal with the changing times so quickly. learning can occur anywhere, whether formally held at school or informally outside of school. robert m. gagne (pribadi, 2009) stated that learning is a natural process that leads to changes that we know, what we can do, and how we behave. slameto (2003) said that learning is a series of physical and mental activities to achieve changes in human behavior due to the interaction of personal experiences related to cognitive, affective, and psychomotor domains. with learning activities, students experience processes and change their abilities from not knowing to know something, on the other hand, learning is the development of new knowledge, skills, or attitudes as people who interact with learning resources (pribadi, 2009). learning resources are all resources used by teachers and students in the learning process in the classroom. this is used as a communication tool, which can interact with students in learning activities and learning processes. learning media is a tool for delivering learning material. this is also a tool to bring problems and be solved by students (sudjana & rivai, 2013). miarso (2009) said that learning media is everything that is used to send messages and stimulate students' thoughts, feelings and desires to encourage the learning process. optimizing the use of ict for 21st century learning is very important to be developed especially to create learning media. the 21st century students live in a boring world of media and spend an average of almost six and a half hours with media and students as multimedia or technology users. recent studies in the use of ict in the form of mobile devices are suitable tools especially for improving the quality of education in developing countries (el-sofany et al, 2014). technology that can be used or utilized in education is a variety of computers, laptops, tablets, mobile phones and many more. tablet is a technology that has a size of 710 inches. the next technology is mobile and is the technology that is closest to students and often used by students in everyday life. cellphone users in indonesia always increase every year. in indonesia currently active mobile users have reached 281.9 million people. this number illustrates that everyone in indonesia has 1.13 units of cellphones. along with the development of the word mobile has turned into a smartphone because this application is growing and increasingly sophisticated. based on the results of surveys and analysis conducted by international data corporation in 2009 to 2013 it can be concluded that smartphone users always increase. in 2009 users reached 2.04 million, in 2012 dramatically increased to 13.20 million users and in 2013 became 15.30 million users and students included. mobile learning is a mobile device such as a smartphone or tablet pc and also known as a download application and has certain functions so that the function of the mobile device itself. according to idrus & ismail (2010) mobile learning is a learning media that utilizes handphone technology. the concept of application learning brings benefits from the availability of teaching materials that can be accessed at any time and the visualization of interesting material. motiwalla (2007) found that mobile learning applications (mla) can be used in class or in distance learning and mla will never replace classroom learning, but if both are combined, learning can be more effective and flexible. mobile learning is a unique type of learning tool to gain access to learning materials, instructions, and questions related to student learning, when and wherever they want to study. learning can develop by paying attention to learning material, so learning becomes widespread, and can encourage students. motivate for lifelong learning. mobile volume 8, no 1, february 2019, pp. 43-56 45 learning is student learning activities and learning in various places or environments by using technology, especially mobile phones with features and applications. students can learn wherever they are and provide interesting material so that students don't get bored. this was created to support education in the digital era. mobile learning is also considered to be the next step of e learning by using wireless devices and other communication technologies so that that they can be used in the learning process anytime and anywhere (ismail & idrus, 2009). seeing the importance of learning resources and media for students in this digital era, mathematic mobile learning really needs to be developed to be able to help teachers in the teaching and learning process and also to improve their learning outcomes. developing this mobile learning program has several programs and in this study it will be focused on developing two mathematic mobile learning applications namely exelearning 2.0 and mit inventor 2. in addition to looking at the validity and effectiveness of the application, it is necessary to look at its uniqueness by examining the comparison of the results of development in order to provide an overview for the researcher who will use both programs. are the two mathematic mobile learning applications developed valid and effective? how unique is the difference between the two applications. 2. method this research is a type of research and development (r & d) used to produce certain products and test the effectiveness of these products. the product produced in this study was mathematics mobile learning application. this research developed two mobile learning products using different software programs. the first product was mobile learning which was developed using the exelearning 2.0 program and the second product was developed using the mit inventor 2 program. the first product research subjects were mathematics teachers and 8th grade students of smpn 6 salatiga consisted of 28 students and the subjects in this study were divided into two namely students in class 8g who used the mobile learning application and those who did not use it. this research was conducted in semester 2 academic year of 2014/2015. the development of mobile learning application in the first product used the development model of assure. the steps of assure model are as follows: 1). analysis step (analyze learners) where this stage analyzes age, sex, and type of learning style; 2). determine step (state) where the purpose is to declare learning standards and objectives; 3). selection step (select strategies, technology, media, and materials) where this stage is done to choose strategies, technology, media and materials with the aim of achieving the set standards and learning objectives; 4).preparation step (utilize technology, media and materials) with activities to prepare technology, media and materials that can support learning activities; 5). usage step (require learner participation) the activity is to test the product where there are three stages in product testing, namely: expert judgment or expert testing carried out with respondents from media experts or supervisors; small group test was conducted with small group respondents, namely several students as users of mobile learning applications; and field testing is a trial at school; 6). evaluation step and revision (evaluate and revise) where evaluation is carried out in two ways, namely assessing the learning outcomes of 8g class students who use the application and who do not use and evaluate mobile learning then if there were improvements, revisions will be made.. the second product research subject was students class x ips 3.2 sma negeri 1 salatiga totaling 24 students. this research was conducted in the second semester of yunianta, putri, & kusuma, development and comparison of mathematic mobile … 46 academic year 2015/2016. all these students will use mathematics mobile learning as a learning supplement. the second product produced in this study was mathematics mobile learning in trigonometry material. the development of this mobile learning application used the addie development model. the development steps are as follows: 1). analysis where there are three activities at this stage, among others, by analyzing the condition of students both in learning and in the use of technology, especially the use of android smartphones, analyzing student learning styles thus later the contents of mobile learning can be tailored to user needs and curriculum analysis to find out more in the content of learning which students follow; 2). design (mobile learning design) where in the design process is a follow-up of the analysis phase by making and developing based on the results of the general analysis of the state of the student, student learning style, curriculum, and material to be taught at the time the research was conducted; 3). develop (mobile learning development) where this stage is the stage of making mathematic mobile learning after being designed based on student needs including the design of the display aspects and material aspects in it, then this application is tested in two stages: a) expert judgment or expert testing with responsive media experts and material experts; b) small group test with limited respondents, namely with several students as users of mathematic mobile learning.; 4). implementation (mobile learning implementation) is the stage of applying mathematic learning to the subject of research; 5). evaluate (evaluation of implementation results) where this stage of evaluation activities includes student learning outcomes after the use of mathematic mobile learning and media effectiveness, as well as student opinions regarding the use of the application. the data analysis in this study aimed to analyze the student learning style questionnaire, determined the validity of the media, the practicality and effectiveness of the mathematics mobile learning application. in addition, the pretest and posttest questions that will be used were alsoanalyzed for validity thus the questions tested to students were valid questions. 2.1. analysis of student learning style data obtained through student learning style questionnaires in qualitative form in the form of letters were converted into quantitative values with the following steps: 1) the type of data taken in the form of qualitative data is then converted into quantitative data with the provisions in table 1. table 1. scalling rules note score always (sl) 5 frequent (sr) 4 sometimes (kd) 3 rare (jr) 2 never (tp) 1 2) after the data is collected, then calculate the average score of each type of learning style (visual, auditory and kinesthetic) with the formula on the likert scale as follows. volume 8, no 1, february 2019, pp. 43-56 47 notes: : average value : total score : number of questions in each category of learning styles 3) based on the average score obtained in each category of learning styles, there are three categories of learning styles namely visual, auditory and kinesthetic and students only have one dominant learning style. indicators of learning styles are taken from deporter & hernacki (2009). learning styles can be determined based on the highest average among the three categories of learning styles obtained by students. 2.2. analysis of application validation data and practical sheets data obtained through the validation sheet of media experts and material experts with scoring tables as shown in table 2. table 2. scalling rules note score very good (sb) 5 good (b) 4 fair (c) 3 less (k) 2 very less (sk) 1 1) quantifying the results of checking in accordance with predetermined indicators by giving a score according to a predetermined weight. 2) make data tabulation. 3) changing the value of each aspect of the criteria in each component of the mathematics application becomes qualitative with the criteria for the assessment category with the provisions in table 3. the categories from table 3 are said to be valid if a minimum is included in the sufficient category. table 3. assessment criteria no interval qualitative category 1 85% ≤ score ≤ 100% very valid 2 70% ≤ score ≤ 85% valid 3 50% ≤ score ≤ 70% less valid 4 0% ≤ score ≤ 50% not valid yunianta, putri, & kusuma, development and comparison of mathematic mobile … 48 2.3. effectiveness of mobile learning application the effectiveness of mathematics mobile learning is determined based on posttest learning outcomes and based on the results of student opinion sheets. the results of student opinion sheets are analyzed qualitatively. the data of students' posttest learning outcomes that have been collected are then analyzed for the significance of the increase from the pretest values calculated by the following n-gain formula. n-gain is the average increase in student grades. the classification of n-gain category as shown in table 4. table 4. classification of n-gain category n-gain score category g ≥ 0.70 high increase 0.30 ≤ g < 0,70 medium increase g < 0.30 low increase based on these categories, mathematics mobile learning is effectively used if at least included in the high increase. 2.4. analysis of posttest problems data before the instrument is used to obtain instrument research data, the instrument in the form of a test question will be tested first to ensure that the instrument is valid for use. validity is the degree of accuracy between the data that occurs in the object of research with the power that can be reported by the researcher. the question of the pretest and posttest is through the validation stage by the validator. 3. results and discussion 3.1. results 3.1.1. description of student learning motivation the results of the development of the first mobile learning product were developed with exelearning 2.0 software programs. this application provides sufficient material content, there are learning videos about circles, and there are exercises for students by filling in the blanks and some in the form of multiple choices. in this application, students' learning styles are considered auditory, visual and kinesthetic. student learning styles are shown in figure 1. the application developed consisted of three parts, material, learning videos and several practices that can be seen in figure 1. volume 8, no 1, february 2019, pp. 43-56 49 figure 1. student learning style for mobile learning development with the exelearning 2.0 application in this class, 28 students participated in the learning process. there were 21 students with smartphones with android os and 7 students did not have them. circle mobile learning application for 8th grade students of smp n 6 salatiga validated by 3 experts. appraisal validation assessment was seen from two aspects, namely material aspects and also aspects of appearance. this validation assessment sheet consisted of 20 indicators and each aspect consisted of 10 indicators. the results of application validation are in table 5. table 5. results of validation of mobile learning application using the exelearning 2.0 program aspect average percentage category material 44.7 89.4% very valid display 42.7 85.4% very valid the menu in this application at the beginning is menu. in the menu there are home, circles and evaluations. in the circle section, there are circle elements, circumference, area, central angle & circumference angle, length of bow & area of juring, and tembereng. furthermore, in the evaluation section there are sub-sections namely true false problem, multiple choice questions and short field questions. the several views in the application can be seen in figure 2. figure 2. display of mobile learning application using exelearning before students use the application, there is a pretest and after the lesson, there is a posttest held. the material provided is a circle. student achievements are shown in table 6. yunianta, putri, & kusuma, development and comparison of mathematic mobile … 50 table 6. student learning outcomes using the mobile learning application from the exelearning program subject average increase of n-gain category early grade final grade using mobile learning 61 88.6 0.7 high not using mobile learning 71.4 82.9 0.4 medium 3.1.2. mathematics mobile learning development product using the mit inventor 2 program the second application was developed using the mit inventor 2 program provided online. this application is made for trigonometry material for mathematics subject. the contents of this mobile learning application include material content, video learning about trigonometric concepts, and student multiple choice questions. in this application also try to be made based on student learning styles. figure 3 shows students' learning styles in the class used. figure 3. student learning styles for mobile learning development with the mit inventor 2 application according to data from figure 3, this application develops more visual parts, and one learning video and 10 practice questions for students. this application is designed with attractive colors to support visual learning styles. this trigonometry learning application is for high school students in grade 3 or xii and is validated by 4 experts. appraisal validation assessment was seen from two aspects, namely material aspects and also aspects of appearance. this validation assessment sheet consisted of 20 indicators and each aspect consisted of 10 indicators. the results of application validation are in table 7. table 7. results of validation of mobile learning application using the mit inventor 2 program aspect average percentage category material 43,5 87% very valid display 40 80% valid volume 8, no 1, february 2019, pp. 43-56 51 the appearance can be shown in figure 4 for video and figure 5 for material content and problem practice. in this application only one learning video is available. practice questions are also limited to 10 questions. figure 4. display of videos on mobile learning application using the mit inventor program figure 5. display of materials and questions on mobile learning application using the mit inventor program students who participated in the learning process for the third application were 24 students. all students used this mathematics mobile learning. student achievements are shown in table 8. table 8. student learning outcomes using mobile learning application from the exelearning program the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. 3.2. discussion before this study there was a study by karabatzaki et al., (2018) about mobile learning for preschool where it was stated that children's cognitive skills development with digital media and have well-designed mobile-based learning activities and games could improve skills of abstract thinking, reflective thinking, and analyzing and evaluating subject average increase category early grade final grade using mobile learning 53.33 86.25 0.71 high yunianta, putri, & kusuma, development and comparison of mathematic mobile … 52 information planning and scientific reasoning. other research by kattayat, josey, & asha (2017) found that the results show that teaching learning processes using mobile apps are helpful to students in improving their learning achievements. in the first mathematics mobile learning application that was developed using the exelearning 2.0 program seen in the results of validation according to table 5 stated that in display and material, this application met the criteria very well. the exelearning 2.0 program allows mobile learning application makers to fill material, images, videos and practice questions easily and varied. the program also provides a variety of practice questions. for example, this application can be filled in with the practice of true false questions, multiple choice questions and short answer questions. the appearance for the application theme is indeed limited, however the choice given is quite a lot, so we make it possible to determine the theme of the application made. the material that is loaded can also vary and easily combine words, images and videos in the application. the programming language used is html 5. table 9. differences between two mathematics mobile learning products aspects seen mathematics mobile learning with exelearning 2.0 program mathematics mobile learning with mit inventor 2 program learning video there are 3 video learning and giving learning videos in this application easier to enter, because it can be directly entered on the application page or inserted. there is one video of learning and giving video learning is not so easy, it needs to have creativity and requires expertise in programming logic to include it. total page there are 13 pages, and the contents of this page are made easier because there are choices of videos, questions, or material to be included there are 10 pages, and the content of this page combines more material in the image but the arrangement is more complicated when combined with videos or questions exercises the exercises are on 10 pages where there are quite a lot of questions provided by the software, there are multiple choices, true and false questions and short questions the exercises are only available on one page with 10 multiple choice practice questions and to make so it should be made to the programming logic display display arrangement is limited by the existence of templates on the software and this requires creativity in providing images or videos that are presented and there are 34 supporting images it is possible to display a creative and extraordinary display depending on the creativity of the maker and there are eight pages that compose the page volume 8, no 1, february 2019, pp. 43-56 53 aspects seen mathematics mobile learning with exelearning 2.0 program mathematics mobile learning with mit inventor 2 program material filling out material is easier because a page is provided and it only needs to fill and organize sentences and images, it requires html 5 programming language expertise for structuring formulas or mathematical operations giving material is limited and capabilities are needed with logic programming, so to fill in more material can be made in the form of images by using programs such as: correl draw in the form of an image, just inserted in the page the second mathematics mobile learning application made with the mit inventor 2 program is equally good. even though mit inventor 2 can be made very well in appearance according to the creativity of the maker, but there are indeed some things that require a longer process to fill the material. this program does not use programming languages but is designed using programming algorithms with programming logic. whether or not to use the program is determined by how experienced people make it. in this application, the material aspect is still inferior to using the exelearning 2 program with good criteria as shown in table 7. the differences between the two mathematics mobile learning products can be seen in table 9. now you need to see the learning outcomes achieved by students who use it. in the first mobile learning application, it shows that student learning outcomes can increase by 70% and students who do not use it only increase 40%. this application impacts an increase in the high category based on n-gain analysis. the second mathematics mobile learning application also has the same impact to improve student learning achievement. this application has an impact on the increase in the high category of student learning outcomes, namely 71%. ghozi (2014) made mathematics mobile learning using java 2 mobile edition and he concluded that the learning process involving mobile learning applications is prospective and feasible to use. zefriyenni & mardhiyah (2017) had the same project to develop this application with netbean software for the equation of two linear variables. they concluded that the application had tested 7 students and they had all students achieved 100% success. furthermore, student responses in zefriyenni, & mardhiyah (2017) noted that students had a positive response when playing mobile learning for spldv material. this also happened in this study for two applications that have been developed. students claimed that this application is very fun because it is designed with images, colors, videos, evaluation questions and menu displays that are easy to learn and more flexible and simple (alamsyah, & ramantoko, 2012; sulisworo, ishafit, & firdausy, 2016). based on the results of the research, these two applications were valid and effective as wellquite successful in improving student learning outcomes. nevertheless, there is a major problem with the use of this mathematics mobile learning application. this is not only the limitation of many applications for the mathematics learning process, but also the problem of "is it possible for students to use smartphones in each of their schools or classes?" if you haven't already, this application can only be used while in certain conditions where students are allowed to access this application the more it will be better. yunianta, putri, & kusuma, development and comparison of mathematic mobile … 54 4. conclusion each application has a difference when it is developed and still depends on which software is used. it has a unique impression between the first and second applications. both mathematics mobile learning products do have differences from the results of their development. this is indeed influenced by what program is used, but in use, the most important thing is that the two products developed have the ability to improve students' mathematics learning outcomes on certain high-category material. therefore, the development of this application is needed further to assist students in learning mathematics. recommendations for teachers or researchers who will make mathematic mobile learning in particular by using exelearning 2.0 and mit office 2 programs, if you want to emphasize the content included and the completeness of the material, you should be able to use exelearning 2 program. this program is easier filled with various material, images , videos and learning activity links. but if the teacher or researcher wants to emphasize the interesting aspects of the display, it is recommended to use mit inventor 2 program because this program is very possible researchers or makers to create according to their tastes and desires but this program requires skills in logic programming. acknowledgements the author would like to thank all of those participating in this research for their help, particularly headmaster and students of sma negeri 1 salatiga and smp negeri 6 salatiga. references alamsyah, a., & ramantoko, g. (2012). implementations of m-learning in higher education in indonesia. in proceedings of 3rd international conference on technology and operation management. deporter, b., & hernacki, m. (2009). discovery learning: membiasakan belajar nyaman dan menyenangkan. bandung: pt. mizan pustaka. el-sofany, h. f., el-seoud, s. a., alwadani, h. m., & alwadani, a. e. (2014). development of mobile educational services application to improve educational outcomes using android technology. international journal of interactive mobile technologies (ijim), 8(2), 4-9. ghozi, s. (2014). pengembangan materi mobile learning dalam pembelajaran matematika kelas x sma perguruan cikini kertas nusantara berau. indonesian digital journal of mathematics and education, 1(1). ismail, i., & idrus, r. m. (2009). development of sms mobile technology for mlearning for distance learners. international journal of interactive mobile technologies, 3(2). idrus, r. m., & ismail, i. (2010). role of institutions of higher learning towards a knowledge-based community utilising mobile devices. procedia-social and behavioral sciences, 2(2), 2766-2770. volume 8, no 1, february 2019, pp. 43-56 55 karabatzaki, z., stathopoulou, a., kokkalia, g., dimitriou, e., loukeri, p. i., drigas, a., & economou, a. (2018). mobile application tools for students in secondary education. an evaluation study. international journal of network & mobile technologies, 12(2). kattayat, s., josey, s., & asha, j. v. (2017). mobile learning apps in instruction and students achievement. international journal of interactive mobile technologies (ijim), 11(1), 143-147. miarso, y. (2009). menyemai benih teknologi pendidikan. jakarta: kencana prenada media group. motiwalla, l. f. (2007). mobile learning: a framework and evaluation. computers & education, 49(3), 581-596. pribadi, b. a. (2009). model desain sistem pembelajaran. jakarta: dian rakyat. slameto (2003). belajar dan faktor-faktor yang mempengaruhinya. jakarta: pt rineka cipta sudjana, n., & rivai, a. (2013). media pengajaran. bandung: sinar baru algesindo. sulisworo, d., ishafit, i., & firdausy, k. (2016). the development of mobile learning application using jigsaw technique. ijim, 10(3), 11-16. zefriyenni, z., & mardhiyah, h. (2017). pengembangan mathematics mobile learning application (mmla)-sistem persamaan linear dua variabel (spldv) untuk siswa kelas 8 sebagai sumber pembelajaran mandiri berbasis android. jurnal teknologi informasi dan pendidikan, 10(2), 25-36. yunianta, putri, & kusuma, development and comparison of mathematic mobile … 56 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p209-218 209 developing of calculus teaching materials based on audiovisual akbar nasrum* 1 , herlina 2 1,2 universitas sembilanbelas november kolaka article info abstract article history: received august 8, 2019 revised sept 21, 2019 accepted sept 26, 2019 millennials community right now prefer video media as a learning resource rather than reading textbooks. to understand the contents of books, we need to have a reasonably high literacy ability. unlike the case with learning videos, explanations accompanied by images in the video can help someone to understand the material in that media. the first goal of this study was to make audiovisual teaching materials that could serve as a supplementary textbook teaching materials. the second goal was to test the effectiveness of teaching materials that have been created. the method used was the development method using the addie model (analyze, design, development, implementation, and evaluation). the process of developing teaching materials began with creating multimedia learning, recording stages, editing stages, and completion stages. material experts and media experts then validated the teaching materials that had been produced and tried out on students. the results of the try out documents to the students showed that there were significant changes both in terms of motivation, enthusiasm for learning, interaction in the classroom, and from student learning outcomes. these results made the student's response to the use of teaching materials excellent. this audiovisual teaching material was worthy and effectively of being used as a learning media, both as a supplementary or primary source. keywords: audiovisual teaching materials, learning videos copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: akbar nasrum, department of mathematics education, universitas sembilanbelas november kolaka, jl. pemuda, kolaka, southeast sulawesi 93561, indonesia. email: akbar.nasrum@gmail.com how to cite: nasrum, a., & herlina, h. (2019). developing of calculus teaching materials based on audiovisual. infinity, 8(2), 209-218. 1. introduction calculus is a compulsory subject given to students at the beginning of the semester. besides being compulsory, this course is essential because the knowledge gained here is the basis for programming advanced calculus courses, the real analysis i, real analysis ii, and differential equations. even though the material in calculus is primary material, this course is a scourge of problems for students so that it has an impact on less than optimal learning outcomes (shodikin, 2017). in order to obtain maximum learning outcomes, the learning process should be supported by adequate learning facilities and resources, especially in terms of teaching materials. however, the importance of calculus material is not directly proportional to the mailto:akbar.nasrum@gmail.com nasrum & herlina, developing of calculus teaching materials … 210 availability of teaching materials such as calculus books at the campus library. students interest in buying books also lacks because the price of this book is quite high. the difficulty of obtaining calculus teaching materials makes students less than optimal in learning. they only rely on the material obtained during the lecture, even though the density of calculus material and the limited time of lectures make this material sometimes incomplete. focusing on the completeness of learning outcomes will cause the learning material to lag. conversely, pursuing the completion of the material in one semester can lead to incomplete learning outcomes. these two variables are essential to note because they can cause failures in courses that require calculus basics. on this basis, we took the initiative to develop calculus teaching materials from textbooks into audiovisual teaching materials. this teaching material can function as a complement or even replace the role of textbooks as the primary source of teaching materials. the process of making teaching materials involves some software, one of which is geogebra software. the representation of the concepts of functions, limits, derivatives, and integrals in calculus is increasingly easily understood if geometrically explained using geogebra (caligaris, schivo, & romiti, 2015). besides takači, stankov, & milanovic (2015) said that using geogebra, student learning achievement in checking and drawing function graphics is better than without using it. teaching materials in the form of videos can be stored on a mobile device that can be carried anywhere (moving learning) making it easier for students to study anywhere and anytime (bano, zowghi, kearney, schuck, & aubusson, 2018). besides, they can construct knowledge from independent learning activities carried out using videos so that the process of developing thinking skills for students occurs. ownership of mobile devices by students supports the creation of outside-class learning using mobile learning (crompton & burke, 2018). the use of video learning in mobile learning is a promising learning strategy, relevant and can improve the quality of education (forbes et al., 2016). however, it is not an easy thing to make exciting learning videos, so it is still rarely used. it takes its creativity to make teaching videos more exciting and varied (christ, arya, & chiu, 2017). teaching materials in the form of videos can also support youtube channels as learning tools in education by uploading videos there (orús et al., 2016). in the era of digital, students tend to prefer to access youtube to search literature videos or educational videos rather than open textbooks. it is not difficult to find good educational videos. surfers can use visitor ratings as an indicator to look for good educational videos (shoufan, 2019). with this teaching video, students are expected to be able to understand the material more quickly because of the presentation of material in teaching materials in the form of visual and sound (audio). according to purwanti (2015), video media tends to be easier to remember and understand because it uses two senses, namely the sense of sight and hearing. unlike the case with a textbook that requires students to read and try to understand the material itself. apart from that, the explanation of still images in textbooks is explained visually moving on audiovisual teaching materials so that they are easier to understand. the explanation in audiovisual teaching materials is designed to resemble explanations in the classroom. in essence, this teaching material is a combination of textbooks and classroom learning. so, the formulation of the problem in this study, i.e., "are the audiovisual teaching materials that have been made feasible and effective to be used as learning media? volume 8, no 2, september 2019, pp. 209-218 211 2. method this research is development research using the addie model (nadiyah & faaizah, 2015). as the name implies, this model consists of several processes, namely: analyze, design, development, implementation, and evaluation. in the analysis phase, researchers must know the student's ability to learn, student attitudes, the conditions of the facilities and infrastructures used for learning, and others. all factors that cause disparities in learning outcomes must be analyzed in order to design instructional media that are suited to the conditions and abilities of students. in the design phase, information that has been obtained from the analysis phase is used to design learning prototypes. how to choose the right media, design a user interface, design a graphic design or display, so users are not bored, and learning objectives can be achieved. the next stage of development is the stage of teaching material production by integrating all technologies that will be used to achieve learning objectives. in this stage, consultation with material experts and learning design experts is needed. the next stage is implementation. in this stage, prototypes of teaching materials that have been produced and examined by experts are then tested on students and ensuring that students can obtain knowledge and skills from the media used. finally, the evaluation stage aims to find out the improvement of student competencies, students' attitudes towards learning activities, and the benefits of the instructors with this learning media. this process produces teaching materials produced in the form of learning videos. the research design used to test the effectiveness of this teaching material was one group pre-test post-test design. the effectiveness of instructional materials was tested by taking 48 students of the 4th semester of mathematics education at the university of sembilanbelas november kolaka as research subjects. forty-eight of these students are all 2017 mathematics education students. sources of data in this study are learning media experts, material experts, and students. data were collected using several instruments including in the form of material evaluation instruments by material experts, evaluation instruments for instructional media experts, learning outcome test instruments consisting of pre-tests and post-tests and questionnaires for student responses to the use of instructional materials. the development of this teaching materials is in dire need of advice from material experts and media experts. in its development, researchers collaborated with material experts and media experts so that the resulting teaching materials could be better. the content focuses on the primary material before entering calculus material, i.e., the introduction of functions. the evaluation results from media experts and material experts are input to improve the quality of teaching materials. furthermore, to see the effectiveness of these teaching materials, the scores of students' pre and post-test were compared. besides, the magnitude of the increase can be measured using n-gain values (marx & cummings, 2007). the last, the researcher gave a questionnaire to students was to see how well the students responded to the use of teaching materials. 3. results and discussion explanation of the results of the following research is two parts. first, it will explain the results of developing teaching materials and then explain the results of the effectiveness of teaching materials. nasrum & herlina, developing of calculus teaching materials … 212 table 1. list of learning video links material name link download definition of function https://youtu.be/5acqu7z87qc linear function i https://youtu.be/e8ykeetjyak linear function ii https://youtu.be/hbxr_jao5z4 quadratic function https://youtu.be/jilm1uhom88 shipting graph https://youtu.be/_aujkivv64c absolute function https://youtu.be/tew7wge6fbk table 2. suggestions for media repair by media and material experts no. aspect recommendation for improvement 1 2 3 4 design sounds and pictures material quality sounds and pictures add learning achievements to the media the image quality is at least 720 pixels so that the image quality does not break when enlarged the presentation time is too fast there is a mismatch between the speaker's sound and the cursor motion on some videos 3.1. results there are several results obtained after the research is carried out. first, the results obtained from the addie process, i.e., several learning videos that can be download from the link in the table 1. the results were obtained through several evaluation processes, both evaluations from material experts and media experts. there are some suggestions for improvement from media experts and material experts in the first stage of evaluation summarized in table 2. these suggestions provide input for researchers to correct some of the deficiencies that exist in the media. after revision, a validation sheet is again given to the material expert and media expert for the final evaluation. the results can be seen in table 3 and table 4. teaching materials that have been made through a consultation process from two expert fields are then tested on a limited basis. two students were taken from the category of a high, medium, and low learning outcomes as a small sample. after media testing on these students, almost no significant problems were found. only the problem of slow understanding of students with low learning outcomes. however, the problem can be solved in a class by re-explaining material that is not understood. table 3. score of the feasibility of learning videos by media experts no. aspect score media expert i media expert ii 1 2 3 design sounds and pictures software engineering 4.25 4.60 4.00 4.00 4.40 4.00 mean 4.28 4.13 https://youtu.be/5acqu7z87qc https://youtu.be/e8ykeetjyak https://youtu.be/hbxr_jao5z4 https://youtu.be/jilm1uhom88 https://youtu.be/_aujkivv64c https://youtu.be/tew7wge6fbk volume 8, no 2, september 2019, pp. 209-218 213 table 4. the score of learning video eligibility by material experts no. aspect score material expert i material expert ii 1 2 3 4 material relevance with syllabus material quality language aspect (sound) functions and benefits 4.67 4.00 4.00 4.00 4.67 4.40 4.00 4.33 mean 4.17 4.35 after limited trials have been conducted, audiovisual teaching materials are ready to be implemented in large classes. forty-eight registered students were used as research subjects but only forty-four people who actively participated in lectures. the final results of the application of media in large classes can be seen in figure 1. figure 1. the average pre-test, post-test, and n-gain besides, to see student responses to the use of instructional video media, a closed questionnaire was given. the results of questionnaire data processing can be seen in the table 5. table 5. student responses to the use of media no indicator category high medium low 1 feeling happy 68.18% 31.82% 0.00% 2 ease of understanding material 78.79% 17.42% 3.79% 3 motivation to follow lessons 68.18% 31.82% 0.00% 4 active in the learning process 63.64% 18.18% 18.18% 5 influence on results 45.45% 50.00% 4.55% 6 interest 97.73% 2.27% 0.00% 0.00 20.00 40.00 60.00 80.00 pre-test post-test n-gain mean 24.37 66.48 0.57 average nasrum & herlina, developing of calculus teaching materials … 214 3.2. discussion 3.2.1. analyze the primary ability of student calculus that researchers pay attention from year to year is the basis for making audiovisual teaching materials. advanced calculus courses, the real analysis i, real analysis ii, and differential equations cannot be taught without this essential knowledge. repeating that fundamental concept in the classroom certainly takes much time so that an appropriate method is needed so that the lecture process is not interrupted. video learning is one of the best choices in various methods. by using video learning media, teaching and learning process can run more effectively. this is consistent with the results of research from nurdin et al. (2019) which says that learning videos were effective and have a positive effect on improving students' understanding of mathematical concepts. video media will also help teachers to make it easier to deliver material and create learning situations that are not monotonous and will help make it easier for students to understand the material (kurniawan, kuswandi, & husna, 2018). because the abilities in class vary, videos are designed in such a way that even people who are unfamiliar with mathematics can easily understand the material presented. 3.2.2. design these design aspects include video design, video format, and video content. these three aspects are adjusted to the needs based on the results at the analysis stage. in terms of video design and video format, it involves some software, i.e., powerpoint, geogebra, camtasia recorder 8.0, and camtasia studio 8.4. the process of display design and animation is made using powerpoint. geogebra software is a complement in explaining calculus material, camtasia recorder is used to record videos, and camtasia studio is used to edit videos and convert recordings to mp4 videos. examples of design and video editing stages can be seen in figure 2. figure 2. design and video editing processes volume 8, no 2, september 2019, pp. 209-218 215 the geogebra seen in the video media in figure 2 is used to facilitate the explanation of calculus materials, especially those related to functional materials. the ease of understanding the material explained by using geogebra is a unique attraction for students. this is consistent with the study of trung (2014) which shows that about 79% of students from the total population studied often use geogebra software independently without being instructed by the teacher to recheck their knowledge. according to fitriyani, (2012), geogebra can also increase student activity and learning outcomes. the use of geogebra is arranging so that it does not interfere with the appearance in the video. after going through the editing process, it finally arrived at the final stage. the final stage produces mp4 videos with 720-pixel quality and uploaded to youtube. this video was set so that search engines can not find it on youtube. the reason was that learning videos were distributed to students according to the schedule set by the lecturer. learning videos in this study were necessary materials that must be well understood by students before entering the core material from calculus. we are still developing similar teaching videos for further research. 3.2.3. development the process of developing teaching materials involves several experts, i.e., two material experts and two instructional media experts. as the first learning media expert, mr. kadaruddin, s.pd., m.pd was chosen who was an expert in the field of learning media while the second media expert was mr. sufri mashuri, s.pd., m.pd. he is a calculus lecturer in the mathematics education study program and also experienced in learning media, so he was chosen as a material expert and media expert. other lecturers who are interested in becoming material experts are mr. ansar, s.si., m.sc who is a calculus lecturer in the faculty of engineering university of sembilanbelas november kolaka. the evaluation values of the three experts were input into developing learning videos. there are some suggestions for improvement from media experts and material experts at the first stage of evaluation summarized in table 2. these suggestions are input for researchers to correct some of the shortcomings in the media. after revision, a validation sheet is again given to the material expert and media expert for the final evaluation. the results can be seen in table 3 and table 4. based on table 3, the average value of each aspect both from the first expert and the second expert is not less than four (the scale of one to five). that is, in terms of the feasibility of the video media, it is declared appropriate to be used as a learning media. while in terms of material, the results of the evaluation can be seen in table 4. based on table 4, the average value of each aspect assessed is almost the same as the assessment of media experts. from the evaluation of two material experts, the average score given by each aspect was not less than 4, which meant that the learning videos from the material side were very valid (farman & yusryanto, 2018). thus, because the video has been validated and declared feasible, the learning video can already be used. 3.2.4. implementation the process of implementing media in large class was carried out after the initial test was given. video media is given to students to watch and learn independently. when the lecture time arrives, the teacher needs to ask which part of the video was difficult to understand. sometimes from some material given, only a small portion is poorly understood, so it needs a few minutes of explanation. the learning process in the nasrum & herlina, developing of calculus teaching materials … 216 classroom becomes more active because students are prepared with material that has been studied earlier. another advantage does not require much time to explain the material taught on that day so that the use of teaching time is more efficient. the lecture process like this was done for four weeks, and the final results can be seen in figure 1. comparison of the average results of the pre and post-test was very significant. these results indicate that the use of audiovisual media can influence on improving learning outcomes. it supported with research conducted by asmara (2015) which says that student learning outcomes using audiovisual learning media are better than not using audiovisual. although not measured in detail, from the way to answer the questions given, it appears that understanding the concept of material after the use of the media for the better. it supported by research conducted by utami, (2013) and nurdin et al. (2019) who say that learning by using audiovisual media can improve understanding of material concepts for students. the quality of learning outcomes can be seen from the magnitude of the n-gain value. the value of 0.57 was included in the category of "moderate" improvement. (hake, 1998). 3.2.5. evaluation from the whole series of process of developing teaching materials, starting from designing to the implementation stage, it has been through various types of evaluation stages, starting from evaluating media experts and material experts to evaluating the use of video media itself. finally, an open questionnaire was given to students to see how much they responded to the use of this video media. based on table 5, there are six indicators assessed. first, from the results of research on student responses, 68.18% of students were happy to learn mathematics using audiovisual media, and 31.82% were mediocre. 78.79% of students can quickly receive lessons, 17.42% are mediocre, and 3.79% of students still do not understand sometimes. 63.64% of students became more active in receiving lessons, 18.18% were less active, and the remaining 18.18% were inactive. 68.18% of students are very motivated to follow the lessons and 31.82% are mediocre. the influence of the use of media on learning outcomes can also be seen in table 5. there are about 45.45% of students whose learning outcomes are getting better, 50% of students who have moderate learning outcomes and 4.55% who have still poor learning outcomes. although the data above varies, for the sixth indicator 97.73% of students expressed interest in following the lesson if it were implemented learning using audiovisual learning media like this and only 2.27% or one less interested person. the result from indicator six shows the positive response given by students towards the use of this media. 4. conclusion from the results of the study, several conclusions can be drawn i.e., we have succeeded in making audiovisual-based learning media that have been validated by media experts and material experts in good categories so that the media can be used as learning material both for primary sources and as a complement to textbooks. the use of audiovisual media can be more beneficial in terms of time management because it does not take much time to explain the material in the video. in addition to being efficient, the use of media has an effect on activeness in the classroom and is also very effectively used in improving student learning outcomes. volume 8, no 2, september 2019, pp. 209-218 217 acknowledgements we would like to thank university of sembilanbelas november kolaka that funded this research through the scheme internal "penelitian dosen pemula" 2018. references asmara, a. p. 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(2020). the effect of geogebra-assisted direct instruction on students’ self-efficacy and self-regulation. infinity, 9(1), 41-48. 1. introduction the rapid advancement of technology has brought many changes in various fields, especially in education. therefore, it becomes very important for educators and students to learn and be able to use technology in learning. the use of technology in learning has a positive effect on improving students' conceptual and procedural abilities (zulnaidi & zakaria, 2012). with the application of educational technology, students can master learning material independently, choose the accuracy of work, review lessons, and know their progress. technology has an important part in the development of education. it is also a part of supporting lecturing activities. a technology tool that helps teachers/lecturers and mailto:zetriuslita@edu.uir.ac.id zetriuslita, nofriyandi, & istikomah, the effect of geogebra-assisted direct instruction … 42 students in learning especially demonstration and visualization mathematical concept is geogebra software. geogebra is one of software to visualize and demonstrate mathematical concepts, especially geometry and algebra. geogebra is recommended to be included in the school curriculum because it has potential in mathematics education (hohenwarter & jones, 2007). for this function, students can use algebraic and geometrical functions simultaneously with interactive dynamics that will enhance their cognitive abilities. besides visualization, geogebra also serves to facilitate students / students to better understand abstract concepts (dikovic, 2009). the use of geogebra by asking probing questions has a positive effect on the exploration phase (hähkiöniemi, 2017; zengin, 2017). in order to make geogebra have an impact on student’s learning outcome, it takes a sense of self-efficacy and self-regulation. self-efficacy and self-regulation or self-regulated learning are related to students' ability to believe in their own abilities and there is no inferiority in dealing with mathematical problems. self-efficacy is an important and main concept to improve understanding and learning outcomes, so that students are able to develop their self-confidence and will correlate with improving learning outcomes (hatlevik, throndsen, loi, & gudmundsdottir, 2018). self-efficacy can be defined as a perception of a person's ability to organize and implement actions to carry out certain skills (zimmerman, 1989). furthermore, regarding self-regulated learning, zimmerman (1989; 1990) defines that self-regulation is an idea that is proactively initiated, and cyclic planned and adapted behavior as feedback from performance in achieving certain goals. the same thing was conveyed by arslan (2014), that self-regulation is a traditional concept related to monitoring and controlling individual performance (iiskala, vauras, lehtinen, & salonen, 2011; isohätälä, järvenoja, järvelä, 2017). based on the opinions above, nofriyandi (zetriuslita, nofriyandi, & istikomah, 2019) compiled self-regulation indicators as follows: (1) learning initiatives, (2) diagnosing learning needs, (3) setting learning goals, (4) organizing and controlling performance / learning, (5) organizing and controlling cognition, motivation, and behavior (self), (6) looking at difficulties as challenges, (7) finding and utilizing relevant sources, (8) choosing and implementing appropriate learning strategies, and (9) ) evaluate the learning process and results. because of the importance of developing students' self-efficacy and self-regulation, it is designed to develop the two attitudes above. with the application of geogebraassisted direct instruction, it is expected that self-efficacy and self-regulation can develop and ultimately improve both attitudes and have an impact on student learning outcomes in mathematics. the objectives in this study are to find out and describe : 1) the achievement of students’self-efficacy through geogebra-assisted direct instruction, 2) the achievement of students' self-regulation through geogebra-assisted direct instruction, 3) the response of mathematics students towards geogebra-assisted direct instruction to the achievement of students' self-efficacy and self-regulation. to find out the answer to the research objectives, the following hypothesis is: (1) the achievement of students’ self-efficacy through geogebra-assisted direct instruction better than conventional learning; (2) the achievement of students' self-regulation through geogebra-assisted direct instruction better than conventional learning. 2. method the research method is a mixed method with sequential explanatory strategy. this method is used because the data is taken from quantitative data and qualitative data, volume 9, no 1, february 2020, pp. 41-48 43 quantitative data is taken, qualitative data is taken. filling in data to complete the results of quantitative data. the research design in this study is a quasi-experimental type with a nonequivalent control group design / with pretest and posttest non-equivalent group design (cohen, manion, & morrison, 2002; creswell, 2014). the population in this study was mathematics education students who took the analytical geometry course, because research is conducted on these subject. the sample was selected using cluster random sampling technique. this way was chosen because the students who take this course consists of 3 classes, so after taking the sample, which was selected as the experimental class is class 3a and class control is class 3b. jumlah sampel and each class totaling 42 people. data collection used self-efficacy and self-regulation questionnaire sheets for geogebra-assisted direct instruction in the experimental class and conventional learning in the control class. data obtained from filling out the student selfefficacy and self-regulation questionnaires were analyzed statistically, both descriptive statistics and inferential statistics. meanwhile, the results of student’s interview were analyzed in a descriptive narrative to complement the results of quantitative analysis. the results of the self-efficacy and self-regulation questionnaire were ordinal data and transformed into interval data using the method of successive interval (msi). furthermore, the normality and homogeneity of the experimental class and the control class were tested and the two similarities were tested using the parametric statistical test, namely the independent samples t-test. if it did not meet normalcy, then the data is processed using a nonparametric test known as the mann-whitney u test. 3. results and discussion 3.1. results students’ self-regulation table 1. description of students’ self regulation in experimental and control classes table 1 show the students’ achievement of self-regulation in experimental class is better than that of students in control class; this can be seen from the mean difference of 15.26. to ensure that it is better to be significant, a statistical test is carried out, nam ely the difference test of the results of students' self-regulated achievement with the first step of the normality and homogeneity test, if the data is normally distributed and homogeneous, then a statistical t test is performed. the criteria for testing students' self-regulated normality, h0 is accepted if the probability value (sig.) is greater than α and h0 is rejected if the probability value is smaller than α (α=0.05). from the normality test results obtained sig = 0.257 for the control class and sig = 0.265 for the experimental class, because the two sig is greater than α = 0.05 means that both classes are normally distributed. the results show the data is normally distributed, then we using the test of homogeneity of the variance of the posttest data using the levene test. homogeneity testing criteria, h0 is accepted if the probability value (sig.) is greater than α and h0 is rejected if the probability value (sig.) is smaller than α (α = 0.05). from the levene test class mean deviation standard minimum maximum variance experimental 117.74 8.36 103.00 137.00 69.81 control 102.48 10.03 82.00 129.00 100.90 zetriuslita, nofriyandi, & istikomah, the effect of geogebra-assisted direct instruction … 44 obtained sig = 0.485> α, meaning that the data on the achievement of self-regulated homogeneous students varied. furthermore, t-test for the students' self-regulation achievement is used to test the hypothesis, namely: h0: the achievement of self-regulation in experimental class is the same as the achievement of self-regulation in control class. h1: the achievement of self-regulation in experimental class is better than the achievement of self-regulation in control class. criteria for testing differences in the results of students' self-regulation achievement, h0 is accepted if the probability value (sig.) is greater than α = 0.05 and h0 is rejected if the probability value is smaller than α = 0.05. table 2. t-test data of students’ self-regulation achievement in experimental and control classes table 2 show the probability value (sig.) is less than significance degree α = 0.05. thus h0 is rejected. thus it can be concluded that students’ of selfregulation are better in experimental class than control class. students’ self efficacy table 3. descriptions of the self efficacy of the experimental class and control class based on table 3 suggests that the result of self efficacy of experimental class students is better than control class students, the difference is 4.43. to ensure this conclusion, a statistical test is carried out, namely a test of the difference in the result of students’ self-efficacy with the first step of the normality and homogeneity test. if the data is normally distributed and homogeneous, then a statistical t test is performed. liliefors normality test results obtained control class data sig. = 0.805 (> 0.05), not normally distributed. however, for the experimental class sig. = 0.016 (<0.05), the distribution is normal. because parametric statistical testing was not met, the test continued with the difference test, using non-parametric statistics namely the mann-whitney u test. the testing hypotheses were: h0: there is no a difference in the mean data (post-test) of the experimental class and the control class on students’ self-efficacy. h1: there is difference in the mean data (post-test) of the experimental class and the control class on students’ self-efficacy. class mean deviation standard sig description experimental 117.74 8.36 0.00 rejected h0 control 102.48 10.03 class mean deviation standard minimum maximum variance experimental 50.33 6.40 42 64 40.92 control 45.90 5.65 32 59 31.89 volume 9, no 1, february 2020, pp. 41-48 45 the criteria for testing differences in students’ self-efficacy results, h0 is accepted if the probability value (sig.) is greater than α, and h0 is rejected if the probability value is smaller than α (α = 0.05). table 4. t-test data post-test self-efficacy of the experimental class and control class students based on table 4 describes that the decision of h0 is rejected, which means that the result (post-test) of the self-efficacy of experimental class students is better than that of control class students. result of interview the results of student interviews about the effectiveness of direct instruction on the achievement of self-regulation and self-efficacy can be seen from the interview excerpt: lecturer : what do you think about geogebra's assisted direct learning in improving your self-regulation and self-efficacy? student 1 : in my opinion, geogebra can develop self-regulation, because of what we already understand from the lecturer, it can be done again at home, discussions with friends and can also improve self-efficacy, because by being told to come forward to try to work on the problems given with the help of geogebra to be more confident, especially if what is presented is true, but even if it is not true, the lecturer does not immediately say wrong, so it does not make us despair. student 2 : geogebra's assisted direct instruction can have an effect on my selfregulation, where the material explained by lecturers can be studied and tried again at home and also has an effect on self-efficacy, with evidence that i already have the courage to come to the front of the class. field and space analytical geometry courses are very suitable to be studied with geogebra, and there is also a feeling of confidence in problem solving, and trying to do it yourself. student 3 : my self-regulation increased because i was challenged to try it on its own, and my self-confidence also improved, although there was still fear / inferiority if i answered the wrong questions given. from the interview with students, it can be concluded that the application of geogebra-assisted direct instruction has an effect on students' self-regulation and selfefficacy. class mean rank rank total sig decision experimental 50.75 2131.50 0.002 rejected h0 control 34.25 1438.50 zetriuslita, nofriyandi, & istikomah, the effect of geogebra-assisted direct instruction … 46 3.2. discussion from the results of research and analysis of the data obtained, it can be explained that the application of geogebra-assisted direct instruction has a positive effect on students' self-regulation and self-efficacy. this means that self-regulated and self-efficacy of students who use geogebra-assisted direct instruction in the experimental class is better than those who do not use geogebra in the control class. this result is obtained from the function of geogebra as a medium of visualization and demonstration of mathematical concepts, make students efficacy and trust the results obtained from the geogebra display, and desires to try themselves so as to make self-confidence increase if no visualization of the concept is given. this is in accord with study result of muslim & haris (2017) that geogebra-assisted learning in geometry material is more effective than conventional learning in terms of self-efficacy. the same thing also obtained by the results of research that if students already have high self-efficacy, it will affect the learning outcomes of mathematics (liu & koirala, 2009). there is a relationship between self-efficacy and selfregulation. so the results of this study reinforce that if students have both, it will have an impact on learning outcomes (los, 2014). one of the advantages is that geogebra demonstrates certain mathematical concepts (hohenwarter & fuchs, 2004). besides, geogebra helps students in the achievement of conceptual and procedural knowledge (zulnaidi & zakaria, 2012). geogebra, with the visualization of the problem given, makes students challenged to find out more and the meaning of the visualization. this feeling of challenge will make curiosity and growing interest impact on learning outcomes. (zetriuslita, wahyudin, & dahlan, 2018). the geogebra application also provides an increase in student mathematical communication (zetriuslita et.al., 2019). the results of the analysis of interviews with students found that geogebra helped them understand the concepts of ellipse, satellite dishes and hyperbole because they could be seen directly in the picture and the desired calculation results. one of the results of the geogebra visualization for ellipse material can be seen in the following figure 1. figure 1. geogebra display results for ellipse material based on figure 1, with the help of geogebra, students can see directly the shape of the ellipse and their eccentricity value. on the left there is a red arrow, it can be seen the results of the calculation of the algebra, so that students only have to analyze the results. geogebra visualization for describe ellips equation 16 𝑥2 + 25 𝑦2 = 400 volume 9, no 1, february 2020, pp. 41-48 47 4. conclusion from data analysis and discussion, we could concluded that : 1) achievement of students’ self-efficacy through geogebra-assisted direct instruction better than conventional learning, 2) achievement of students’ self-efficacy through geogebra-assisted direct instruction better than conventional learning, 3) the response of mathematics students towards geogebra-assisted direct instruction to the achievement of students' selfefficacy and self-regulation are very good and positive. confidence and certainty in what is obtained will make students challenged and excited to solve the problems given by the lecturer. it is recommended that this learning be applied better and developed for other materials and other learning model, such as model problem based learning. acknowledgements the authors would like to express my deepest gratitude to kemenristekdikti who funded this research. i also wish to thank my colleagues as the researcher’s team for the help and support in finishing this research. references arslan, s. 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(2) mahasiswa pada level kemampuan sedang yang menjawab dengan benar soal pada indikator 1 sebanyak 5 orang (5,68%), indikator 2 sebanyak 75 orang (85,23%), dan indikator 3 sebanyak 8 orang (9,09%); (3) mahasiswa pada level kemampuan rendah yang menjawab dengan benar soal pada indikator 1 sebanyak 0 orang (0%), indikator 2 sebanyak 13 orang (68,42%), indikator 3 sebanyak 5 orang (26,32%); (4) secara keseluruhan, indikator 1 sebanyak 9 orang (7,69%), indikator 2 sebanyak 96 orang (82,05%), dan indikator 3 sebanyak 18 orang (15,38%). kata kunci : kemampuan berpikir kritis matematis, level kemampuan mahasiswa abstract this research aims to describe the ability of students to solve any problem at integral calculus course which is based indicators of mathematical critical thinking ability and level mathematical ability of students. research method used in this study is a qualitative research. subjects in this study were students of the 2nd semester 2014/2015 academic year at mathematics education fkip uir pekanbaru. this population of this study were 115 students participating in integral calculus course. subjects were divided into three groups: students in high-level abilities (10), students in medium-level (88), and students in low-level (17). data collection techniques in this study using the test and interview techniques. processing of data validity using triangulation techniques (data reduction, data presentation, and conclusion). based on the results of the study are found that: (1) students at a highlevel correctly answering questions on the indicator 1 of 4 students (40%), indicator 2 of 8 students (80%), and 3 indicators as much (50%); (2) students at the level of ability is being correctly answering questions on the indicator 1 5 students (5.68%), indicator 2 many as 75 students (85.23%), and indicators 3 of 8 persons (9.09%) ; (3) students at a low level ability correctly answering questions on a first indicator from 0 (0%), indicator 2 as many as 13 students (68.42%), indicator 3 by 5 votes mailto:zetri.lita@gmail.com mailto:ariawanrezi@rocketmail.com infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 57 (26.32%); (4) overall, the indicators one by 9 votes (7.69%), indicator 2 as many as 96 students (82.05%), and a third indicator as many as 18 students (15.38%). keywords: mathematical critical thinking ability, level mathematical ability of students i. pendahuluan dalam setiap kurikulum pendidikan nasional indonesia, mata pelajaran matematika selalu diajarkan disetiap jenjang pendidikan, tidak terkecuali di perguruan tinggi. hal ini menunjukkan bahwa diharapkan dengan mempelajari matematika, maka ketersediaan akan sumber daya manusia indonesia yang handal, yakni mampu berpikir kritis, sistematis, logis, kreatif, dan cermat dapat terpenuhi. salah satu aspek penting dalam matematika di perguruan tinggi adalah kemampuan berpikir kritis. hal ini sejalan dengan apa yang direkomendasikan oleh committee on the undergroude program in mathematics dalam karlimah (2010) yaitu “ enam rekomendasi dasar untuk jurusan, program, dan mata kuliah dalam matematika. salah satu rekomendasinya menjelaskan bahwa setiap mata kuliah dalam matematika hendaknya merupakan aktivitas yang akan membantu mahasiswa dalam pengembangan analitis, penalaran kritis, pemecahan masalah, dan keterampilan komunikasi. klurik dan rudnick (sabandar, 2008) menyatakan bahwa yang termasuk berpikir kritis dalam matematika adalah berpikir yang menguji, mempertanyakan, menghubungkan, mengevaluasi semua aspek yang ada dalam suatu situasi maupun dalam suatu masalah. hassoubah, 2004), berpikir kritis adalah berpikir secara beralasan dan reflektif dengan menekankan pada pembuatan keputusan tentang apa yang harus dipercayai atau dilakukan. mason (2010) dalam lunenburg (2011) menyatakan bahwa konsep berpikir kritis salah satu trend yang paling signifikan dalam pendidikan dan memiliki hubungan yang dinamis bagaimana guru mengajar dan peserta didik belajar. lunenburg menambahkan setelah kita memahami konten yang tidak terpisahkan dari pemikiran yang menghasilkan, mengatur, menganalisis, mensin-tesis, mengevaluasi, dan mentransformasinya. sejalan dengan itu marzano (1989) berpikir kritis adalah sesuatu yang masuk akal, berpikir reflektif yang difokuskan pada apa keputusan yang diyakini, dikerjakan dan diperbuat. selanjutnya facione (2011) bahwa konsep dasar dari berpikir kritis adalah interpretasi, analisis, evaluasi, menyimpulkan, penjelasan dan kepercayaan diri. sedangkan berpikir kritis menurut onosko and newmann (1994) bagaimana menantang peserta didik dalam menginterpretasikan, menganalisis dan memanipulasi informasi. oleh karena itu keterampilan berpikir kritis diperlukan ketika kita mencoba memahami informasi yang akan digunakan untuk mencetuskan ide atau gagasan. (firdaus et.al (2015). berpikir kritis menurut johnson (2007) merupakan sebuah proses terarah dan jelas yang digunakan dalam kegiatan mental seperti memecahkan masalah, mengambil keputusan, membujuk, menganalisis asumsi, dan melakukan penelitian ilmiah. senada dengan itu, ennis yang dikutip lipman (2003) mengemukakan aspek dalam berpikir kritis adalah focus (fokus), reasons (alasan), inference (simpulan), situation (situasi), clarity (kejelasan), dan overview (tinjau ulang). infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 58 berdasarkan pernyataan di atas, dapat disimpulkan bahwa kemampuan berpikir kritis dapat membuat dan melatih seseorang untuk melakukan (doing math) dalam pembelajaran matematika. kemampuan berpikir kritis matematis adalah kemampuan matematika tingkat tinggi yang dalam penelitian ini diukur dengan menggunakan indikator: (1) kemampuan mengidentifikasi dan menjastifikasi konsep, yaitu kemampuan memberikan alasan terhadap penguasan konsep; (2) kemampuan menggeneralisasi, yaitu kemampuan melengkapi data atau informasi yang mendukung; (3) kemampuan menganalisis algoritma, yaitu kemampuan mengevaluasi atau memeriksa suatu algoritma. berdasarkan latar belakang di atas, maka rumusan masalah dalam penelitian ini adalah: bagaimanakah kemampuan berpikir kritis mahasiswa dalam menyelesaikan soal uraian kalkulus integral ditinjau berdasarkan indikator kemampuan berpikir kritis matematis dan level kemampuan mahasiswa? adapun tujuan dalam penelitian ini mendiskripsikan dan menganalisis kemampuan berpikir kritis mahasiswa dalam menyelesaikan soal uraian kalkulus integral ditinjau berdasarkan indikator kemampuan berpikir kritis matematis dan level kemampuan mahasiswa. ii. metode penelitian metode penelitian yang digunakan dalam penelitian ini adalah kualitatif. subjek dalam penelitian ini adalah mahasiswa semester 2 program sudi pendidikan matematika fkip uir semester genap yang sedang menempuh mata kuliah kalkulus 2. pemilihan subjek penelitian ini didasari oleh beberapa pertimbangan, yaitu: (a) mahasiswa semester 2 sudah mendapatkan mata kuliah kalkulus 1, sehingga dapat dikatakan mereka sudah memenuhi syarat untuk mempelajari kalkulus 2; (b) mudah diwawancarai sehingga akan diperoleh data akurat yang dibutuhkan dalam penelitian ini. iii. hasil penelitian dan pembahasan pengambilan data pada penelitian ini dilakukan di ruangan a6.09 a6.14 kampus fkip uir pekanbaru pada tanggal 11-13 mei 2015 pada jam 09.00-12.00 wib. setelah subjek penelitian mengerjakan lembar instrumen berpikir kritis matematis, selanjutnya peneliti menentukan subjek yang akan diwawancara berdasarkan tingkat kemampuan akademik. setelah proses wawancara selesai, maka peneliti melakukan analisis data penelitian. analisis data penelitian dilakukan dengan memaparkan jawaban subjek penelitian secara tertulis dan kemudian dilanjutkan dengan memaparkan hasil wawancara peneliti dengan subjek. terakhir peneliti akan melakukan hasil triangulasi dari data yang telah diperoleh. tabel berikut adalah paparan kemampuan berpikir kritis mahasiswa dalam menyelesaikan soal uraian kalkulus integral ditinjau dari tiap indikator dan level kemampuan matematis mahasiswa. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 59 tabel 1. kemampuan berpikir kritis matematis mahasiswa berdasarkan indikator kemampuan berpikir kritis matematis dan level kemampuan level kemampuan indikator kemampuan berpikir kritis matematis 1 2 3 b % s % b % s % b % s % tinggi 4 40% 6 60% 8 80% 2 20% 5 50% 5 50% sedang 5 5,68% 83 94,31% 75 85,23% 13 14,77% 8 9,09% 80 90,91% rendah 0 0% 19 100% 13 68,42% 6 31,58% 5 26,32% 14 73,68% jumlah 9 7,69% 108 92,30% 96 82,05% 19 16,24% 18 15,38% 97 82,90% keterangan: indikator 1: kemampuan mengidentifikasi dan menjastifikasi konsep, yaitu kemampuan memberikan alasan terhadap penguasaan konsep. indikator 2: kemampuan mengeneralisasi, yaitu kemampuan melengkapi data atau informasi yang mendukung. indikator 3: kemampuan menganalisis algoritma, yaitu mengevaluasi atau memeriksa suatu algoritma. b: menjawab benar s: menjawab salah berdasarkan tabel di atas terlihat bahwa secara keseluruhan jumlah mahasiswa yang menjawab dengan benar paling banyak adalah untuk indikator 2 yaitu sebanyak 96 orang atau 82, 05%. sedangkan jumlah mahasiswa yang menjawab dengan benar paling sedikit adalah untuk indikator 1 yaitu sebanyak 9 orang atau 7, 69%. hal ini mengindikasikan bahwa mahasiswa sudah mampu untuk melakukan generalisasi yang ditandai dengan mampu untuk melengkapi data atau informasi yang mendukung. selanjutnya sebagian besar mahasiswa dapat dikatakan belum memiliki kemampuan untuk mengidentifikasi dan menjastifikasi konsep yang diperlukan untuk menyelesaikan masalah dan juga belum memiliki kemampuan menganalisis yang ditandai belum mampu mengevaluasi kebenaran dari sebuah jawaban yang disajikan. dari lembar jawaban mahasiswa, peneliti mendapatkan informasi bahwa sebagian besar mahasiswa tidak dapat menyelesaikan soal untuk indikator 1 dengan benar, disebabkan oleh: (1) mahasiswa tidak dapat membuat sketsa kurva dari persamaan yang diberikan; (2) mahasiswa tidak dapat menentukan batas-batas yang merupakan perpotongan dari persamaan yang diberikan; (3) mahasiswa tidak dapat menentukan konsep luas mana yang akan digunakan, apakah luas di atas sumbu–x, luas di bawah sumbux atau luas diantara dua kurva. sedangkan untuk indikator 3, mahasiswa tidak dapat menyelesaikan soal dengan benar disebabkan oleh: (1) mahasiswa masih belum memahami tentang konsep integral trigonometri sehingga tidak dapat menentukan apakah jawaban yang disajikan sudah benar atau belum; (2) ada sebagaian yang sudah bisa menentukan jawaban yang disajikan salah, tetapi tidak bisa menyatakan mana bagian yang salah dan tidak dapat menyatakan jawaban yang benarnya. selanjutnya berdasarkan lembar jawaban mahasiswa untuk indikator 2, peneliti menemukan bahwa hampir semua mahasiswa bisa melengkapi data yang diberikan, hal ini mengindikasikan bahwa mahasiswa sudah memahami konsep integral rasional bentuk faktor linier. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 60 untuk lebih memberikan informasi tentang pernyataan diatas dapat dilihat dari triangulasi hasil jawaban, wawancara yang diperoleh dari mahasiswa level kemampuan tinggi, sedang dan rendah. a. analisis data mahasiswa level kemampuan tinggi setelah diperoleh hasil analisis jawaban tertulis dan analisis data wawancara, selanjutnya dilakukan perbandingan untuk mengetahui valid tidaknya data yang diperoleh. berikut adalah rangkuman kemampuan berpikir kritis matematis mahasiswa level kemampuan rendah berdasarkan data tertulis dan data wawancara dan hasil triangulasi. tabel 2. triangulasi dari hasil jawaban dan wawancara mahasiswa dari level kemampuan tinggi indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara 1. kemampuan mengidentifikasi dan menjastifikasi konsep, yaitu kemampuan memberikan alasan terhadap penguasaan konsep  jelas dalam menuliskan jawaban  memberikan respon terhadap soal tes yang diberikan  tidak menentukan titik puncak dari persamaan parabola yang diberikan  tidak menentukan titik potong persamaan garis yang diberikan dengan persamaan parabola  mencoba membuat sketsa gambar dari persamaan parabola dan garis yang diberikan, tetapi masih salah.  salah dalam menentukan batas integral tentu, yang berakibat salah dalam menentukan luas daerah yang diarsir.  dapat menyebutkan apa saja yang ditanyakan dari soal  informasi yang diberikan tidak cukup untuk menyelesaikan soal  kesulitan dalam menggambar kurva dari persamaan yang diberikan yang diakibatkan tidak mengetahui titik potong dari persamaan parabola yang diberikan dengan garis.  alasan yang diberikan dalam menentukan luas daerah yang dibatasi oleh parabola dan garis masih salah karena karena tidak bisa menentukan batas integral yang digunakan. 2. kemampuan menggeneralisasi, yaitu kemampuan melengkapi data atau informasi yang mendukung  jelas dalam menuliskan jawaban  melengkapi data yang diberikan, walaupun dibagian akhir selesaian terdapat sedikit kesalahan.  dapat menyebutkan apaapa saja yang harus dilakukan dalam menyelesaikan persoalan  dapat menyelesaikan persoalan dengan melengkapi data yang diberikan, namun belum benar karena ada beberapa hal yang lupa dan kurang teliti infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 61 indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara 3. kemampuan menganalisis algoritma, yaitu kemampuan mengevaluasi atau memeriksa suatu algoritma  jelas dalam memberikan jawaban  tidak memberikan evaluasi terhadap sebuah pernyataan yang diajukan.  memberikan koreksian terhadap pernyataan yang diberikan, namun masih salah.  tidak memberikan evaluasi terhadap sebuah pernyataan yang diajukan.  mengetahui bahwa persoalan merupakan integral tirgonometri, tetapi tidak dapat memberikan koreksian dimana letak kesalahan, sehinggaperbaikan diberikan masih salah. kesimpulan 1. mahasiswa tidak dapat mengidentikasi konsep yang dibutuhkan untuk menyelesaikan persoalan, namun mahasiswa tidak dapat melakukan jastfikasi dan memberikan alasan dalam menyelesaikan persoalan yang diberikan. 2. mahasiswa dapat melakukan generalisasi dengan melengkapi data atau informasi yang mendukung, walaupun melakukan sedikit kesalahan yan disebabkan oleh faktor kurang teliti 3. mahasiswa tidak dapat menganalisis dan melakukan evaluasi dari persoalan yang diberikan, tetapi mahasiswa dapat menentukan jenis persoalan yang diberikan b. analisis data mahasiswa level kemampuan sedang setelah diperoleh hasil analisis jawaban tertulis dan analisis data wawancara, selanjutnya dilakukan perbandingan untuk mengetahui valid tidaknya data yang diperoleh. berikut adalah rangkuman kemampuan berpikir kritis matematis mahasiswa level kemampuan rendah berdasarkan data tertulis dan data wawancara dan hasil triangulasi. tabel 3 triangulasi dari hasil jawaban dan wawancara mahasiswa dari level kemampuan sedang indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara 1. kemampuan mengidentifikasi dan menjastifikasi konsep, yaitu kemampuan memberikan alasan terhadap penguasaan konsep  tidak memberikan jawaban  dapat menyebutkan apa saja yang diketahui dan yang ditanyakan, tetapi tidak bisa menyelesaikan persoalan yang ada, karena tidak mengetahui konsep selesaiannya. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 62 indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara 2. kemampuan menggeneralisasi, yaitu kemampuan melengkapi data atau informasi yang mendukung  menyelesaikan persoalan dengan menggunakan konsep integral fungsi rasional  tidak sampai menemukan nilai-nilai kooefisien a, b, c, dan d, sehingga tidak dapat menyelesaikan persoalan dengan benar.  menyatakan konsep yang digunakan untuk menyelesaikan persoalan, tetapi tidak mengetahui nama konsep yang digunakan dan alasan penggunaan konsep tersebut.  memiliki sedikit pengetahuan dalam menggunakan konsep tersebut.  tidak dapat menyelesaikan persoalan yang diberikan. 3. kemampuan menganalisis algoritma, yaitu kemampuan mengevaluasi atau memeriksa suatu algoritma  tidak memberikan evaluasi terhadap selesaian yang disajikan.  tidak menyelesaikan persoalan dengan benar.  memberikan evaluasi terhadap selesaian yang diberikan, namun masih salah.  tidak memahami bagaimana menyelesaiakan persoalan yang ada. kesimpulan : 1. mahasiswa tidak dapat mengidentifikasi dan menjastifikasi pengetahuan yang dibutuhkan untuk menyelesaikan soal, serta belum mampu memberikan alasan dengan benar terhadap selesaian yang diberikan. 2. mahasiswa tidak dapat melakukan generalisasi dengan melengkapi data atau informasi yang mendukung dengan benar. 3. mahasiswa tidak dapat mengevaluasi dengan benar persoalan yang diberikan, tetapi tidak mampu menganalasis persoalan yang diberikan, serta tidak mampu menyelesaiakan persoalan dengan benar. c. analisis data mahasiswa level kemampuan rendah setelah diperoleh hasil analisis jawaban tertulis dan analisis data wawancara, selanjutnya dilakukan perbandingan untuk mengetahui valid tidaknya data yang diperoleh. berikut adalah rangkuman kemampuan berpikir kritis matematis mahasiswa level kemampuan rendah berdasarkan data tertulis dan data wawancara dan hasil triangulasi. tabel 4 triangulasi dari hasil jawaban dan wawancara mahasiswa dari level kemampuan rendah indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara 1. kemampuan mengidentifikasi dan menjastifikasi konsep, yaitu kemampuan  tidak menentukan titik potong antara parabola, garis dan sumbu y.  hanya membuat sketsa  dapat menyebutkan apa saja yang ditanyakan, dan hanya sedikit mengetahui infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 63 indikator kemampuan berpikir kritis matematis hasil tes tertulis hasil wawancara memberikan alasan terhadap penguasaan konsep persamaan garis, namun tidak membuat sketsa parabola sehingga tidak dapat menentukan mana daerah yang diarsir untuk dihitung luas daerahnya.  mencoba menyelesaikan dengan menggunakan integral tentu, namun batas integralnya masih salah, dan persamaan yang digunakan untuk menghitung luas daerahnya masih salah. langkah-langkah untuk menyelesaikan persoalan.  tidak bisa membuat sketsa parabola, sehingga tidak dapat menentukan daerah yang akan dihitung luas daerahnya.  tidak memahami bagaimana menggunakan konsep integral dalam menghitung luas daerah. 2. kemampuan menggeneralisasi, yaitu kemampuan melengkapi data atau informasi yang mendukung  mencoba merespon dengan cara melengkapi data diberikan secara langsung, tapi menggunakan konsep atau aturan integral yang lain, sehingga tidak dapat melengkapi data yang diberikan.  tidak memahami konsep apa yang akan digunakan untuk menyelesaikan persoalan, sehingg tidak bisa melengkapi data. 3. kemampuan menganalisis algoritma, yaitu kemampuan mengevaluasi atau memeriksa suatu algoritma  tidak memberikan evaluasi terhadap permasalahan yang diberikan  memberikan respon dengan mencoba memberikan koreksian yang tidak jauh berbeda dengan peneliti sajikan, namun masih salah.  tidak memahami konsep integral fungsi trigonometri, sehingga koreksian yang diberikan tidak menggambarkan pemahaman seutuhnya. kesimpulan : 1. mahasiswa belum mampu untuk mengidentifikasi konsep dan menjastifikasi konsep yang akan digunakan untuk menyelesaikan persoalan yang diberikan. 2. mahasiswa tidak mengetahui konsep apa yang akan digunakan untuk dapat melengkapi data yang diberikan. hal ini mengindikasikan bahwa mahasiswa belum memiliki kemampuan untuk menggeneralisasi konsep. 3. mahasiswa tidak memahami konsep integral trigonometri, sehingga mahasiswa tidak memberikan evaluasi yang benar. hal ini mengindikasikan bahwa mahasiswa belum memiliki kemampuan untuk mengevaluasi dan memeriksa sebuah persoalan yang ada. berdasarkan paparan di atas, maka dapat peneliti simpulkan bahwa mahasiswa belum memiliki kemampuan untuk mengidentifikasi dan menjastifikasi konsep yang diperlukan untuk menyelesaikan masalah. selain itu, mahasiswa juga belum memiliki kemampuan menganalisis atau mengevaluasi sebuah algoritma. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, februari 2016 64 sedangkan apabila dilihat dari level kemampuan matematis, mahasiswa pada level kemampuan tinggi, paling banyak menjawab dengan benar adalah soal untuk indikator 2 yaitu sebanyak 8 orang (80%), sedangkan yang paling sedikit menjawab dengan benar adalah soal untuk indikator 1 yaitu sebanyak 4 orang (40%). untuk mahasiswa pada level kemampuan sedang, paling banyak menjawab dengan benar adalah soal untuk indikator 2 yaitu sebanyak 75 orang (85,23%), sedangkan yang paling sedikit menjawab dengan benar adalah soal untuk indikator 1 yaitu sebanyak 5 orang (5,68%). mahasiswa pada level kemampuan rendah, menjawab dengan benar paling banyak adalah untuk soal indikator 2 yaitu sebanyak 13 orang (68,42%), sedangkan untuk soal indikator 1, semua mahasiswa pada level kemampuan rendah tidak satu orang pun yang menjawab dengan benar. berdasarkan hasil analisis lembar jawaban dan hasil wawancara, peneliti mendapatkan beberapa informasi, diantaranya: (1) untuk soal indikator 1, mahasiswa pada level kemampuan tinggi dan sedang sudah bisa menentukan konsep apa yang akan digunakan untuk menyelesaikan persoalan, namun mahasiswa tersebut menemui kesulitan dalam menggambar sketsa kurva dan menentukan batas-batas integralnya. sedangkan mahasiswa pada level kemampuan rendah, tidak bisa menentukan konsep apa yang akan digunakan untuk menyelesaikan persoalan yang ada; (2) untuk soal indikator 2, mahasiswa pada level kemampuan tinggi, sedang, dan rendah sebagian besar sudah bisa melengkapi data yang diberikan, hal ini disebabkan sebagian besar mahasiswa dari masing-masing level kemampuan sudah memahami konsep integral rasional, sebagian kecil melakukan kesalahan hanya pada menentukan selesaian akhirnya saja; (3) untuk indikator 3, mahasiswa pada level kemampuan tinggi sudah bisa menyatakan bahwa selesaian yang diberikan salah dan bisa menyatakan mana bagian yang salah serta sudah bisa menyatakan jawaban yang benarnya, namun sebagian kecil masih salah dalam menentukan integral trigonometrinya. sedangkan mahasiswa pada level kemampuan sedang, hanya bisa menyatakan jawaban yang diberikan salah, namun belum bisa menentukan bagian mana yang salah dan menyatakan jawaban yang benarnya. mahasiswa pada level kemampuan rendah sama sekali belum bisa menentukan apakah jawaban yang diberikan masih salah atau sudah benar. iv. kesimpulan secara keseluruhan dan berdasarkan level kemampuan mahasiswa (tinggi, sedang, dan rendah), mahasiswa mampu menjawab benar dengan persentase paling tinggi adalah pada soal untuk indikator 2 yaitu sebanyak 96 orang (82,05%), artinya baik secara keseluruhan maupun berdasarkan level kemampuan (tinggi, sedang, dan rendah), mahasiswa sudah memiliki kemampuan untuk melengkapi data atau melakukan generalisasi. sedangkan baik secara keseluruhan maupun berdasarkan level kemampuan (tinggi, sedang, rendah), mahasiswa mampu menjawab dengan benar paling sedikit adalah pada soal untuk indikator 1 yaitu sebanyak 9 orang (7,69%). dapat disimpulkan bahwa, mahasiswa baik secara keseluruhan maupun berdasarkan level kemampuan matematis (tinggi, sedang, rendah), sudah memiliki kemampuan menggeneralisasi, namun belum memiliki kemampuan untuk mengidentifikasi dan menjastifikasi konsep serta belum memiliki kemampuan menganalisis atau mengevaluasi sebuah algoritma. infinity jurnal ilmiah program studi matematika stkip siliwangi bandung, vol 5, no. 1, 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(1997). adolescence. london: mc-graw-hill, inc. somakim.(2010). peningkatan kemampuan berpikir kritis dan self-efficacy matematik siswa sekolah menengah pertama dengan penggunaan pendekatan matematika realistik. disertasi pada sekolah pascasarjana universitas pendidikan indonesia. disertasi tidak diterbitkan. sugiyono.(2011). metode penelitian pendidikan (pendekatan kuantitatif, kualitatif, dan r&d. bandung: alfabeta. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p147-158 147 challenges during the pandemic: use of e-learning in mathematics learning in higher education muhammad irfan, betty kusumaningrum, yuyun yulia, sri adi widodo* universitas sarjanawiyata tamansiswa, yogyakarta, indonesia article info abstract article history: received jul 19, 2020 revised aug 20, 2020 accepted aug 26, 2020 on march 16, 2020, many universities in indonesia began implementing online-based learning to replace lectures in the classroom. this is done as a way to reduce the transmission of the covid-19 outbreak in indonesia. there is an opinion that with the implementation of online learning, especially in mathematics education study programs, there are many obstacles when learning takes place. this study aims to determine the obstacles that arise after the implementation of online learning in mathematics learning in higher education. this research is a qualitative case study, assisted by an online survey. the researcher collected data through an online survey consisting of 27 questions. the survey is aimed at lecturers who teach in mathematics education study programs in indonesia. the survey contains structured questions and leads to three parts, namely; basic skills challenges, teaching and learning challenges, and university challenges. the 27 questions contained questions about the ability of platform mastery to support online learning owned by each lecturer. the research involved 26 lecturers from universities in sumatra, java, kalimantan, and sulawesi. the results of this study reveal that all lecturers affected by the pandemic use a learning management system (lms) based website as a means of online learning. the learning management system-based platform is the most widely used (google class and edmodo) while video conferencing is the second choice (zoom and skype). what is interesting is that the lms available on campus is less attractive to lecturers. however, there are obstacles faced such as the limitations of writing mathematical symbols and the limited basic capabilities of the learning management system and multimedia software to support online learning. keywords: pandemic, covid-19, online learning, mathematics learning copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: sri adi widodo, department of mathematics education, universitas sarjanawiyata tamansiswa jl. batikan uh iii/1043, umbulharjo, yogyakarta 55165, indonesia email: sriadi@ustjogja.ac.id how to cite: irfan, m., kusumaningrum, b., yulia, y., & widodo, s. a. (2020). challenges during the pandemic: use of e-learning in mathematics learning in higher education. infinity, 9(2), 147-158. 1. introduction indonesia confirmed the first case of covid-19 on march 2, 2020 (setiawan, 2020; setiawan & ilmiyah, 2020; tosepu et al., 2020). two weeks ago, many universities in indonesia issued a learning policy at home and implemented online learning. until may 16, 2020, the number of confirmed cases was 16,496 positive (kemenkes ri, 2020). with the https://doi.org/10.22460/infinity.v9i2.p147-158 irfan, kusumaningrum, yulia, & widodo, challenges during the pandemic: use of e-learning … 148 implementation of the learning policy at home by tertiary institutions, many lecturers began to use the learning management system (lms) platform as well as video conferences that could assist in mathematics learning activities (gunawan et al., 2020; sulisworo et al., 2020). in indonesia, the most widely used lms platforms include: google classroom, edmodo, elearning portals for each tertiary institution, etc. as for video conferences, you can use zoom, webex, google meet, microsoft teams, etc. online learning in a pandemic is an alternative solution (basilaia & kvavadze, 2020; bauerlein, 2008; laprairie & hinson, 2006; taha et al., 2020). the use of online learning is indeed practical because it can be used anywhere and anytime (bourne et al., 2005; means, 2010; nakamura et al., 2018; özyurt et al., 2013). however, do not close your eyes that the implementation of online learning raises its problems (hung & chou, 2015; smart & cappel, 2006; van bruggen, 2005). therefore, higher education that has limited or no experience of e-learning or e-learning resources experiences difficulties, especially, when lecturers lack knowledge of how to use online applications (kim & bonk, 2006; zaharah & kirilova, 2020). implementation of online learning in higher education does have advantages and disadvantages. the advantages of online learning are that it is flexible and can be widely used, while the drawback is that it is very potential to do plagiarism practices, internet signal strength, and devices that support (arkorful & abaidoo, 2015; irfan, 2015). since covid19 pandemic occurred in indonesia, research on covid-19 began to be carried out. in the field of mathematical modeling, many experts predict when the pandemic reaches its peak when it ends, and the transmission model of the spread of viruses (kim et al., 2020; ndaïrou et al., 2020; nuraini et al., 2020; peirlinck et al., 2020; rahimi & abadi, 2020; resmawan & yahya, 2020; soewono, 2020; tang et al., 2020). whereas in learning mathematics in schools, research on obstacles in the use of e-learning that occurred in schools (mailizar et al., 2020; mulenga & marbán, 2020). much research on barriers caused by the application of online learning in nonpandemic situations (ali & magalhaes, 2008; beetham & sharpe, 2007; eady & lockyer, 2013; karasavvidis, 2010). while research on barriers to the use of online learning during the pandemic is still not widely done. mailizar et al. (2020) initiated research on barriers to the use of e-learning in indonesia, but the participants involved were mathematics teachers. this research was conducted in indonesia at the time of the pandemic period and focused to find out the obstacles that arise after the implementation of online learning in mathematics learning in higher education. most of the studies conducted did not focus on learning mathematics in tertiary institutions (ali et al., 2018; donnelly & mcsweeney, 2008; kabilan & khan, 2012). this poses many challenges but at the same time highlights the importance of investigating e-learning barriers for mathematics education lecturers during a pandemic. this research was conducted in indonesia which focuses on the challenges and obstacles faced by lecturers who teach in mathematics education study programs during a pandemic. therefore, to see the challenges and obstacles faced by lecturers who teach in mathematics education study programs during the pandemic, researchers feel the need to focus on three aspects, namely basic skills challenges, teaching and learning challenges, and university challenges. the findings from this study will help advance our understanding of the obstacles to e-learning integration amid the covid-19 pandemic in the context of developing countries at the tertiary level. therefore, this study adds valuable insights to the e-learning literature and provides important suggestions for improving e-learning practices. to achieve this goal, this study aspires to answer questions (1) what are the challenges faced by lecturers in implementing e-learning during a pandemic? and (2) what is the basic ability possessed by lecturers to support e-learning? volume 9, no 2, september 2020, pp. 147-158 149 2. method 2.1. research design this research is qualitative research with a type of case study. the cases studied are the challenges faced by lecturers in implementing e-learning during a pandemic and what basic abilities lecturers have to support e-learning. in line with (creswell, 2012a; 2012b), qualitative research is exploratory in nature, which helps researchers to find out more about the challenges faced by mathematics education lecturers during the pandemic in teaching and learning activities. this research cannot help in making a decision or coming to a conclusion (generalization). however, this research can help understand how mathematics education lecturers in indonesia experience obstacles in undergoing teaching and learning activities using e-learning during the pandemic. 2.2. participants random sampling was used in this study. this is because the researcher cannot control who and from the institution, the respondent is from. researchers only limit and ensure that the subjects used in this study are lecturers in mathematics education in indonesia. researchers do not limit whether they are from state universities or not, respondent's age, length of work, and gender. this is different from research (mailizar et al., 2020) whose participants are mathematics teachers and are categorized based on length of work, gender, and also certification. the research involved 26 lecturers from universities in sumatra, java, kalimantan, and sulawesi. 2.3. research instruments the instrument in this study was made by researchers and was discussed in a group discussion forum (fgd-online). for this reason, researchers collected data through an online survey consisting of 27 questions. the survey is aimed at lecturers who teach in mathematics education study programs in indonesia. of the 27 questions consisted: 3 question contains participant's profile such as name, email address and place of study of the participant; 3 questions contained information about people who were positively affected by covid-19 in the college's; 3 questions about the college's response and each lecturer in responding to pandemics in learning; and the remaining questions were about the ability to master the platform that supports learning online owned by each lecturer consisting of basic skills challenges, teaching and learning challenges, and university challenges. 2.4. data collection and analysis data was collected using an online survey. online surveys are used for reasons of the flexibility of compatibility with lecturers' online work during a pandemic. also, online surveys are easily managed and accessed using various devices (fraenkel et al., 2012). the survey was distributed after passing the evaluation process from the results of the fgd online which involved 4 e-learning experts from 3 universities in indonesia. furthermore, online surveys are disseminated through whatsapp groups, e-mails, and also facebook in march 2020. google form was chosen by the researcher to make an online survey because of its ease of use. after the respondent fills in the survey, the respondent will get a recapitulation of the results and can then be confirmed by the researcher. the questionnaire is open for three weeks. the data obtained are then grouped based on basic skills challenges, irfan, kusumaningrum, yulia, & widodo, challenges during the pandemic: use of e-learning … 150 teaching and learning challenges, and university challenges. the data obtained is then interpreted and described by researchers. 3. results and discussion 3.1. results the results obtained from filling out an online survey show that learning management system-based platforms are the most used (google classroom and edmodo) while video conferencing is the second choice (zoom and skype) (see figure 1). the webex platform and google meeting no one uses it on the research subjects. but learning mathematics using conferencing, subjects preferred to use zoom and skype. figure 1. the platform that is often used in e-learning the results of a survey of respondents used as participants in this study found that in the basic skills challenges section, students generally could use the online learning platform used during the pandemic covid-19 (see table 1). table 1. the survey results of respondents in basic skills challenges no. question response (%) yes maybe no 1. have you ever used the application before? 61.5 0 38.5 2. are you good at presentation applications (such as microsoft powerpoint, openoffice impress)? 88.5 0 11.5 3. are you familiar with spreadsheet applications (such as microsoft excel, openoffice calc)? 80.8 3.8 15.4 4. do you master digital image recording applications (such as: with a digital camera and scanner)? 73.1 7.7 19.2 5. do you master multimedia compilation applications (such as adobe flash)? 23.1 23.1 53.8 volume 9, no 2, september 2020, pp. 147-158 151 no. question response (%) yes maybe no 6. do you master mind / concept mapping applications (such as inspiration, mindmapple, mindjet)? 15.4 11.5 73.1 7. do you master digital video applications (such as camtasia, adobe premiere, moviemaker, imovie)? 34.6 15.4 50 8. do you master web design applications (such as adobe dreamweaver, frontpage)? 7.7 11.5 80.8 9. do you master learning management systems (such as moodle, edmodo, khanacademy)? 38.5 15.4 46.2 10. do you master applications for online meetings (such as zoom, webex, google meeting)? 57.7 3.8 38.5 11. do you feel comfortable using digital technology? 88.5 3.8 7.7 table 1 show that the respondents master presentation applications (such as microsoft powerpoint, openoffice impress) and also image managers (questions 1-4). however, respondents have weaknesses in making animation and the ability for objectoriented programming (adobe flash and web design) and video editing (questions 5-9). respondents find it easier to use a ready-to-use and familiar lms such as edmodo or google classroom. also, video conferencing is still the main choice when teaching. the results of respondents surveys that were used as participants in this study found that in the teaching and learning challenges section, among them most of the students had no training to use online learning platforms. but in general, students are open to the existence of new digital technology, so they always learn independently and find out about digital technology. especially in a covid-19 pandemic condition like this, students are required to learn independently including in learning new information technology (see table 2). table 2. the survey results of respondents in teaching and learning challenges no. question response (%) yes maybe no 12. are there courses that cannot apply for your study program online learning? 18.2 9.1 72.7 13. do you always learn new digital technology? 61.5 34.6 3.8 14. do you always find out about new digital technology? 61.5 34.6 3.8 15. can you combine course content with technology and appropriate teaching approaches? 65.4 34.6 0 16. do you understand and can conduct performance assessments (for example student performance)? 88.5 11.5 0 irfan, kusumaningrum, yulia, & widodo, challenges during the pandemic: use of e-learning … 152 table 2 show that the problems faced by lecturers during online learning are limitations in delivering material, especially pure mathematics. this is reasonable because to teach the material, special software (eg mathtype) is required. in addition, the computer programming course also encountered problems, because lecturers found it difficult to check the obstacles faced by students, and students found it difficult to convey their problems. this is because programming courses are related to syntax, computer specifications, software, and algorithms. of course, this becomes something complex. the results of a student survey that were used as participants in this study found that in the university challenges section. the most interesting thing is found as many as 73.1% of respondents stated that tertiary institutions have e-learning websites but only 34.6% use them (see table 3). table 3. the survey results of respondents in university challenges no. question response (%) yes maybe no 17. does the university where you work have elearning facilities? 73.1 3.8 23.1 18. have you used it for learning? 34.6 30.8 34.6 3.2. discussion in addition to the challenges faced by lecturers, this study also aims to find out whether the basic abilities possessed by lecturers to support the application of e-learning. the findings show two important points of interest. first, this research shows that the obstacles to implementing e-learning in tertiary institutions include: lecturers have mastered the basic skills to support e-learning learning (al-rahmi et al., 2015; ash et al., 2003; davies et al., 2017; govindasamy, 2001; pundak et al., 2010; trelease, 2015), but precisely the obstacles to the mathematical content (sözgün et al., 2018; vrugt & oort, 2008). the lecturers find it difficult when they have to teach online on mathematics (adnan & boz, 2015; kurt, 2017; lin et al., 2017). also, many lecturers use learning management systems that are publicly available (eg edmodo and google classroom) rather than e-learning developed by their tertiary institutions. the findings show that lecturers were ready to use elearning before this pandemic. therefore, when this pandemic comes, they are not so panicked to do online learning. this contrasts with the results of research (ali & magalhaes, 2008; assareh & bidokht, 2011; childs et al., 2005) which show that there are many obstacles in the application of e-learning during a pandemic. this pandemic brings us to an unusual life, including in learning mathematics in college. surely, the findings of this study cannot be used as a benchmark when learning in normal (non-pandemic) situations. also, this research focuses on mathematics learning where the implementation of e-learning presents new challenges to lecturers due to difficulties in explaining mathematical concepts online (frid, 2002; nakamura et al., 2018). second, this research shows that lecturers prefer to use e-learning platforms that are widely available (eg edmodo and google classroom) or use video conferencing (zoom or skype) rather than using e-learning developed by universities. this shows that the lecturers are not satisfied with the features and facilities of the university e-learning. some suggested by lecturers includes: there is an attendance system that can be recorded properly and can be exported in the form of excel, the presence of video conferencing features, and an volume 9, no 2, september 2020, pp. 147-158 153 assessment system. this suggestion makes great sense because it is indeed a necessity when learning (albelbisi & yusop, 2018; ali et al., 2018; donnelly & mcsweeney, 2008). this study examines the challenges faced by lecturers when implementing e-learning during the pandemic. it is not easy for lecturers who are used to teaching in class and must be replaced by using e-learning. many obstacles are faced, one of which is the availability of features in the academic portals of each tertiary institution, the limited interaction between lecturers and students, and limitations in writing mathematical symbols. this certainly can be used as further research on the challenges and obstacles faced by students in using elearning, the development of e-learning systems of each tertiary institution, and also the resolution related to mathematical symbols that tend to be difficult if written on several elearning platforms. 4. conclusion during the covid-19 pandemic, universities in indonesia have implemented many online-based learning policies. this is a form of a rapid response from universities in indonesia to minimize covid-19 transmission in the campus environment. however, this policy still provides several obstacles that arise from both lecturers and students. in this study, it shows that the challenges faced by lecturers in implementing online learning include: limitations in presenting material, especially when courses have many mathematical equations and programming languages. besides, the lecturers are not good at video editing or animation using various animation maker software. they are limited to presenting material using powerpoint and text. overall, to use online learning, lecturers must at least master presentation software, text processing, assessment, and video conferencing. this study proves that higher education policies by implementing online learning are not accompanied by the ability to use platforms that can support online learning. this research can be used as a reference to explore further the obstacles faced by students when online learning is implemented. acknowledgments the authors would like to thank the institute of research, development, and community service, universitas sarjanawiyata tamansiswa for providing funding for this research. references adnan, m., & boz, b. 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(2019). development of web game learning materials for primary school students. infinity, 8(2), 199-208. 1. introduction along with the progress of times, the development of technology has increasingly affected many realms of life including education that now has widely utilized technology as a means of communication among teachers and students (bainbridge, 2013; bates, 2005; mumtaz, 2000). education as an information activity, and can be disseminated to the next generation of a nation through information technology, is able to turn challenges into opportunities, and master various methods in solving problems (kalelioglu & gülbahar, 2014). problem solving is one of the two skills a person must have to face the future (pomalato, 2005). this skill is considered important as to make a decision, one must first understand what problems they face in order to find a way out. mailto:yudhi@staff.uksw.edu wahyudi, ambarwati, & indarini, development of web game learning … 200 thus, problem solving skills need to be developed since the early age on. this can be done by starting to familiarize child to solve problems, one of them through learning activities at school. mathematics is one of the subjects that can facilitate students to hone their problem solving skills (schoenfeld, 2016). however, mathematics is often still considered a difficult and boring subject (rahayu & afriansyah, 2015). this assumption forms a negative understanding of students in mathematics (gazali, 2016). to eliminate this misunderstanding, the teacher as a facilitator must be able to create an interesting learning atmosphere. however, what is happening now, there are still many teachers who teach without providing opportunities for students to learn on their own how to solve a problem. teachers tend to give "short cuts" for granted to students. often, the teachers keep focusing on transfering the knowledge stated on the text book instead of enhancing students’ problem solving skillls by providing certain problematical cases. this is indeed out of balance because the assessment onlyconcern on cognitive abilities, while the problem solving skills are ruled out. in fact, students' ability to solve problems is important and needs to be developed. although some teachers have provided opportunities for students to hone their skills in solving problems, many students still feel bored with their learning activities. this is because the teacher still use the old teaching methods which are not balanced with the current demands. the teacher should be able to choose the right learning media that are in accordance with the characteristics of students. also, , the media chosen by the teacher must also meet the requirements of good learning media. in addition to fulfilling good media requirements, the media used must also be in line with the demands of education today, where learning has begun to be integrated with technology (yunianta, putri, & kusuma, 2019). the teacher should choose the media that is in accordance with this demand. one of the media that can be utilized is web game technology or often called online gaming. research conducted by newzoo, indonesia is a country with the highest number of players playing on mobile devices and pcs. as much as 61 percent of game players in indonesia are actively playing games on more than two platforms every month. while 46 percent of indonesian game players are confirmed to play games on mobile devices and pcs. considering that game users are increasing, teachers can use game as a learning medium. games are developed based on fun learning, where students will be faced with a number of instructions and rules of the game (panggayudi, suweleh, & ihsan, 2017). through the instructions and rules of the game, students can find a way out and solve every problem passed in the game being played. this activity can attract students' attention and foster a great curiosity about how they can complete the game and find a way to reach the destination. learning using media is suitable for elementary school children, given that they are entering the phase of having great curiousity. in addition, children in elementary school are still very happy to play thus the material packaged in game form can reduce students' boredom compared to teacher-oriented learning . the purpose of this study is to develop a web game learning materials to improve problem-solving skills in two-dimensional figure. the expectation is that by using this media, students’ mathematical problem solving skills especially regarding two-dimensional figure could improve. volume 8, no 2, september 2019, pp. 199-208 201 2. method 2.1. stages of development this research is a type of research development or r & d using the assure development model (smaldino, lowther, russell, & mims, 2008). this model is divided into 6 stages, namely: 1) analyze learners, 2) state objectives, 3) select strategi, technology, media, and materials, 4) utilize media and materials, 5) require learner participation, 6) evaluated and revise (smaldino et al., 2008). the assure model is shown in figure 1. figure 1. assure development model based on the chart in figure 1, the stages of assure model development can be explained as follows. 2.1.1. analyze learners (analyze students’ characteristics) the initial stage that needs to be done is to identify students including their characteristics, initial abilities, and learning styles. to obtain thedata, interviews will be conducted with teachers and students, and questionnaires to students are provided. these data are used as preliminary data to develop learning media thus appropriate methods and learning activities can be determined. the subjects chosen in this study were 4th grade students at sd n sugihan 01. 2.1.2. state objectives (setting learning objectives) the second stage is to set learning objectives to be achieved. before setting learning objectives, the researchers conducted an analysis of the curriculum used in schools, core competencies (ki) and basic competencies (kd) in accordance with the revised 2013 curriculum in 2017. the learning objectives need to be determined in order to choose the correct media, regulate the learning environment that is in line with the demand objectives, determine assessment / evaluation techniques and instruments. 2.1.3. select method, media, and materials the third stage in this development model is choosing methods, media, and teaching materials that will be used during the learning process. this activity needs to be done because those three components can help students achieving the learning objectives wahyudi, ambarwati, & indarini, development of web game learning … 202 that have been made before. determination of all three was based on students’ characteristics and learning objectives designed. 2.1.4. utilize media and materials the next stage will be applying the methods, media, and teaching materials that have been chosen. before all of them were implemented, validation / expert test (material experts, media experts, learning experts, and expert questions) has to be carried out first to see the quality and feasibility of media developed. after obtaining expert validation, the media can be used in the learning process at the chosen elementary school. 2.1.5. require learner participation the fifth stage was involving students in learning activities. here, they will get learning activities that has been previously designed. the learning process will use media learning web games to help improving theirproblem solving skills. the purpose of using this media was to help students understand two-dimensional figure material using problemsolving steps. 2.1.6. evaluation and revise after the learning media has been designed and developed, the next step is to evaluate the media used. this activity was conductedto obtain data related to the advantages and disadvantages of the media being developed. the results will be used as input to improve the media, thus it is ready to use in every school. 2.2. techniques and data collection instruments the validity (feasibility), practicality, and effectiveness of the product will surely be tested. the validity of the product can be seen by means of expert judgment. expert assessment includes learning experts, learning media experts, learning material experts and problem experts. expert assessment data was done using a likert scale with a range of 1-5 with criteria not good (1) to very good (5). these results were used to see the feasibility (validity) of the products produced. to see the practicality of the product carried out by a limited test to 10 students and conducted interviews to see students' responses to the products used. effectiveness was seen from the comparison of student learning outcomes when working on the pretest and posttest questions. 2.3. data analysis technique data obtained from expert assessment results (product validation) were then analyzed using percentage descriptive and categorical analysis techniques to see the feasibility of the media. initially the results obtained from the product assessment in the form of scores were presented in the form of a closed questionnaire which was then summed and averaged between the results of the assessment from media experts, material experts, matter experts and learning experts. the score obtained was then percentage using the formula. note ap : percentage number actual score : score given by expert validator ideal score : maximum score between the number of items and maximum score for each item. volume 8, no 2, september 2019, pp. 199-208 203 table 1. product development quality category interval category 81-100% very good 6180% good 4160% fair 2140% low 120% very low based on the categories (table 1), web game learning materials is declared feasible if the average percentage reaches a high category (≥ 61%). practical testing was done by looking at students' responses known from the interview activities. the effectiveness test was done by comparing student learning outcomes before and after being given the pretest and posttest questions. this test will later be processed using one group pretest-posttest. final scores before and after using the media were compared using the paired-sample ttest with the help of spss 16. 3. results and discussion 3.1. results the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. this study produced a webgame-based media learning product to improve problem-solving skills in twodimensional figure material. the results of media development using the assure development model consisting of 6 stages are explained as follow: 3.1.1. analyze learners concerning students’ characteristics of learning styles, it was found that 19 students (73%) had a visual learning style. this data will be used as the basis for determining the appearance and facilities available in the web game. other aspects measured was the students' initial ability to construct two-dimensional figure material, the initial ability to operate a computer. the results obtained from the analysis of the three aspects can be seen in table 2. table 2. results of analyze learners no competency total total not good less good good 1 understanding of two-dimensional figure material 21 2 3 26 2 able to operate a computer 2 4 20 26 based on table 2, it was implied that most students (73%) agreed that they had a visual learning style. as much as (81%) students showed disagreement in understanding two-dimensional figure material and as much as (77%) agreed of being able to operate a computer. meanwhile, the interview results revealed that half of the students in the class claimed to be bored with math because they were only fixated on the book. they were given too many questions without any fun activities. this then impacted on student learning outcomes. in addition, minimum use of media was also one of the reasons why wahyudi, ambarwati, & indarini, development of web game learning … 204 students were not interested in learning. this was then used as the basis for developing webgame-based learning media with the aim of putting studentsin a pleasant atmosphere. besides that, it can help them improve their problem solving abilities. 3.1.2. state objectives after analyzing the students' initial abilities, the next step was to determine the learning objectives. before the learning objectives were determined, basic competencies (kd) were first determined and also learning indicators can be formulated in table 3. table 3. basic competencies and indicators basic competencies indicator explain and determine the circumference and area of a square, a rectangle, and a triangle and a power of two with a square root. describe the formula around and area of a square, rectangle and triangle. calculate the area and circumference of a square, rectangle and triangle. solve problems related to area and circumference of square, rectangle and triangle. re-prove the calculation relating to the area and circumference of a square, rectangle and triangle. resolve a problem related to the circumference and area of a square, rectangle, and triangle including involving the power of two with a square root. design the model according to the area and circumference of the building that has been calculated. the next step was formulating the leaarning objectives based on the basic competency and its indicators. the following is the formulation of the objectives of mathematics learning in the material of two-dimensional figure: (1) by playing the game, students can describe the formula around and the area of a square, rectangle and triangle correctly; (2) by playing the game, students can calculate the area and circumference of the square, rectangular and triangle shapes in the game correctly; (3) by playing games and designing models, students can solve problems related to the area and circumference of square, rectangle and triangle correctly; (4) with group activities, students can prove again the calculations related to the area and circumference of the square, rectangle and triangle; (5) with group activities, students can design models according to the area and circumference of the building that has been calculated. 3.1.3. select method, media, and materials the next stage in the assure development method was to choose methods, media and teaching materials. the method chosen in thelearning activities were lectures, discussions, assignments and questions and answers. the selected medi was webgamebased learning media and teaching materials used were student books and mockups to check students' understanding after playing using webgame media. 3.1.4. utilize media and materials after drafting in stage 3, the next step was to arrange the design of learning activities. the tools needed included lesson plans, student worksheets, media (webgame learning media). before being used, the devices underwent series of expert tests such as media, materials, questions and learning experts. the following results of the expert test are presented in table 4. volume 8, no 2, september 2019, pp. 199-208 205 table 4. expert test results no indicator ideal score actual score ap category 1 learning materials 60 48 80% good 2 learning materials 70 52 72.4% good 3 question 55 41 74.5% good 4 learning 60 49 81.7% very good based on the media eligibility criteria that have been developed, the developed media fell into good and very good category (ap > 61%) thus the media were feasible to use. furthermore, through student responses, the media were seen practical. interviews were conducted with 10 students about what they thought after using webgame media in learning. the interview results can be seen in table 5. table 5. student response results no question score 1 what do you think about using web game media in learning? 10 2 are you interested in participating in learning using web game media? why? 10 3 by learning using web games, can you better understand the material easily? why? 7 4 what do you think about using web game media to solve problems related to flat, rectangular, and triangular material? 8 5 do you prefer to use web media games in learning? 10 total 45 percentage 90% based on the results of student responses after using the webgame media, 90% students stated that it was very good, thus it can be concluded that the webgame-based media were very helpful for the students to comprehend the two-dimensional figure material. 3.1.5. require learner participation the next step was to involve students in learning activities. students must be actively involved in learning to be able to see the effectiveness of the media that has been developed in order to achieve learning goals. learning activities were carried out in accordance with the designs that have been made before. webgame display that can improve problem solving capabilities can be seen in figure 2a, figure 2b, figure 2c, and figure 2d. figure 2a. example of game display at the stage of understanding problem. figure 2b. example of game display at the stage of planning completion wahyudi, ambarwati, & indarini, development of web game learning … 206 figure 2c. example of game display implementing completion figure 2d. examples of game display at the stage of look again. the effectiveness of the model can be seen from the results of the pretest compared with the results of the posttest. based on the paired samples test results, the results obtained show the sig. (2-tailed) equal to 0.000 or less than 0.05, it can be concluded that there are significant differences between learning outcomes. 3.1.6. evaluation and revise based on the results of expert testing and implementation of learning there were several things that must be revised and evaluated. in the media section, there were still a lot of typospunctuation, and unclear game instructions. the display of the call out was also too monotonous, in addition to learning devices such as lesson plans, problems in the selected problem based learning learning model were not shown. 3.2. discussion the percentage shown from the results of the evaluation by material experts was 80% and hence the media were considered ‘very good’ to use. the essence of learning mathematics using logic and thinking emphasized in the use of webgame learning media produced a level of product validity that is in accordance with the opinion (suhendri, 2011). then based on the results of evaluations by media experts, the percentage was 72.4%, which is considered ‘good’ to use. webgame-based learning media in twodimensional figure material has the same benefits as opinions (sanaky, 2009), which can attract student learning interest and can provide varied methods and media. then based on the results of the evaluation by the expert of questions, the percentage obtained was 74.5%, showing that it was actually ‘very good’ to use. finally, based on the results of evaluations by learning experts, the percentage of results was 81.7%, the score was concluded ‘very good’ to use. based on the results of interviews conducted by researchers, the percentage was 90%.the score that researchers gained was very good, thus it can be concluded that students were helped in understanding two-dimensional figure material and gained practicality when using webgame-based media. the pre-test results were 57.05% and after using the media, it changed into 83.9% based on post-test data. this has proven to be effective in improving problem solving skills in students. the pre-test and post-test data obtained were tested by paired samples t test with the results of the data showing the sig. volume 8, no 2, september 2019, pp. 199-208 207 (2-tailed) equals 0,000 or less than 0.05, thus there are significant differences between the learning outcomes in the pre-test and post-test data. based on the results of research that has been carried out web game learning materials to improve the ability to solve problems in two-dimensional figure material can improve problem solving abilities. this can be seen from students being happy when using instructional media to help them understand two-dimensional figure material. mathematics learning is a learning activity designed by teachers to develop students' creative thinking, and to help students improving their ability to build new knowledge as an effort to improve mastery of mathematical material (sutanto, 2016). at first the students' ability to understand the material of two-dimensional figure was still low, and students were also still confused to understand the material if two-dimensional figure. this is because students were bored with learning activities that werer fixated on books without any interesting activities. in addition, the lack of media used by the teacher when learning takes place, is also one of the reasons for the decline in student interest of learning. media-based learning activities also gave students the opportunity to learn math from the contextual problems in mission-solving activities. this made them challenged to continue to find solutions to problems in the mission. this was what gave students the opportunity to learn mathematics from solving problems in a structured and challenging activity. this was in accordance with the research results of darling-hammond (2008), where the teacher should always give students the opportunity to solve the problem in a structured and challenging way. however, there are still constraints in the use of media games. one of them is that some students have not been able to use this media games due to the lack of ict skills among them. they focuses on learning using the application so that problem solving skills are not maximally improved. in additionn, it takes a good internet network so that the web application can run well and hence the students can learn optimally and smoothly. it also needs teachers who master this application, so that they can guide students to study with this web application. 4. conclusion based on the results of the research and discussion that have been done, it can be concluded that the webgame-based learning media is valid, practically seen from the expert test and student responses after using webgame media and effective to be used in classroom learning activities seen from student learning outcomes that have improved before the pretest and after posttest. based on the results of the study, it is recommended that teachers can use creative media that utilize technology so that students become accustomed to using computers or laptops not only to play but also enrich their insights by visiting the webgmae link "adventure of risa". in addition, using media web games students can distinguish differences after using webgame-based learning media with learning that is usually done in class. the teacher can also compare which learning is better, without learning media or by utilizing learning media. acknowledgements we gratefully acknowledge the useful comments of dr. suryasatriya trihandaru, m.sc. and mozes kurniawan, m.pd lecturer in universitas kristen satya wacana. gratefully acknowledge for universitas satya wacana who has provided financial support and facilities for learning facilities in the classroom. wahyudi, ambarwati, & indarini, development of web game learning … 208 references bainbridge, w. s. 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https://journal.lppmunindra.ac.id/index.php/formatif/article/view/61 https://journal.lppmunindra.ac.id/index.php/formatif/article/view/61 https://journal.lppmunindra.ac.id/index.php/formatif/article/view/61 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/1121 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/1121 http://www.e-journal.stkipsiliwangi.ac.id/index.php/infinity/article/view/1121 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p31-40 31 students’ mathematical thinking skill viewed from curiosity through problembased learning model on integral calculus zetriuslita*, rezi ariawan universitas islam riau, indonesia article info abstract article history: received apr 25, 2020 revised sep 22, 2020 accepted sep 23, 2020 this study aims to find out the improvement of students’ mathematical critical thinking viewed from curiosity through teaching materials with problem based learning model. this is quasi-experimental research. the samples are, where the overall population consists of 75 samples divided into two classes. this research employed a simple random sampling technique. the instruments were the mathematical critical thinking skill test and curiosity questionnaire. the technique of data collection was carried out with test and non-test techniques. the data were analyzed through a two-way anova test. based on the analysis and interpretation of the research findings, it was found that: there was an improvement in students’ mathematical critical thinking skill from high curiosity by using teaching materials with problem based learning model; 1) there was an improvement from average curiosity through learning with integral calculus teaching materials; 2) there was an improvement from low curiosity; 3) there was no improvement from the level of curiosity (high, medium, low) through problem-based learning. 4) there was no influence between the level of curiosity and learning in improving students’ mathematical critical thinking skills. do more in-depth studies related to improving mathematical critical thinking skills by reviewing other affective aspects keywords: curiosity, integral calculus, mathematical critical thinking, problem-based learning, teaching material copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: zetriuslita, department of mathematics education, universitas islam riau, jl. kaharuddin nasution no.113, bukit raya, pekanbaru, riau 28284, indonesia. email: zetriuslita@edu.uir.ac.id how to cite: zetriuslita, z., & ariawan, r. (2021). students’ mathematical thinking skill viewed from curiosity through problem-based learning model on integral calculus. infinity, 10(1), 31-40. 1. introduction calculus is a course that nearly exists in every university, especially in mathematics and science majors. it is also one of the compulsory courses and skills for students majoring in science, especially in mathematics. they are expected to excel in this important subject. as a result, some researchers have carried out a research to evaluate and improve students’ ability in learning calculus (hartono & noto, 2017; lumbantoruan, 2017; mutakin, 2013; parma & saparwadi, 2015; ramdani, 2012; romadiastri, 2013; sopiany & rikayanti, 2018; usman et al., 2015). https://doi.org/10.22460/infinity.v10i1.p31-40 zetriuslita & ariawan, students’ mathematical thinking skill viewed from curiosity … 32 furthermore, the researcher has also conceived her own studies related to calculus, namely: an analysis of students’ mathematical critical thinking, connecting on integral calculus problem solving based on the students’ academic level, gender, cognitive style (ariawan & nufus, 2017; zetriuslita et al., 2016a; 2016b). practicality teaching material based on the problem based learning to improve mathematical critical thinking ability (zetriuslita & ariawan, 2016). based on the previous studies, the researcher focused on mathematical critical thinking ability. critical thinking ability is quite necessary in calculus. klurik and rudnick (zetriuslita et al., 2018) argued that critical thinking in mathematics involves testing, questioning, connecting, evaluating all aspects in any situations or problems. it doesn’t only demand knowledge and comprehension but also more than that. the given problem consists of analytical ability. ennis (2011) stated that critical thinking is a summary of logical thinking (logic) and reflective which emphasizes on decision making about what we believe or do. according to johnson (2007), it’s a focused and clear process which is employed in mental activities such as problem solving, decision making, persuasion, assumption analysis, and conducting scientific research. in line with the statement, ennis (2011) mentioned that the aspects of critical thinking are focus, reasons, inference, situation, clarity, and overview. from the statements above, it indicates that critical thinking can make and train someone to do math. students don’t learn mathematical rules or formulas by heart but learn mathematics by action and active discovery. consequently, critical thinking is very important to develop. in general, developing critical thinking or improving mathematics learning outcome can be achieved by using teaching materials or learning media (bien et al., 2019; dewi, 2016; guntur et al., 2017; hikmawati et al., 2013; melisa, 2014; putra et al., 2017; saparwadi & yuwono, 2019). based on the researcher’s teaching experience, there was a problem related to the lack of determination and curiosity. it was seen from the students’ unwillingness to find out the answers to the given problems, especially in calculus integral subject. they didn’t try to ask the teacher about the problem they didn’t understand. this level of curiosity can affect the students’ activity in learning and also lead to their comprehension. carin (1997) defined curiosity as one’s willingness and need to obtain answers from a question or things that spark curiosity. curiosity can foster internal motivation for learning and understanding something, therefore it can be developed in the learning process (ameliah & munawaroh, 2016; mardhiyana, 2017; nurkamilah, 2017; solehuzain & dwidayanti, 2017). in accordance with the findings, the researcher did some studies in regards to students’ mathematical curiosity (zetriuslita et al., 2017). from the review of related literature and identification of the problem above, students’ curiosity is a cognitive aspect that seeks attention and serious consideration. subsequently, the researcher’s interest focuses on improving students’ mathematical critical thinking ability viewed from curiosity. the researcher applied teaching materials based on problem based learning. the researcher suggested the following hypotheses, such as: (1) there is a significant difference in terms of high curiosity between students who use calculus integral teaching materials based on problem-based learning model and those who follow conventional learning; (2) there is a significant difference in terms of medium curiosity between students who use calculus integral teaching materials based on problem-based learning model and those who follow conventional learning; (3) there is a significant difference in terms of low curiosity between students who use calculus integral teaching materials based on problembased learning model and those who follow conventional learning; (4) there is a significant difference in terms of curiosity (high, medium, low) between students who used calculus integral teaching materials based on problembased learning model and those who follow volume 10, no 1, february 2021, pp. 31-40 33 conventional learning; and (5) there is an interaction effect between level of curiosity and learning towards the improvement of mathematical critical thinking ability. 2. method this study is a quantitative research. quantitative research is the process of using data in numerical form as a means to find out knowledge (creswell & creswell, 2017; sugiyono, 2011). the type of research is quasi-experimental research with pre-test and posttest non-equivalent group design described in table 1. table 1. pre-test and post-test non-equivalent group design (cohen et al., 2007) pre-test treatment posttest experiment o1 x o2 control o1 o2 description: o1 : pretest of experiment classes and control classes o2 : postest of experiment classes and control classes x : treatment using integral calculus-based teaching materials problem based learning : treatment with does not use teaching materials problem-based integral calculus the samples of this research were the 2nd semester students of mathematics education fkip uir who took integral calculus subject. there were two classes. saturated sampling was used because the samples were selected from the overall population by using simple random sampling technique. the lottery showed that 2a was chosen as the control class (conventional learning) and 2b as the experiment class (using calculus integral teaching materials with problem-based learning model). this research was conducted in the even semester of academic year 2018/2019. the instruments of this research consisted of mathematical critical thinking ability worksheet and curiosity questionnaire. the data collection was carried out by using both instruments for test and non-test technique respectively. the test about critical thinking ability was given in the first and last meeting, while the questionnaire of curiosity was distributed in the first meeting only. the data obtained from the test and questionnaire was analyzed by using descriptive statistics and inferential statistics with spss version 22 software. in order to determine the level of curiosity, the researcher used the following interval in table 2. table 2. interval group level of curiosity (zetriuslita et al., 2016b) group level of curiosity interval high 𝑥 ≥ �̅� + 𝜎 medium �̅� + 𝜎 < 𝑥 < �̅� + 𝜎 low 𝑥 ≤ �̅� − 𝜎 description: x = students’ curiosity score �̅� = mean score of students’ curiosity σ = standard deviation of students’ curiosity score zetriuslita & ariawan, students’ mathematical thinking skill viewed from curiosity … 34 meanwhile, the improvement of mathematical ability intended is normalized gain obtained from the pre-test and post-test results, with the following formula by meltzer (2002). 𝑁𝐺𝑎𝑖𝑛 = 𝑃𝑜𝑠𝑡𝑒𝑠𝑡 − 𝑃𝑟𝑒𝑡𝑒𝑠𝑡 𝐼𝑑𝑒𝑎𝑙 𝑆𝑐𝑜𝑟𝑒 − 𝑃𝑟𝑒𝑡𝑒𝑠𝑡 in order to the test hypothesis of the research, t-test, one-way anova test and two-way anova test were used. 3. results and discussion 3.1. results the data analysis of improving mathematical critical thinking ability was obtained from n-gain result. it was classified into the level of curiosity, starting from grouping the students based on their curiosity level. from the result of curiosity score, the data were obtained as follows: the data of improving mathematical critical thinking ability viewed from curiosity is presented in table 3. table 3. description of students’ total in terms of curiosity level of curiosity learning pmbakipbl ptmbakipbl high 5 (13.16%) 5 (13.5%) medium 26 (68.42%) 27 (72.97%) low 7 (18.42%) 5 (13.51%) total 38 (100%) 37 (100%) description: pmbakipbl : learning using integral calculus teaching materials based on problembased learning ptmbakipbl : learning does not use integral calculus teaching materials based on problem-based learning table 3 shown that in experimental class is better than control class. students in the experimental class and the control class with curiosity were not significantly different from the level of curiosity. this indicates that the curiosity of the experimental class and control class students is almost the same. basically, if the curiosity is not much different, then the resulting mathematics learning outcomes should not be different either. however, with different treatment with the use of problem-based learning materials, the learning outcomes can be presented in the following table. by using spss version22, the calculation of hypotheses 1, 2, and 3 are presented in the table 4. volume 10, no 1, february 2021, pp. 31-40 35 table 4. the improvement of mathematical critical thinking level of curiosity statistics (t-test) conclusion high sig. (2-tailed) = 0.000 < 0,05 hypothesis is accepted medium sig. (2-tailed) = 0.000 < 0.05 hypothesis is accepted low sig. (2-tailed) = 0.000 ≤ 0.05 hypothesis is accepted table 4 show that on average the improvement of students’ mathematical critical thinking ability from high level of curiosity, medium and low who followed problem based learning is better than studentswho follow conventional learning. then, hypothesis was tested by using one-way anova test (table 5). table 5. one-anova test of improvement critical thinking based on curiosity sum of squares df mean square f sig. between groups 0.003 2 0.002 0.50 0.952 within groups 2.287 72 0.032 total 2.290 74 from table 5, it was obtained the sig = 0,952 ≥ α, with α= 0,05. it means h0 is accepted and h1 is rejected. when h1 is rejected, the hypothesis “there is a significant difference of improving mathematical critical thinking ability between students from curiosity level who follow problem based learning and conventional learning” is rejected. in other words, there is no significant difference of improving the students’ critical thinking ability viewed from level of curiosity. next, hypothesis 5 was tested by using two-way anova test with the spss version 22 software. the result was obtained as follows in table 6. table 6. two-way anova test source type iii sum of square df mean square f sig. corrected model 1.271a 5 0.254 17.204 0.000 intercept 9.226 1 9.226 624.620 0.000 curiosity 0.001 2 0.000 0.021 0.980 kelas 0.949 1 0.949 64.239 0.000 curiosity*kelas 0.067 2 0.033 2.263 0.112 error 1.019 69 0.015 total 18.307 75 corrected total 2.290 74 table 6 show that sig. = 0.112 > 0.05, where hypothesis h1 is rejected, meaning there is no interaction effect between level of curiosity and learning towards the improvement of mathematical critical thinking ability. the curiosity has no significant effect on the improvement, it’s indicated with sig. = 0.980 > 0.05 (table 6), meanwhile the class or in this case learning has a significant effect on the improvement shown by sig = 0.000 < 0.05. zetriuslita & ariawan, students’ mathematical thinking skill viewed from curiosity … 36 3.2. discussion the result of inferential statistics show that the improvement of mathematical critical thinking ability for the students who follow problem based learning in integral calculus is higher than those who follow conventional learning. it means that the use of integral calculus teaching materials with problem based learning is effective in improving students’ mathematical critical learning ability. this is in line with a research finding by guntur et al. (2017) which stated that the effect between intensively, rarely, and never using comics towards students’ selflearning. dewi (2016) said that the average rank of experimental group is higher than that of control group. in other words, students’ learning outcome in learning with screencast-o-matic media is better than students who follow conventional learning. so this research reinforce the assumption that using screencasto-matic is effective to increase students’ learning outcome in integral calculus subject. hikmawati et al. (2013) stated that there was an influence of the use of instructional media and cognitive styles on mathematics learning outcomes of grade viii madrasah tsanawiyah students and there was no interaction between the use of instructional media and cognitive style on mathematics learning outcomes of grade viii madrasah tsanawiyah students. bien et al. (2019) stated that the use of teaching calculus textbooks was effective in increasing the ability to understand student concepts. farhan and retnawati (2014) stated that pbl can make students more active when they are working on worksheets based on pbl made by teachers. gordah and fadillah (2014) stated that increasing students' mathematical representation ability through the use of differential calculus teaching materials based on the open ended approach was classified as moderate, there was no difference in increasing students' mathematical representation ability in terms of gender (male and female); 3) there is a difference in the improvement of students 'mathematical representation ability in terms of the initial level of students' ability (upper, middle, and lower). meanwhile, if viewed based on the level of curiosity, thenthere is no difference in the increase in mathematical critical thinking ability reviewed based on curiosity (high, medium, low) between students who learn by using integral calculus based on problem based learning with those not. furthermore, there is also no interaction effect between learning and curiosity on the improvement of mathematical critical thinking ability. there are several analyzes that researchers can convey why there are no differences in enhancement and interaction, including: (1) the questionnaire researchers do before curiosity study or research is carried out, with the aim only to determine the condition of student curiosity. even though if the questionnaire was given after they learned by using integral calculus based learning materials based on problem based learning, maybe their curiosity would be better; (2) the structure of teaching materials is designed to improve critical thinking ability, so that attention to students' curiosity lacks attention. the results of this study are in line with research conducted by zetriuslita et al. (2017) which states that after being given treatment, mathematical curiosity of students does not/ experience a significant increase. this is due in the learning process, lecturers are not maximized in developing students' mathematical curiosity, and other factors are the lack of maximum learning tools for student worksheets or lkm. furthermore solehuzain and dwidayati (2017) stated that there was a significant influence between curiosity on students' mathematical creative thinking abilities on the problem based learning model of learning with open-ended problems. the variable of curiosity affects the variable of mathematical creative thinking ability by 77.4% and the rest is influenced by other factors. volume 10, no 1, february 2021, pp. 31-40 37 4. conclusion the conclusion of this research are: (1) improvement of mathematical critical thinking ability of students who learn to use integral calculus based problem based learning is better than improvement in mathematical critical thinking ability of students who learn not to use integral based learning calculus based on problem based learning both in terms of high, medium, high curiosity levels low; (2) there is no increase in mathematical critical thinking ability of students who learn to use integral based learning calculus based on problem based learning compared to the students who learn not to use integral based learning calculus based on problem based learning based on curiosity (high, medium, low ); (3) there is no effect of interaction between levels of curiosity and learning on improving students' mathematical critical thinking ability. there are a number of things that researchers can advise on the results of this study, namely: it is recommended that a more in-depth study is conducted related to improving mathematical critical thinking skills by reviewing other affective aspects, a study in the form of further development of calculus root material which is not only based on active learning models, but by using the help of mathematical software such as matlab, geogebra and others. studies are more extended, not only focus on integral calculus, but focus on the study of calculus (differential calculus, and calculus of many variables or advanced calculus). future studies must use a comprehensive research method in the form of a mix method, which combines data obtained qualitatively with data obtained quantitatively. acknowledgments the author would like to thank universitas islam riau for the support in funding and the research team for their commitment in this research. references ameliah, i. h. 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(2018). association among mathematical critical thinking skill, communication, and curiosity attitude as the impact of https://doi.org/10.21580/phen.2013.3.1.179 https://doi.org/10.22460/infinity.v5i1.p56-66 https://doi.org/10.5539/ies.v10n7p65 zetriuslita & ariawan, students’ mathematical thinking skill viewed from curiosity … 40 problem-based learning and cognitive conflict strategy (pblccs) in number theory course. infinity journal, 7(1), 15-24. https://doi.org/10.22460/infinity.v7i1.p15-24 https://doi.org/10.22460/infinity.v7i1.p15-24 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p87-98 87 design of learning materials on circle based on mathematical communication cita dwi rosita 1 , tri nopriana *2 , isna silvia 3 1,2,3 universitas swadaya gunung djati article info abstract article history: received dec 15, 2018 revised jan 26, 2019 accepted jan 28, 2019 mathematical communication skills is an important role in mathematics learning. however, the importance of mathematical communication skills has not been fully realized in learning, especially circle material. design a learning material based on mathematical communication is one way to develop this ability. the preliminary study produced the findings of an epistemological learning obstacle so that students' mathematical communication skills were still in the low category. this study aimed to analyze learning obstacle and designing learning materials based on the material mathematical communication circle. this study is design research that contained two stages of didactical design research (ddr), didactic situation and metapedadidactional stage. research result obtained are students difficulties in relating the material defining elements of the circle with their own language, identifying circle elements that were known and explaining through pictures, calculating the circumference and distance of circular objects based on problems, calculating the surface area of circular objects based on the problems, and rearrange the formula which states the relationship of circle elements. to solve that learning obstacle, we recommend some learning trajectory in a circle that useful for teachers. this design of learning material is valid and practical to implement in the classroom. keywords: didactical design research circle learning obstacle mathematical communication copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: tri nopriana, departement of mathematics education, universitas swadaya gunung djati, jl. pemuda no.32, cirebon, indonesia email: trinopriana@unswagati.ac.id how to cite: rosita, c. d., nopriana, t., & silvia, i. (2019). design of learning materials on circle based on mathematical communications. infinity, 8(1), 87-98. 1. introduction mathematical communication skills are one of five abilities recommended by the national council of teachers of mathematics (hiebert, 2003). this is in line with baroody's (1993), there are two important reasons why mathematical communication is one of the focuses on mathematics learning. first, mathematics is basically a language for mathematics itself. second, learning and teaching mathematics are social activities that involve at least two parties, namely teachers and students. in addition, it was also strengthened by minister of education and culture regulation no.58 at 2014 which states mailto:trinopriana@unswagati.ac.id rosita, nopriana, & silvia, design of learning materials on circle … 88 that the purpose of mathematics learning is to communicate ideas, reasoning, and be able to compile mathematical evidence by using complete sentences, symbols, tables, diagrams or other media to clarify the situation or problem. according to greenes & schulman (1996), mathematical communication is (1) a central force for students in formulating concepts and strategies; (2) capital of success for students towards approaches and solutions in mathematical exploration and investigation; (3) a forum for students to communicate with their friends to obtain information, share thoughts and discoveries, brainstorm, assess and sharpen ideas to convince others. based on the explanation, it can be said that mathematical communication skills are mathematical abilities that must be realized in mathematics learning. geometry is a branch of mathematics in the scope of learning material in schools. in our environtment, geometry can be found on objects in the form of spheres, tubes, boxes, lines, circles and so on. according to bobango (abdussakir, 2012) states that the purpose of geometry learning is students gain confidence about their mathematical abilities, become good problem solvers, can communicate mathematically, and can reason mathematically. in line with that, walle & jhon (2001) suggested the reason for the importance of studying geometry including geometry being able to provide more complete knowledge about the world and play an important role in learning other concepts in mathematics learning. based on the description above, it can be concluded that geometry is very important to learn, to foster the ability, and to communicate a mathematical idea. in connection with that, one of the materials in the scope of geometry in the eighth grade of junior high school that needs attention is the topic of the circle. the basic competencies that must be achieved by the eighth grade students on the topic of circle are students required to be able to express a description of the circle material that has been studied using their own language, explain the concept of broad and circumferential circles with real objects or images, connect a real situation about the circle into a mathematical model, and formulate definitions and generalizations of the relation of the center angle, bow length, and the area of the circle’s section. however, the importance of mathematical communication skills has not been fully realized in mathematics learning. based on horizon research center, inc. (tiffani, surya, panjaitan, & syahputra, 2017) suggests that at the junior high school level, student achievement related to problem-solving abilities, reasoning abilities, and communication skills in helping students to think mathematically is only 30%. this is in line also as expressed by munawaroh, rohaeti, & aripin (2018) the percentage of student error in solving the problem of mathematical communication is equal to 38% at inappropriate data category, 34% at inappropriate procedure category, and 26% at omitted data category. based on some of these explanations, it shows that mathematical communication skills have not been realized in the learning of mathematics, including in circle material. one of the causes is due to the learning barriers experienced by students. students find it difficult to explain a mathematical idea related to circles with pictures, construct conjectures, and relevant generalizations of the circle’s concepts. learning barriers here are learning barriers commonly known as the epistemological learning obstacle sulistiawati, suryadi, & fatimah (2015) explained the cause of the epistemological learning obstacle is due to a lack of conformity of teaching materials presented to students with the condition of students in learning mathematics material. one way that is thought to be able to overcome the epistemological learning obstacle is by designing a learning materials. this is in line with rosita (2016), who argued that educators should provide and develop learning materials that are in accordance with the characteristics and social environment of students. in this study, the learning materials that will be designed are printed learning materials in the form of modules. learning materials that are designed in the form of volume 8, no 1, february 2019, pp. 87-98 89 modules can help students to understand mathematical concepts independently with minimal assistance from teachers, because the material is arranged based on student’s learning obstacle. the design of learning materials is thought to be a solution for building students' mathematical communication skills in mathematics learning. the research that supports this solution is the research conducted by rosita, nopriana, & dewi (2017), arguing that a learning material design can complete the ability of mathematical understanding both classical and individual. noto, pramuditya, & fiqri (2018) stated that the limit learning material for algebraic functions based on mathematical understanding can minimize learning obstacle after being implemented. the previous research shows that students' mathematical abilities can be built with the didactical design in the form of learning materials arranged by the teacher. therefore, the purpose of this study is to design circle learning materials that develop mathematical communication skills and learning trajectories to understand circle material. 2. method this study uses a qualitative method in the form of didactical design research. didactical design research (ddr) which consists of three phases, namely the didactic situation analysis before learning, metapedadidactic analysis, and retrospective analysis (suryadi, 2013). the stages carried out in this research are the didactic situation analysis stage before learning and the metapedadidactic analysis phase. didactical situation analysis phase before learning in the form of extracting information about learning obstacle material circles and formulating anticipatory didactical pedagogical (adp) as a framework for learning materials. meanwhile, the metapedadidactic phase is the implementation of learning materials and the preparation of learning trajectories related to circle material. subjects in this study were 47 students of majalengka 6th junior high school which was divided into two groups, namely the subject for extracting information about learning obstacle, circle material and subject for the implementation of learning materials. information gathering about learning obstacle was carried out for grade ix students who had already received circle material and the implementation of learning materials was done for class viii students who had not studied circle material. data collection techniques used in this study are triangulation techniques. the combination of techniques used in this study is tests, interviews, and questionnaires. the instrument tests used was a mathematical communication ability test questions for learning circle material learning obstacle information, interviews are used to track data that is not obtained through tests. module validation questionnaire, and module practical questionnaire for teachers and students given to test the validity and practicality of modules that have been made used. 3. results and discussion 3.1. learning obstacle and didactical situation related to circle material the stage of didactical situation analysis before learning are students’ learning obstacle analysis on circle related to their mathematical communication. grouped learning obstacle then find a solution to overcome the learning obstacle. the solution is to design learning materials based on learning obstacle related to the material in the eighth grade of junior high school which will be included in the learning materials using the following rosita, nopriana, & silvia, design of learning materials on circle … 90 didactical situations. the results of the analysis of the learning obstacle and didactical situation were explained at table 1, table 2, and table 3. table 1. learning obstacle related to circle element number didactical situation didactic anticipation 1 a student divides the area in a circle using 6 bowstrings. how many areas can the student make? draw the circle area in question! a student divides the area in a circle using 6 bowstrings. how many areas can the student make? draw the circle area in question like the description of the image shown below. the maximum area of 1 bowstring is 2 area. 2 in the picture below, the point o is the center point of the circle, points c, d, e, f, g, i and j lie on the circle and the point h lies inside the circle. cd and ij line segments are circle diameters, ef line segments and gh are not circular diameters. based on the description, write in your own language what is meant by the center and diameter of the circle! observe the following picture. the picture below is a picture of a bullet-shaped game forming a circle based on the description, write in your own language what is meant by the center and diameter of the circle! in question number 1, students still have difficulty in describing the ideas specified in the problem because they do not understand well about the concept of bowstring and areas that can be drawn and model images that are suitable for getting relevant solutions. in problem number 2, students still have difficulty in writing down the definitions of elements that have been identified by their own language. the way to overcome this learning obstacle is to provide a stimulus in the form of questions/statements true/false to students so that students can be directed to define themselves about the elements of the circle. the existence of a "definition column" in the module to provide opportunities for students to redefine the elements of the circle that have been learned using their own language and discuss with friends to share with each other the definitions that have been prepared by each student. in addition, a "creation column" is also provided in the module for students to train students in communicating various possibilities from an explanation of an idea and provide opportunities for students to present their ideas, as well as the teacher's direction to determine the relevant solution of a circle problem through discussion small group. volume 8, no 1, february 2019, pp. 87-98 91 table 2. learning obstacle related to circumference and area of the circle number didactical situation didactic anticipation 3 a bicycle wheel has a radius of 21 cm. when the bicycle is paddled, the wheel rotates 50 times. determine the circumference and distance traveled by the bicycle wheel! a bicycle wheel has a radius of 21 cm. when the bicycle is paddled, the wheel rotates 50 times. determine the distance traveled by the bicycle wheel using the circumfeence formula relationship! 4 a pool is known as a circle. if the distance of a pole that is right in the middle of the pond to the edge of the pool is 14 meters, determine the surface area of the circle-shaped pond! a pool is known as a circle. if the distance of a pole that is right in the middle of the pond to the edge of the pool is 14 meters as illustrated in the sketch below. determine the area of the pool based on the situation! in problem number 3 above, students still have difficulty in connecting the real situation to calculate the distance traveled by the wheel based on the circumference that has been calculated previously. so there are some students who are only able to answer around the wheel without the distance from the wheel based on the given situation. in problem number 4, students still have difficulty in identifying the known elements in real situations, namely the error in setting the radius in diameter based on the real situation given. the way to overcome these various learning obstacles is to remind them of the concept of distance and circumference in daily life through question and answer. the teacher also gives direction to students to link concepts that are known to the real situation through questions and answers and a series of materials in the module. the existence of giving some pictures of the real form of the elements of the circle in the module. the teacher also provides a variety of real situations in the form of drill exercises on a series of module materials. table 3. learning obstacle related to determine the relationship of angle at circle number didactical situation didactic anticipation 5 shown ab=15 cm. calculate the arc length by completing the following fields. ... dan ... , then the cd arc length can be calculated using the shown aob=〖30〗^°, cod=〖120〗^°, ab=15 cm. calculate the arc length by completing the following fields. ... dan ... , then the cd arc length can be calculated using the arc length relationship with ................., namely as follows. rosita, nopriana, & silvia, design of learning materials on circle … 92 number didactical situation didactic anticipation arc length relationship with ................., namely as follows. so that, . . . is . . . so, the relationship between the central angle and the arc length can generally be written as sehingga, is . . . cm so, the relationship between the central angle and the arc length can generally be written as in problem above, students still have difficulty in compiling arguments to determine the relationship of the center angle with the arc length, and determine the generalization of the relationship of the center angle and the length of the circle arc based on the steps of the previous settlement. the way to overcome these various learning obstacles is to provide guidance to students to develop generalizations from the concept of a circle through discovery guided by the "let's find" activity and help students to establish relevant generalizations through discussions between students regarding the results of the generalization that has been done. 3.2. result validation of learning materials on circle based on mathematical communication one way to find out whether a module that is designed to be feasible or not to use is to do a validation test by the validator. the indicators used in this module validation cover several aspects, namely aspects of relevance, adequacy, completeness of presentation, systematics of presentation, student-centered orientation, linguistics, bruner's learning theory, and mathematical communication. below is presented the validation results of the module by the validator in table 4. table 4. result of validation expert validator percentage validation criteria validation validator 1 96 % very valid validator 2 78 % valid enough validator 3 90 % very valid validator 4 93 % very valid average 89 % very valid based on the results of the validation of the four experts, the average percentage of 89% with the validation level is very valid so the circle module based on mathematical communication is appropriate to be used in mathematics learning with small revisions. the volume 8, no 1, february 2019, pp. 87-98 93 suggestions for improvement of the validator for the module are described in the following explanation. validator 1, the criteria for learning materials are very valid and provide suggestions for presenting more creative questions in the module. validator 2, it is obtained that the criteria for learning materials are quite valid and provide suggestions for multiplying practice questions per sub-chapter, presenting real pictures related to the active phase, and completing indicators of mathematical communication skills in all exercises and questions in the module. validator 3, the criteria for learning materials are very valid and provide suggestions to correct some errors in the use of words in the module. furthermore, validator 4, the criteria for learning materials are very valid and provide suggestions for adding non-routine questions to sample questions and exercises and updating illustrations that are not clear. 3.3. result validation of learning materials on circle based on mathematical communication after the module is revised based on advice from experts, then the circle module based on mathematical communication is tested for practicality. aspects used in the module practicality test include aspects of ease, aspects of time efficiency, and aspects of benefits. the module practicality test was carried out by a junior high school mathematics teacher and 15 students from majalengka 6 state middle school. the following module practicality results are presented in table 5. table 5. result of validation expert subject percentage practicality criteria practicality average practicality by teachers 93 % very practical average practicality by students 89 % very practical based on the calculation of the percentage of module practicality test by the teacher and students, the average percentage is 93% and 89% so that the practicality of the module is very practical. calculation of the percentage of the practicality of learning materials for teachers in each aspect and obtained a result of 95% in terms of convenience, 100% in the aspect of time efficiency, and 83% in aspects of benefits. thus, facilitation aspects are very practical criteria, time efficiency aspects are very practical criteria and aspects of benefits are very practical criteria. while the calculation of the percentage of practicality for students in each aspect and obtained a result of 91% in the convenience aspect, 82% in the aspect of time efficiency, and 88% in aspects of benefits. thus, facilitation aspects are very practical criteria, time efficiency aspects are very practical criteria, and benefits aspects are very practical criteria. 3.4. learning trajectory of circle material based on the analysis of learning obstacle in the circle material, the learning trajectory design (learning trajectory) needs to be done as a reference in learning circle material. trajectory learning is a certain learning path that is facilitated through a series of learning activities according to students' abilities. the preparation of learning trajectory has the potential to bring up an alternative presentation of learning materials that are more in line with the needs of students because they have considered the various thinking processes of students (dedy & sumiaty, 2017). through this approach, the didactic concept of the circle concept is arranged in the hope that students can understand the concept as a whole in order to minimize the learning obstacle that occurs when students go through the learning trajectory. in the preparation of the learning trajectory, there is the term main rosita, nopriana, & silvia, design of learning materials on circle … 94 concept and pressure point. the main concept shows a common thread of concepts in circle material. while the pressure point is an important point of the main concept that makes the flow of learning flow or does not cause jumping thinking. this is so that the learning material presented is in line with a series of didactic situations, namely the pattern of student-material relations through the help of teacher presentations developed on students' learning trajectory. trajectory learning of circle material is grouped into three types, namely learning trajectory material of circle elements, learning trajectory of material around an area of the circle, and learning trajectory of the material relation between center angle, arc length, and circle area of the circle which are presented in the following figure. figure 1. learning trajectory related circle element learning trajectory to understand the elements of a circle begins by giving students the opportunity to observe the surrounding environment related to the shape of a circle. next, students express their own language about the definition of a circle in a section accompanied by several reasons for their observations. when students are able to define themselves appropriately, students identify similarities and differences from circular and non-circular objects so that circle characteristics and circle elements can be found. students learn to find forms of relationships between elements of circles based on their own observations. volume 8, no 1, february 2019, pp. 87-98 95 figure 2. learning trajectory related circumference and area of circle in this learning trajectory the process of understanding students towards the concept of circumference and the area of a circle is built through discovery activities that are associated with the concept of comparative values and the area of rectangles. figure 3. learning trajectory related relation of center angle, bow length, rosita, nopriana, & silvia, design of learning materials on circle … 96 the student learning trajectory in understanding the concept of the relationship between central angles, arc length, and circle circle area is carried out through the process of discovery by involving students' understanding of the concept of comparable worth as prerequisite material. the three learning trajectories that are built in understanding students in circle material are based on constructivism and brunner learning theories. the effort was made to create a learning community for students through the process of rediscovering. based on the results of the study of johar, patahuddin, & widjaja (2017) that contextual problems may be used to examine students' interest in problems and motivating students to work on it. this is in line with the opinion of hwang et al. (2007) that learning activities in which there are problem solving, discussing each other and providing responses among fellow students, can improve the ability to express mathematical ideas with various forms of representation. linking the prerequisite material to the concepts students will learn is in line with hung (1997) who states that in learning, students need to be motivated and guided by instructors to construct their own ideas, concepts, and understandings based on the prior knowledge they already have. martin, et al. (2005) emphasize that a set of mathematical understandings is an understanding obtained when a group of students work together in completing mathematical tasks. development of knowledge through observation and discovery is intended to add to the learning experience of students. jonassen (2010) explains, one of the factors that influence students' ability to solve problems is prior experience. the prior experience role for a problem solver as a basis for interpreting the problem, provides signs about what should be avoided and predicts the consequences of decisions or actions taken. this is in line with the meaning of full learning according to von glasersfeld & steffe (1991) that meaningful learning will not be realized only by listening to lectures from the instructor, meaningful learning experiences for students can be given one of them through teaching assignments that are more oriented to students' thinking abilities, and students who also do the thinking process. 4. conclusion based on the results of the analysis that has been done, it can be concluded that at the stage of didactic situation analysis before learning produces a learning obstacle experienced by students is an epistemological type of learning obstacle. the learning obstacle that occurs in the circle material is the related learning obstacle, defining the elements of the circle with their own language. based on the results of the validation of the four experts, the validation level is very valid so the circle module based on mathematical communication is appropriate to be used in mathematics learning with small revisions. meanwhile, at the metapedadidactic stage, the results of the module practicality test obtained by the teacher and students obtained very practical module criteria, so that the module is easy to use, useful, and the time used becomes more efficient in mathematics learning. researchers also compiled learning trajectories related to circle material including learning trajectory material circle elements, learning trajectory material around and wide circle, as well as learning trajectory material relationship between center angle, arc length, and circle circumference area. this trajectory learning can be used as a reference in studying circle material to facilitate the diversity of student learning trajectories. volume 8, no 1, february 2019, pp. 87-98 97 references abdussakir, a. (2012). pembelajaran geometri sesuai teori van hiele. madrasah: jurnal pendidikan dan pembelajaran dasar, 2(1). baroody, a. j., & coslick, r. t. (1993). problem solving, reasoning, and communicating, k-8: helping children think mathematically. prentice hall. dedy, e., & sumiaty, e. (2017). desain didaktis bahan ajar matematika smp berbasis learning obstacle dan learning trajectory. jrpm (jurnal review pembelajaran matematika), 2(1), 69-80. greenes, c., & schulman, l. (1996). communication processes in mathematical explorations and investigations. pc elliott and mj kenney (eds.). hiebert, j. (2003). what research says about the nctm standards. a research companion to principles and standards for school mathematics, 5-23. hung, d. w. l. (1997). meanings, contexts, and mathematical thinking: the meaningcontext model. the journal of mathematical behavior, 16(4), 311-324. hwang, w. y., chen, n. s., dung, j. j., & yang, y. l. (2007). multiple representation skills and creativity effects on mathematical problem solving using a multimedia whiteboard system. journal of educational technology & society, 10(2), 191-212. johar, r., patahuddin, s. m., & widjaja, w. (2017). linking pre-service teachers’ questioning and students’ strategies in solving contextual problems: a case study in indonesia and the netherlands. the mathematics enthusiast, 14(1), 101-128. jonassen, d. h. (2010). learning to solve problems: a handbook for designing problemsolving learning environments. routledge. martin, t. s., mccrone, s. m. s., bower, m. l. w., & dindyal, j. (2005). the interplay of teacher and student actions in the teaching and learning of geometric proof. educational studies in mathematics, 60(1), 95-124. munawaroh, n., rohaeti, e. e., & aripin, u. (2018). analisis kesalahan siswa berdasarkan kategori kesalahan menurut watson dalam menyelesaikan soal komunikasi matematis siwa smp. jpmi (jurnal pembelajaran matematika inovatif), 1(5), 993-1004. noto, m. s., pramuditya, s. a., & fiqri, y. m. (2018). design of learning materials on limit function based mathematical understanding. infinity journal, 7(1), 61-68. rosita, c. d. (2016). the development of courseware based on mathematical representations and arguments in number theory courses. infinity journal, 5(2), 131-140. rosita, c. d., nopriana, t., & dewi, i. l. k. (2017). bahan ajar aljabar linear berbasis kemampuan pemahaman matematis. unnes journal of mathematics education research, 6(2), 266-272. sulistiawati, s., suryadi, d., & fatimah, s. (2015). desain didaktis penalaran matematis untuk mengatasi kesulitan belajar siswa smp pada luas dan volume limas. kreano, jurnal matematika kreatif-inovatif, 6(2), 135-146. tiffani, f., surya, e., panjaitan, a., & syahputra, e. (2017). analysis mathematical communication skills student at the grade ix junior high school. ijariie-issn (o)-2395-4396, 3. rosita, nopriana, & silvia, design of learning materials on circle … 98 suryadi, d. (2013). didactical design research (ddr) dalam pengembangan pembelajaran matematika. in prosiding seminar nasional matematika dan pendidikan matematika, 3-12. von glasersfeld, e., & steffe, l. p. (1991). conceptual models in educational research and practice. the journal of educational thought (jet)/revue de la pensée educative, 91-103. walle, j. a., & jhon, a. (2001). geometric thinking and geometric concepts. elementary and middle school. mathematics: teaching developmentally, 4th ed. boston: allyn and bacon. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p117-128 117 enhancement of students’ mathematical communication under metacognitive scaffolding approach sufyani prabawanto universitas pendidikan indonesia article info abstract article history: received july 15, 2019 revised july 30, 2019 accepted sept 27, 2019 this research aims to investigate the enhancement of students‟ mathematical communication under metacognitive scaffolding approach. this research used a quasi-experimental design with pretest-posttest control. the subjects were pre-service elementary school teachers in bandung. in this study, there were two groups of subjects: experimental and control groups. the experimental group consists of 60 students under metacognitive scaffolding approach, while the control group consists of 58 students under direct approach. based on the prior mathematical ability, the students were classified into three levels, namely high, midlle, and low. data collection instrument used mathematical communication test. the conclusions of the research are: (1) there is a significant difference in enhancing mathematical communication ability between students who attended the course under metacognitive scaffolding approach and those under direct approach, and (2) there was no significant interaction effect between teaching approaches and ability levels based on prior knowledge in enhancing students‟ mathematical communication. keywords: mathematical communication, metacognitive approach. copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: sufyani prabawanto, department of mathematics education, universitas pendidikan indonesia, jl. dr. setiabudi no.229, bandung, west java 40154, indonesia email: sufyani@upi.edu how to cite: prabawanto, s. (2019). enhancement of students‟ mathematical communication under metacognitive scaffolding approach. infinity, 8(2), 117-128. 1. introduction mathematical communication is an important part of learning mathematics. it is explicitly stated in regulation of national education minister of republic of indonesia number 22 (departemen pendidikan nasional ri, 2006). however, some of the survey results (mullis et al., 2004, oecd, 2005, 2007, 2010) showed that indonesian students' mathematical communication ability were quite low compared to some other countries. if it was seen from the content of mathematics in the surveys, a lot of the topics had been studied by the students while they were in elementary school. thus the students' lack of mathematical communication was related to lack of their mathematical communication in elementary school. sufyani, enhancement of students’ mathematical communication … 118 elementary students' mathematical ability related to the teachers‟ ability. this linkage revealed in study reports. in their study report, hill, rowan, & ball (2005) stated that the mathematical ability of elementary school teachers was significantly related to students' mathematical achievement. passos (2009) reported that there was a relationship between elementary school teacher competence and student achievement in reading and math. therefore, it could be said that the development of mathematical communication of preservice elementary school teachers is very important. one effort to develop their mathematical communication ability was looking for factors expected to enhance their mathematical communication ability. one of the factors was a teaching approach. carpenter & gorg (2000) recommended an approach, namely metacognitive that includes strategy, planning, monitoring, and evaluation during the learning process. although the approach had been recommended by experts, this approach still has drawbacks, such as when a student realized that he/she did not find a way to solved mathematical problems, he/she would pause in his own confusion. yee (2002) reported that the metacognitive approach (without scaffolding) was not able to raise students' success in learning mathematics. instead, peters (2011) revealed that if the scaffolding approach run itself (not involving metacognitive) the students were weak in developing their own ways to solve problems. to overcome this, metacognitive approach needs to be combined with scaffolding. for learning in the classroom that involves a lot of students, usually more than 30 people, metacognitive scaffolding approach was almost impossible to be implemented. therefore, this approach needed to be combined with cooperative learning. in cooperative learning, the lower mathematical ability students could learn of mathematical work habits of higher mathematical ability students, and in the process of explaining the material, the higher mathematical ability students could develop mastery stronger and deeper understanding for themselves about the mathematical tasks. however, cooperative learning was not entirely a positive impact on students' academic performance (hecox, 2010; iqbal, 2004). iqbal (2004) reported that low ability students of middle school at rawalpindi got a benefit from cooperative learning, while high ability students did not have it, although they remain in position at the top in math achievement. hecox (2010) found no difference in scores florida comprehensive assessment test (fcat) among fourth-grade students who obtained teaching under cooperative learning and students who obtained traditional teaching in polk county, florida. thus, metacognitive scaffolding approach in cooperative learning pattern was expected to enhance the students‟ mathematical communication ability. despite many studies that examine the influence of metacognitive approach, scaffolding or cooperative learning to academic performance in certain educational levels, but it did not found studies that aimed to review the enhancementof mathematical communication on low, middle and high ability students under metacognitive scaffolding approach in cooperative learning pattern, hereinafter referred to as metacognitive scaffolding approach. thus, the purpose of this study was to find the students' mathematical communication after obtaining metacognitive scaffolding approach. the research questions were, “is there a difference of enhancement of mathematical communication ability between students who acquire teaching under metacognitive scaffolding approach and students who acquire teaching under direct approach?” and “is there an interaction effect between teaching approaches (metacognitive scaffolding approach and direct approach) and students‟ prior mathematical ability (high, middle, and low) to the enhancement of mathematical communication ability?” to address that question, enhancement of mathematical communication of students taught by metacognitive scaffolding approach compared with that of a control group. besides that, volume 8, no 2, september 2019, pp. 117-128 119 the enhancement both of students taught by metacognitive scaffolding approach and that of a control group based on the prior mathematical ability compared each others. in this study, the hypotheses were, “there is a difference of enhancement of mathematical communication ability between students who acquire teaching under metacognitive scaffolding approach and students who acquire teaching under direct approach” and “there is an interaction effect between teaching approaches and students‟ prior mathematical abilities toward enhancement of mathematical communication ability”. in this study, mathematical communication was considered as students‟ ability to understand and state mathematical ideas both in writing, and drawing. the indicators used in this study were: (1) stating a picture or diagram into mathematical ideas, (2) stating a daily occurrence in the mathematical symbols, and (3) explaining the idea, situation, or a mathematical relation with graphs or algebraic. metacognitive scaffolding approach was considered as a teaching approach that was characterized by activities: (1) teacher raised a mathematical problem, (2) students tried to solve the problem; and (3) teacher provided temporary metacognitive assistance, which is gradually reduced and eventually the student can independently take full responsibility for mathematical tasks that must be completed; whereas direct approach, was consider as a teaching approach that was characterized by activities: (1) explanation or manipulation concept by teacher, (2) providing an opportunity for students to ask, (3) demonstrating completion of example problems, (4) giving exercises to be completed by the students, (5) asking some students to write again their answer on the board, (6) commenting on student answers, and (7) providing homework assignments if it deemed necessary. 2. method 2.1. research design the method used in this study was aquasi-experimental method. there were two groups of students. as the experimental group was students who acquire teaching mathematics under metacognitive scaffolding approach, while the control group were students who acquire teaching mathematics under direct approach. this study implemented a pretest and posttest for both groups of students.thus the research design was the design of the control group pretest-posttest, and expressed as follows. o x o o o description: o : pretest-posttest onmathematical communication. x : treatment in the form of learning with metacognitive scaffolding approach. the research design involved two factors, namely learning approaches and student groups based on factors prior mathematical abilities. the first factor consisted of metacognitive scaffolding and direct approaches. the second factor consisted of a group of students based on prior mathematical ability (high, middle, and low). this research design could be described as the relationship between the factors as presented in table 1. sufyani, enhancement of students’ mathematical communication … 120 table 1. average of mathematical communication gain based on teaching approach and prior mathematical ability prior mathematical ability teaching approach metacognitive scaffolding approach (b1) direct approach (b2) high (a1) a1b1 a1b2 middle (a2) a2b1 a2b2 low (a3) a3b1 a3b2 2.2. sample the research was conducted at the elementary school teacher education program at a university in bandung. thus, population of the study was all students of elementary school teacher education program who received mathematics education course, at a university in bandung. whereas the sample was 118 students; 60 students as an experiment groupand 58 students as a control group. 2.3. research procedure research activities initiated by determining the study sample. after the sample was set, each student was given a prior mathematical ability test. the test is intended to classify students based on prior mathematical abilities (high, midlle, and low). after the experimental and the control groups were formed, the students were given the pretest about the mathematical communication ability. after providing a treatment, posttest on mathematical communication was given for the students. for data analysis, researchers used the help of statistical package for social science (spss) for windows version 20 software. 2.4. data analysis there were two main hypotheses to be tested. the first one was related to test two independent samples with the interval ratio of measurement. the data was analyzed by t test, t‟ test, and mann-whitney. in the second hypothesis, data could be tested using twoways anova if the conditions were available. if the conditions were not available, the interaction effect would be seen by the diagram and one way anova or kruskal-wallis. 3. results and discussion 3.1. results descriptive statistical analysis of the results of students‟ mathematical communication ability was presented in table 2. volume 8, no 2, september 2019, pp. 117-128 121 table 2. descriptionstatisticsof students„ mathematical communicationability (mca) group level pretest posttest gain of mca mean sd mean sd mean sd experiment mix 25.83 13.57 50.17 14.98 24.33 10.95 low 18.61 7.63 45.56 11.36 26.94 12.85 middle 24.40 12.10 47.60 15.22 23.20 8.40 high 35.59 15.30 58.82 15.16 23.24 12.24 control mix 33.71 9.49 46.98 11.28 13.28 7.75 low 29.21 7.50 43.68 11.28 14.47 9.11 middle 31.04 7.37 44.79 9.15 13.75 6.30 high 42.81 8.16 53.75 11.62 10.94 7.79 ideal maximum score (ims) =100. from the table 2, it was appeared that the enhancement of communication ability that students acquire teaching under metacognitive scaffolding approach was relatively higher than students who acquire teaching under direct approach, the well-viewed as a whole and viewed based on the level of prior mathematical ability. inferential statistical analysis of the results of students‟ mathematical communication ability to experimental and control groups were presented in table 3. table 3. difference of students‟ mathematical communications ability (mca) between experiment and control groups (the level of significance α = 0.05) variable group difference test test statistic sig. conclusion mca-1 (pretest) mix experiment m-w test 0.000 different control gain of mca mix experiment m-w test 0.000 different control mca-1 (pretest) low ability experiment t-test 0.000 different control gain of mca low ability experiment t-test 0.002 different control mca-1 (pretest) middle ability experiment m-w test 0.017 different control gain of mca middle ability experiment m-w test 0.000 different control mca-1 (pretest) high ability experiment t‟-test 0.100 not different control mca-2 (posttest) high ability experiment t-test 0.291 not different control from the table 3, it could be stated that there was difference in enhancing of mathematical communication ability significantly between students who attained teaching under metacognitive scaffolding approach (experimental group) and students who attained teaching under direct approach (control group), the well-viewed as a whole (mix) and sufyani, enhancement of students’ mathematical communication … 122 viewed based on the prior mathematical ability levels (low and middle). if these results were associated with the results in table 2 it can be concluded that the enhancement of students‟ mathematical communication who attained teaching under metacognitive scaffolding approach were higher than students who attained teaching under direct instructional approach. the interaction effect between teaching approaches and prior mathematical ability toward enhancement of students‟ mathematical communication ability would be tested by using two ways anova. before using the two ways anova, it was necessary to be viewed whether the data of each factor was distributed normally. the result of distribution normality was presented in table 4. table 4. data distributionnormality on mathematical communication ability based on group and prior mathematical ability (the level of significance α = 0.05) group ability distributionnormality test implications of the use of two ways anova sig. conclusion experiment 0.006 not normal two ways anova is not used control 0.006 not normal low 0.163 normal middle 0.072 normal high 0.022 not normal from table 4 it appeared that the condition for using two ways anova was not sufficient. therefore, the interaction effect was analyzed using diagram 1 and table 5. diagram 1. interaction effect teaching approach and prior mathematical ability on enhancement of students‟ mathematical communication ability volume 8, no 2, september 2019, pp. 117-128 123 the diagram 1 indicated that there was interaction effect between teaching approaches and prior mathematical ability toward enhancement of students‟ mathematical communication ability. to confirm the presence of this interaction effect is significant, it was necessary to be tested the difference of the gain among mathematical ability levels (low, midlle, and high), both in the experimental and the control groups, as presented in table 5. table 5. test of difference of mathematical communication ability gain (gain of mca) based on ability levels of experimental and control groups (the level of significance α = 0.05) variable ability level difference test statistic test sig. conclusion mca gain of experimental group low kruskalwallis 0.332 not different middle high mca gain of control group low kruskalwallis 0.288 not different middle high from table 5, it could be seen that there were no differences of enhancement of students‟ mathematical communication ability in experiment group among the low, midlle, and high level students. the same result was occured in the control group. it can be concluded that there is no significant interaction effect between teaching approaches and prior mathematical ability toward enhancement of students‟ mathematical communication ability. 3.2. discussion the results showed that the enhancement of communication ability of students who acquired teaching under metacognitive scaffolding approach is significantly higher than students who acquired teaching under direct approach. thus, it could be said that teaching mathematics under metacognitive scaffolding approach significantly positive impact on enhancement of students‟ mathematical communication ability. clark, jacobs, pittman, & borko (2005) stated that giving problems that triggered discussion was a strategy in developing mathematical communication. by giving mathematical problems, students in small groups tried to understand and showed the model of the problem solution, and the solving models developed students would be object of discussion, or revision of the understanding of mathematical problem faced. it appeared students in some groups argued for their opinion, while in other group it appeared that a student explained his idea to friends in his group. thus, through cooperative learning, students not only got an understanding of mathematical solving, but they also had opportunity to represented and evaluated mathematical ideas to be tested and compared it with the mathematical idea of their friend. teacher encouraged students to actively engaged in discussions, and provided assistance if there was a group of students came to a halt in understanding or solving the problem. the presence of these metacognitive questions encouraged students to identify the problem, identify relevant information, display ideas, and explain mathematical ideas to the friends group. the findings of this study were consistent with there commendation sufyani, enhancement of students’ mathematical communication … 124 clark et al. (2005) which stated that teachers should encourage students to actively explain the mathematical ideas. although it was not as high as in teaching under metacognitive scaffolding approach, there was a significant enhancement of mathematical communication ability of students who obtained teaching under direct approach. teaching under this approach is characterized by teacher activities, such as explain a concept. when the teacher explains a concept, students gain an understanding of the concept needed to build mathematical communication ability. new knowledge acquired by students encourage them to match the existing cognitive structure and frequently it was preceded by cognitive conflict (disequilibrium); furthermore through student‟s question and teacher‟s answer, this conflict can be resolved, so that the cognitive structures remain in equilibrium. it encouraged the development of thinking and understanding of concepts, which is required to solve mathematical problems, especially with problems related to mathematical communication ability. the presence of solving mathematical problem samples by teacher could encourage students learned meaningfully, because those samples based on the concepts that was already explained by their teacher and understood by the students could be used as a model completion by students in solving mathematical problems, especially those related to mathematical communication ability. in addition, the time required on teaching under direct approach was efficient, so that teachers can provide additional practice materials. the presence of the materials made students had chance to practice more and represent mathematical ideas in a variety of forms, so that adds to the experience in the face of problems related to mathematical communication. according to pressley (1995), with experiences, gradually students were able to implement relevant strategies flexibly and precisely. although the samples of solving mathematical problems could encourage mathematical communication ability, it could also lead students tend to imitate those procedures. as a result, students solved the problems easily if the problems were similar to the samples, but he had trouble when facing new problems. furthermore, in learning the direct approach, there is avery limited social interaction, almost nothing, especially the interaction among students. meanwhile, vygotsky (1980) considers that an individual's cognitive development depends on social interaction. therefore, it could be predicted that the presence of mathematical problems, metacognitive questions and cooperative learning on teaching under metacognitive scaffolding approach on the one hand, and the explanation of the concepts and samples of solving mathematical problems on teaching under direct approach on the other hand, are factors that could explain one of the results of this study, enhancement of students‟ mathematical communication ability who acquired teaching under metacognitive scaffolding approach was higher than students who acquired teaching under direct approach. data analysis showed that there was no interaction effect between learning approaches (scaffolding metacognitive and the direct) and students‟ prior mathematical ability (high, midlle, and low) toward the enhancement of students‟ mathematical communication. in teaching under metacognitive scaffolding approach, a teacher posed metacognitive questions when students had difficulty on understanding or solving the problem. the questions posed would be used by high ability students to link the mathematical problems encountered with mathematical ideas that will be displayed. this supported the study of clark et al. (2005) argued that the posing of a problem was strategy that could develop students‟ mathematical communication. besides that, there were elements of cooperative learning, that enable high ability students explained their mathematical ideas to friends in the group. volume 8, no 2, september 2019, pp. 117-128 125 the result of this study also showed that there was a significant enhancement of student mathematical communication ability for low and midlle ability students who obtained teaching under metacognitive scaffolding approach. on teaching under this approach, low and midlle ability students had been supported by high ability students. the support was obtained through discussions, explanations, and examples of mathematical representation. the interesting finding in this study was that the low and midlle ability students who obtained teaching under metacognitive scaffolding approach got benefit as much as high ability students who obtained teaching under the same approach. this finding could also be interpreted in the context of the theory of piaget and vygotsky. in terms of the theory of cognitive development (piaget, 1970), metacognitive scaffolding approach played an important role in improving the cognitive development of low ability students. cognitive conflicts on high ability students might be initiated by teacher‟s metacognitive questions that impacted on tension on the students and that gave a mismatch between what had been understood and the fact that be faced. it result an disequilibrium condition in the cognitive system then they tried to overcome it through thinking. in this case, the approach encouraged high ability students to overcome the disequilibrium condition by relying on their knowledge and experiences. for low and midlle ability students, cognitive conflicts may be acurated because of the cooperative learning with high student. in cooperative learning, low and midlle ability students were faced with a mathematical representation from high ability student. the mathematical representation as often not in accordance with the ideas developed by the low and midlle ability students. to resolve this conflict, they could ask for an explanation from the high ability students, so that a balance in their cognitive system was recovered (re-equilibrium). in teaching under metacognitive scaffolding approach, low and midlle ability students did not feel awkward discussed and applied their mathematical ideas to his friends, including to the high ability students. effort of low and midlle ability students gave their argumentations were encouraged activate their prior knowledge with new mathematical problems. thus, the discussion could activate of their schemata, so that allow the students elaborate and provide representation of the problem and the solution, either in the form of drawing, diagrams, as well as mathematical sentences. the idea of a cooperative learning approach in teaching under metacognitive scaffolding approach related to zone of proximal development (zpd) of vygotsky (1980). through cooperative learning with students obtained model of representation and solving problems from the high ability students, so that the low and midlle ability students were being able to achieve the level of mathematical communication that cannot be achieved without the metacognitive scaffolding approach. thus, the approach gave low and midlle ability students could fully explore their potential capabilities, thus changing the position of the level of potential development into level of actual development, and the level of potential development moving into his new position. it could be predicted that the enhancement of mathematical communication ability of high ability students who obtained teaching under metacognitive scaffolding approach triggered by readiness of their mathematical knowledge and metacognitive assistance from their teacher, while the low and midlle ability students were triggered by the interaction with high ability students. in teaching under direct approach, high ability students showed an enhancement of their mathematical communication ability. this enhancement seemed to be related to their readiness of mathematical knowledge and learning experiences. in addition, this enhancement was apparently due to an explanation of mathematical concept and representation of solving problems. explanation of mathematical concepts through illustrations that were easy to be understood and providing representation of solving problems step by step, would give a positive effect on mathematical communication of sufyani, enhancement of students’ mathematical communication … 126 high ability students. if in the metacognitive scaffolding approach, there was an aspect of metacognitive assistance, then in the direct approach, there were aspects of a concept explanation and an example of representation of solving a problem. thus it was predicted that the metacognitive assistance on teaching under metacognitive scaffolding approach on the one hand , and the readiness of a concept explanation and the example on solving a problem on teaching under direct approach on the other hand were factors that could explain one of the results of this study, which was no difference of enhancement of mathematical communication ability between high ability students who obtained teaching under metacognitive scaffolding approach and those who obtained teaching under direct approach. this evidence suggested that teaching by applying principles of constructivism was not always better compared to teaching by applying principles of behaviorism. conversely, teaching by applying principles of behaviorism is not always worse compared to teaching by applying principles of constructivism. for low and midlle ability students who obtained teaching under direct approach, the research result showed that there was a significant enhancement of students‟ mathematical communication ability. explanation of mathematical concepts through illustrations that were easy to be understood and providing representation of solving problems step by step, would give a positive effect on mathematical communication. explanation of the concept and representation of solving problem were suspected as factors indetermining the enhancement of mathematical communication of low and midlle students that obtained teaching under direct approach. however, although apparently students might understood only on the specific problems. therefore, it could be indicated that the presence of cooperative learning in teaching under metacognitive scaffolding approach on the one hand, and the explanation of concepts and the examples of representation of solving problems on the other hand are factors that could explain one of the results of this study, that was the enhancement of mathematical communication ability of low and midlle ability students who obtained teaching under metacognitive scaffolding approach higher than those who obtained teaching under direct approach. 4. conclusion based on the results of the study, the conclusions are there is a difference in mathematical communication ability between students under metacognitive scaffolding teaching approach and students under direct teaching approach. the enhancing in mathematical communication ability of students under metacognitive scaffolding teaching approach higher students under direct teaching approach. furthermore, there is no interaction effect learning approaches and prior mathematical skills on the enhancement of students‟ mathematical communication. these study results are only based on specific aspects of mathematical ability, the subject is limited, and the topic is narrowed. even so, it is clear that the approach was effective in supporting students‟ mathematical communication ability. in addition, the implementation of learning with metacognitive scaffolding approach did not require expensive. therefore, the recommedation is that the teaching under metacognitive scaffolding approach can be tried in other aspects of mathematical ability, other topics or other subject matter. volume 8, no 2, september 2019, pp. 117-128 127 references carpenter, j & gorg, s, (2000), principles and standards for school mathematics. reston, va: national council of teachers of mathematics. clark, k. k., jacobs, j., pittman, m. e., & borko, h. (2005). strategies for building mathematical communication in the middle school classroom: modeled in professional development, implemented in the classroom. current issues in middle level education, 11(2), 1-12. departemen pendidikan nasional ri (2006). peraturan menteri pendidikan nasional republik indonesia nomor 22 tahun 2006 tentang standar isi untuk satuan pendidikan dasar dan menengah. lampiran 3: standar kompetensi dan kompetensi dasar mata pelajaran matematika untuk sma/ma. hecox, c. c. (2010). cooperative learning and the gifted student in elementary mathematics. retrieved march 9, 2012 from https://digitalcommons.liberty.edu/cgi/viewcontent.cgi?referer=https://scholar.goog le.com/&httpsredir=1&article=1385&context=doctoral hill, h. c., rowan, b., & ball, d. l. (2005). effects of teachers‟ mathematical knowledge for teaching on student achievement. american educational research journal, 42(2), 371-406. iqbal, m. (2004). effect of cooperative learning on academic achievement of secondary school students in mathematics (doctoral dissertation, university of arid agriculture). mullis, i. v., martin, m. o., gonzalez, e. j., & chrostowski, s. j. (2004). timss 2003 international mathematics report: findings from iea's trends in international mathematics and science study at the fourth and eighth grades. timss & pirls international study center. boston college, 140 commonwealth avenue, chestnut hill, ma 02467. oecd. (2005). pisa 2003 technical report. retrieved april 25, 2012, from http://www.oecd.org/education/school/programmeforinternationalstudentassessmen tpisa/35188570.pdf oecd. (2007). pisa 2006 results. retrieved april 25, 2012, from https://www.oecd.org/education/school/programmeforinternationalstudentassessme ntpisa/pisa2006results.htm oecd. (2010). pisa 2009 results: executive summary. retrieved april 25, 2012, from https://www.oecd.org/pisa/pisaproducts/46619703.pdf passos, a. f. j. (2009). a comparative analysis of teacher competence and its effect on pupil performance in upper primary schools in mozambique and other sacmeq countries (doctoral dissertation, university of pretoria). peters, g. (2011). advantages & disadvantages of scaffolding in the classroom. retrieved april 25, 2012, from https://www.theclassroom.com/advantages-disadvantagesscaffolding-classroom-8008434.html piaget, j. (1970). science of education and the psychology of the child. trans. d. coltman. pressley, m. (1995). more about the development of self-regulation: complex, long-term, and thoroughly social. educational psychologist, 30(4), 207-212. sufyani, enhancement of students’ mathematical communication … 128 vygotsky, l. s. (1980). mind in society: the development of higher psychological processes. harvard university press. yee, f.p. (2002). using short open-ended mathematics questions to promote thinking and understanding, singapore: national institute of education. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p1-14 1 the construction learning media and level of students’ mathematical communication ability dian fitri argarini* 1 , nok izatul yazidah 2 , anik kurniawati 3 1.2.3 insititut keguruan dan ilmu pendidikan budi utomo malang article info abstract article history: received aug 16, 2019 revised jan 24, 2020 accepted jan 26, 2020 this study aims to look at the effect of the use of instructional media on student learning achievement in terms of students' mathematical communication. the learning media in this study are textbooks with a constructivism approach that has been validated and tested previously. this study will compare the learning achievements of students who learn using constructivism learning models with constructivism media, constructivism learning models without media, and direct learning. this is a quasiexperimental research with a 3 × 3 factorial design. it involved junior high school students in malang district as the research population. based on the hypothesis, it is revealed that : (1) students who learn using constructivism approach with constructivist media had better performance than other groups, (2) students with high mathematical communication had higher learning achievement than students with moderate and low communication skills, ( 3) based on the category of high, moderate and low mathematical communication, students with constructivist learning and constructivist media gained better achievements, (4) in the constructivist learning group using constructivist media, constructivist learning without media, and direct learning, students with high mathematical communication gained better achievement than students with moderate and low mathematical communication keywords: constructivism, mathematical communication, media copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: dian fitri argarini, department of mathematics education, insititut keguruan dan ilmu pendidikan budi utomo malang jl. simpang arjuno no.14b, kauman, malang city, east java 65119, indonesia email: kejora.subuh14@gmail.com how to cite: argarini, d. f., yazidah, n. i., & kurniawati, a. (2020). the construction learning media and level of students’ mathematical communication ability. infinity, 9(1), 1-14. 1. introduction one way to measure the success of a classroom learning is to look at students’ achievement. learning achievement is one of the results that the students achieved from a learning process by students and teachers to achieve an educational goal (paulpandi & govindharaj, 2017). to assess student achievement teachers usually conduct regular evaluations and assessments. it is teacher’s authority to evaluate the method used and its implementation, because evaluation has no specific benchmark, except that it is based on learning objectives designed by the teacher himself. in general, to see student achievement mailto:kejora.subuh14@gmail.com argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 2 in learning a certain subject, the teacher can rely on students’ cognitive aspects, whether it increases, remains the same, or decreases. based on the explanation from the research and development (r & d) department, the ministry of education and culture stated that the results of the 2019 unbk (national exam) decreased, since the students’ overall mathematics score only reached 46 points, below the 55 point standard. what are the factors affecting the declining achievement? in addition to changing the assessment system, it is important to highlight teacher performance in designing and implementing the learning process to suit the students need. this is so because teacher quality, academic climate, and student achievement in understanding mathematics have a significant relationship (daso, 2013). high student achievement as seen from their cognitive aspects is very closely related to understanding students’ mathematical concepts. students are required to understand mathematical concepts in order to solve the problems given. comprehension on concepts indicates students’ ability to explain concepts, use concepts and develop matters relating to concepts (duffin & simpson, 2000). ferrinimundy (2001) states that ‘students’ understanding of mathematical concepts can be seen from (1) students’ ability to define concepts both verbally and in writing, (2) students’ understanding of examples and not examples, (3) using mathematical symbols to explain concepts, (4) changing forms of a representation, (5) ability to recognize and interpret a concept, (6) ability to identify characteristics and ability to explain, and (7) ability to compare several concepts. skemp (2006) divides concept understanding into two types, namely instrumental understanding and relational understanding. instrumental understanding is indicated by students’ ability to memorize formulas and use them, without the ability to explain their reasons. meanwhile, rational understanding can be seen from the ability of students to understand more complex concepts and relate several concepts to solve mathematical problems. based on skemp's opinion, it is conclusive that students' relational understanding is more meaningful than instrumental understanding (qohar, 2011). the importance of understanding concepts greatly affects the level of student mathematics achievement. therefore, teachers are expected to create a good academic atmosphere to improve student mathematics achievement. mathematical achievement in this study is seen from the obtained results after carrying out mathematical achievement tests in geometry. in addition to understanding mathematical concepts that needs improvement, teachers must also understand other factors that can influence student achievement in designing the learning process and learning objectives to achieve the maximum target. other factors are indicated by various angles, for example cognitive abilities, cognitive styles, learning styles, spatial abilities, mathematical communication skills or other reviews. one aspect to highligt in this research is students' mathematical communication skills. in general, it is possible to interpret communication as a way of conveying messages from the sender of the message to the recipient of the message to inform opinions either directly (verbally) or with the help of the media (tinungki, 2015). mathematical communication skills are the ability of students to use mathematics as written language in the form of vocabulary, notations, and mathematical structures in understanding problem solving. mathematical communication is influenced by three aspects, namely aspects of drawing, aspects of mathematical expressions, and aspects of written texts. on the other hand, there are five aspects of communication, namely representation, listening, reading, discussion, and writing. representation is the ability of students to translate a problem or a diagram from a physical model into symbols or words. listening is students’ ability to hear a question of a certain problem. reading, in this volume 9, no 1, february 2020, pp. 1-14 3 ability is related to several aspects owned by students, namely remembering, understanding, comparing, finding, analyzing, organizing, and applying what is contained in the reading. discussion is the ability of students to express ideas related to problems and material provided. in contrast, writing is the ability to express and reflect ideas in the form of writing that is carried out consciously. there are two reasons that underlie the importance of communication in mathematics, namely mathematics as language and mathematics as a social activity. in addition to understanding the level of students’ mathematical communication, it is certain that the teacher must design learning to maximize the learning process to achieve the highest learning achievement. one of such attempts can be done using the right learning model to help students’ better master the material when the learning process takes place. one learning model offered is a constructivism learning model. constructivism can be interpreted as knowledge gradually built by humans expanded through a limited context. to do constructivist learning at school, a teacher needs access to models and strategies that they can apply effectively with relative ease (boddy, watson, & aubusson, 2003) constructivist-based learning aims to (1) provide experience with the knowledge construction process, (2) provide experience and appreciation for various perspectives, (3) conduct learning in a realistic and relevant context, (4) voice opinions in the learning process, (5) apply what is gained during the learning process into social experience, (6) encourage the use of some representations, and (7) encourage self-awareness to carry out the process of knowledge construction (koohang, riley, smith, & schreurs, 2009). in this study, students were taught using a constructivism model with material related to geometry. the researcher chose some material in the field of geometry for junior high level and compiled a constructivism-based learning media to help maximize the learning process with a constructivist learning model. the national education association (nea) defines the media as anything that can be manipulated, seen, heard, read, or talked about along with the instruments used for these activities. media also means something that carries information between sources and recipients (nurseto, 2012). media can be classified into several types based on the form and method of presentation, including (1) graphics, printed materials and still images, (2) silent projection media, (3) audio media, (4) audio-visual silent media, (5) live audiovisual/film media, (6) television media, and (7) multi media. in this study, the researcher used printed books media known as "smart books". this smart book media contains geometry materials for students in junior and senior high school levels. smart book is a textbook developed by researchers. this textbook is a constructivism-based textbook. the constructivism approach is chosen so that students are able to build on the material or initial knowledge they have with the newly received knowledge. the smart book contains mathematical geometry material received at junior high school level, namely (1) lines and angles, (2) straight line equations, (3) rectangles and triangles, (4) pythagorean theorems, (5) circles, (6) geometry transformation, (7) build flat side space, and (8) build curved side space. this smart book media was developed with a plomp development model, which consists of three stages. the first stage is known as preliminary research that aims to make initial observations about matters related to the development of constructivist-based smart books. the second stage is the prototype design stage, by developing constructivist-based smart book and formative valuation. the last stage is the assessment stage conducted by a summative or semi-summative evaluation. this smart book media has been validated both in the fields of material, media, and language by expert fields. after being validated and revised, the media was also tested on students to ensure its feasibility to use. the use of argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 4 constructivist-based media is expected to improve student mathematics learning achievement. based on the above mentioned description, this study aims to find out: (1) which method give better mathematical achievement by comparing students who learned using constructivist learning models assisted by constructivist media (smart books), and those who learned using constructivist learning models without media assistance, or direct learning model. (2) which student has better performance among students with high, moderate, or low mathematical communication skills. (3) in each of the categories of mathematical communication skills, which one provides better learning achievement among students who learned using constructivist learning models assisted by constructivist media (smart books), constructivist learning models without the help of media, or direct learning models. (4) in each learning model, which one has better learning achievement among students with high, moderate, or low mathematical communication skills. 2. method in this study, subjects will also be grouped based on their mathematical communication skills. classification of students' mathematical communication skills is done by considering 5 aspects of mathematical communication skills according to baroody namely, (i) making a representation of an idea or problem, (ii) listening to the topics being discussed, (iii) reading, (iv) discussion to express ideas , and (v) writing as an effort to convey ideas through various media. in this study, a preliminary test of students' mathematical communication skills will be conducted to group the research subjects. mathematical communication skills of students will be measured through the ability of students to express their mathematical communication skills in writing in solving mathematical problems. for each mathematical problem, measurement of written communication skills is carried out by taking into account several aspects and indicators according to the table 1. table 1. aspects and indicators of mathematical communication skills material measured communication aspects indicator triangles and squares states and illustrates mathematical ideas in the form of mathematical models students can express and illustrate ideas and problems given in the form of images students can state the problems given in the form of mathematical models in the form of equations and solve them volume builds side space flat states and illustrates mathematical ideas in the form of mathematical models students can state and illustrate ideas and problems related to the ballroom of flat side spaces students can state the problems given in the form of mathematical models in the form of equations and solve them surface area build flat side space states and illustrates mathematical ideas in the form of mathematical models students can state an image into an idea or mathematical problem related to congruence. then students can solve these problems volume 9, no 1, february 2020, pp. 1-14 5 after the mathematical communication ability test questions are made based on the grid in table 2, the test is given to students. the results of students' mathematical communication ability tests in each class of learning models are grouped into three groups namely, high mathematical communication skills, medium mathematical communication skills, and low mathematical communication skills. the research design based on the learning model and the results of tests of mathematical communication skills can be seen in table 2. this is a type of quasi experimental research using the factorial 3 × 3 design. it involved all junior high school students in malang district as the research population. the research sampling was done using stratified cluster random sampling techniques. in this sampling technique, the population was divided according to strata, which were drawn randomly from sample sub-populations (budiyono, 2003). from the sampling technique, the researcher selected tarbiyyatus shibbyan middle school, kh. amir wajak, and tambaksari ybpk middle school. for each school, 2 experimental groups and 1 control group were taken. in the first experimental group, students learned using a constructivism model assisted by smart book media with a constructivism approach. the second experimental group conducted learning with a constructivism model without media assistance. meanwhile, the control class received learning materials using a direct learning model. table 2. research design learning model (a) mathematical communication (b) high (b1) moderate (b2) low (b3) constructivism model using constructivism media (a1) (ab)11 (ab)12 (ab)13 constructivism model without media (a2) (ab)21 (ab)22 (ab)23 direct learning model (a3) (ab)31 (ab)32 (ab)33 information: ab11 : student learning achievement with high mathematical communication skills who get a model of cooperative learning assisted by constructivist media (smart book). ab12 : student learning achievement with moderate mathematical communication skills who get a model of cooperative learning assisted by constructivist media (smart book). ab13 : student learning achievement with low mathematical communication skills who get a model of cooperative learning assisted by constructivist media (smart book). ab21 : student learning achievement with high mathematical communication skills who get cooperative learning models without the media. ab22 : student learning achievements with moderate mathematical communication skills who get cooperative learning models without the media. ab23 : student learning achievement with low mathematical communication skills who get cooperative learning models without the media. ab31 : student learning achievement with high mathematical communication skills who get a direct learning model ab32 : student learning achievement with moderate mathematical communication skills who get a direct learning model. argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 6 ab33 : student learning achievement with low mathematical communication skills who get a direct learning model the study involved 99 students as research samples, consisting of 33 students in the experimental group one, 33 students in the experimental group two, and 33 students in the control group. this study used two independent variables, namely the learning model and students’ mathematical communication skills and one dependent variable namely mathematics learning achievement of students. data were collected using the test documentation method. documentation methods were used to collect data on students' initial abilities, test methods were used to collect final learning achievement data, and mathematical communication tests were used to collect data on students’ mathematical communication skills. this study used a description test on geometry material that the students received as research instrument. the second instrument, a mathematical communication test, was used to distinguish between high, moderate, or low mathematical communication groups. the instrument testing was conducted at smp kh. amir wajak with 20 students as respondents. the learning achievement test instrument referred to the criteria namely content validity, differentiation (d=0.3), difficulty level (0.3≥p≥0.7) and reliability (r_11≥0.7). of the 15 items in the description that were tested, 12 items were used as test instruments for students’ mathematical achievements. the mathematical communication test was used as a test instrument that has been validated by material experts to suit for use in this research study. the prerequisite test for the analysis is the normality test with liliefors and the homogeneity test with the bartlett test. the data analysis test used twoway variance analysis with unequal cells. 3. results and discussion 3.1. results this research is an experimental smart book textbook that has been developed previously. what distinguishes this book from mathematics or other geometry books? first, the smart book gives a very easy apperception to readers in everyday life and is assisted with relevant images at the beginning of each chapter (figure 1). figure 1. the "apperception" section of the smart book volume 9, no 1, february 2020, pp. 1-14 7 second, smart book has a section where students discuss. in this section besides discussing with peers, the reader is also expected to be able to construct old knowledge and new knowledge (figure 2). this is because geometry material is always obtained at every level of education. figure 2. the "let's discuss" section of the smart book the third difference from the smart book is that there is a self-evaluation section for readers. in the self-evaluation section, the reader is expected to write down what has been learned in the previous sub-chapter and write what the weaknesses of the reader while studying the material (figure 3). this section can be used as a benchmark or note for the teacher to implement improvements for students in understanding further material. figure 3. the "self evaluation" section of the smart book after arranged the book and focused group discission (fgd) together with mathematics education lecturers at ikip budi utomo malang, the smart book was then validated by experts in terms of material, media, and language. following the final validation results obtained after the revision of the experts, the score is obtained from a scale of 5.0 (table 3). argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 8 table 3. smart book’s validation results validator score material expert 4,09 media expert 4,03 linguist 4,50 average 4,207 after validation with the material experts, the smart book can already be experimented on kh. amir wajak junior high school’s students. the balance test results were obtained from the initial data analysis. the initial data used was the students’ math scores in the previous semester's grades. based on the balance test of the initial data, it was revealed that the three populations had the same initial ability. the experiment obtained the student’s mathematical achievement data. the mean mathematical achievement of students in the experimental group 1, experimental group 2, and the control group can be seen in table 4. table 4. the mean of each cell from the data model of learning and mathematical communication learning model mathematical communication marginal average high moderate low constructivism model with constructivism media 73,40 69,38 63,80 68,91 constructivism model without constructivism media 70,09 64,75 60,50 65,24 direct learning model 69,33 58,13 58,22 61,21 marginal average 70,97 63,78 60,93 before carrying out a two-way analysis of variance, a normality test and a homogeneity test were used as a prerequisite for the analysis of variance. a summary of the normality test is presented in table 5. table 5. summary of normality test results normality test lobs l0,05;n decision summary constructivism and media 0,0940 0,1542 h0 accepted normal constructivism without media 0,0972 0,1542 h0 accepted normal direct 0,0905 0,1542 h0 accepted normal high mathematical communication 0,0963 0,1566 h0 accepted normal moderate mathematical communication 0,0821 0,1419 h0 accepted normal low mathematical communication 0,1404 0,1674 h0 accepted normal volume 9, no 1, february 2020, pp. 1-14 9 based on table 5, it can be seen that the sample comes from a population that is normally distributed, both in terms of learning models and in terms of students’ mathematical communication. the next step is to do a homogeneity test. the summary of homogeneity test can be seen in table 6. table 6. summary of homogeneity test samples k decision summary learning model 3 3,5025 5,991 h0 accepted homogeneous mathematical communication 3 3,8818 5,991 h0 accepted homogeneous based on table 6, it is clear that the data in each learning model and students’ mathematical communication skills have homogeneous variance. furthermore, a two-way variance analysis test was carried out with unequal cells. a summary analysis of variance analysis is presented in table 7. table 7. summary analysis of two-way variance source jk dk rk fobs fα decision learning model (a) 781,1563 2 390,5782 4,9622 3,11 h0a rejected mathematical communication (b) 1709,1189 2 854,5594 10,8569 3,11 h0b rejected interaction (ab) 172,7210 4 43,1802 0,5486 2,49 h0ab accepted error 7084,0249 90 78,7114 total 9747,0211 98 the two-way variance analysis with unequal cells based on table 7 summarizes that (1) in terms of the main effects between lines (a), exposing students to constructivism learning models assisted by media constructivism, constructivist learning models without media, and direct learning models leads to a different effect on students’ mathematics learning achievements. (2) in terms of the main effect between columns (b), high, moderate, and low mathematical communication skills have different effects on students' mathematics learning achievement. (3) in terms of the interaction effect (ab), it can be concluded from the table that there is no interaction between the learning model and students 'mathematical communication abilities on students' mathematics learning achievement. based on the two-way anava analysis, it was found that h0 a was rejected. h0 a hypothesis states there is no difference in the effect between the use of learning models on learning achievement, then based on table 7 h0a is rejected this means, at each constructivism learning model groups have different effects on students' mathematics learning achievement. thus, it is necessary to do further tests after the analysis of variance by the scheffe’ method for inter-line warranty testing. the summary calculation of the average test between lines is presented in table 8. argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 10 table 8. summary of the multiple comparison tests between lines comparison h0 h1 fobs 2f0,05;2;90 decision 6,8183 6,19540 h0 rejected 3,4050 6,19540 h0 accepted 12,4190 6,19540 h0 rejected based on the two-way analysis, it is found that h0b was rejected. the h0 b hypothesis show there is no difference in the effect of students' mathematical communication abilities on learning achievement, so based on table 7 h0b is rejected, meaning that high, medium, and low mathematical communication abilities have different effects on students' learning achievement in mathematics.thus, it is necessary to do further tests after the variance analysis with the scheffe method to compare between columns. the summary calculation of the average intermediate test between columns (table 9). table 9. summary of multiple comparison tests between columns comparison h0 h1 fobs 2f0,05;2;90 decision 11,2643 6,19540 h0 rejected 6,7275 6,19540 h0 rejected 18,8677 6,19540 h0 rejected 3.2. discussion the comparison test between lines in each category of the learning model is concluded that students who obtain constructivism learning models assisted by constructivism learning media (smart books) provide better mathematics learning achievement than students who obtain constructivism learning model without media and direct learning. whereas students who learned with constructivism learning models without media and direct learning models get the same learning outcomes. this is in accordance with the research conducted by rudiyanto (2008), which stated that the use of media-assisted learning models with constructivist strategies had valid and effective effects centered on student interests, and were able to improve student learning outcomes. rudiyanto (2008) stated that the use of media can help students in understanding the concepts of the material taught by the teacher. thus, the purpose of using media to facilitate the learning process and help the achievement of learning objectives is met. the group model of constructivism learning without media provides the same learning achievement as students who get a direct learning model. this is due to researchers’ limitations in controlling the course of learning during treatment. implementation of the designed constructivism learning models is less than the maximum in the aspect of group discussion. students seemed passive during learning implementation since they found it difficult to find a solution without learning media. on the other hand, the learning process of students who get direct learning was very centered on the teacher. as a result, students who tended to be passive only received materials and experienced one-way learning. based on the problem in the constructivism learning model group without the media and the direct learning model group of geometry material, students did not weel absorb the conceptual understanding and resulted in lower learning achievement than students who obtained learning with constructivist media-assisted constructivism models (smartbooks). volume 9, no 1, february 2020, pp. 1-14 11 the comparison tests between columns in each category of students' mathematical communication skills (table 2 and table 7), it was concluded that students with higher mathematical communication skills had better mathematics learning achievement than students with moderate and low mathematical communication skills. students with high mathematical communication are having better mathematics learning achievement than students with low mathematical communication. this result is in accordance with the research hypothesis. this is because students with high mathematical communication are better able to understand the problems associated with mathematical symbols and symbols given by researchers. the results of this study are in line with the results of research conducted by tinungki (2015), which states that an increase in students 'mathematical communication skills is directly proportional to an improvement in the learning process that also affects an increase in students' mathematics learning achievement. based on these findings, it can be concluded that the understanding of students' mathematical communication is needed by the teacher in order to create an atmosphere of effective learning process and to obtain the maximum possible learning objectives. based on the two-way variance analysis, it was found that h0 ab was accepted. the h0 ab hypothesis states that there is no interaction between the use of learning models with students' mathematical communication skills on learning achievement. based on table 7, it can be concluded that h0ab is accepted so that there is no interaction between the learning model and students' spatial ability on student mathematics learning achievement. thus, there was no interaction between the learning model and students' mathematical communication skills regarding mathematics learning achievement on geometry material, so there was no need to do a double comparison test between cells. that is, conclusions on special effects (on each learning model and on each mathematical communication ability) will be in accordance with the conclusions on the main effects. that is (1) on the level of high, moderate, and low matematic communication skills, students' mathematics learning achievements obtaining a constructivist learning model assisted by constructivism media is better than students who obtained learning with a constructivism model without media and direct learning models. in addition, the achievement of students who obtained learning with constructivism learning models without media and direct learning models gets the same good results. based on this, it can be seen that the third hypothesis in this study is not all in accordance with the results of the study, namely students with moderate and low mathematical communication skills who obtain constructivism learning models without media and direct learning models provide the same results of mathematical achievement. the failure to fulfill this hypothesis may be due to the fact that researchers are not able to fully control the condition of students both in terms of health and students’ internal motivation when taking tests and taking lessons in class and due to lack of learning duration. this might be the reason why the learning achievement of students who received the constructivism model without media and the direct learning model equally good for students who have moderate and low mathematical communication skills. (2) in the media-assisted constructivism model, constructivism without the media or the direct learning model, the learning achievement of students who have high mathematical communication skills is better than students who have medium and low mathematical communication skills. in addition, the achievement of students who have moderate mathematical communication skills is better than students with low mathematical communication. based on the hypothesis analysis, it can be seen that the fourth hypothesis in this study is not all in accordance with the results of the study. that is, in the direct learning model, the achievement of students with moderate mathematical communication skills is no better than students with low mathematical communication. the failure of the hypothesis is likely because students with low mathematical argarini, yazidah, & kurniawati, the construction learning media and level of students’ … 12 communication abilities have better enthusiasm to try to understand the material well than students with moderate mathematical communication skills. in addition, researchers are not fully able to control the condition of students both during learning and during the test. this is the reason why students' achievement of moderate mathematical communication skills is no better than students with low communication skills. 4. conclusion based on the results obtained as follows: (1) the learning achievement of students who obtain constructivism learning models assisted by constructivism media (smart book) is better than students' achievement who obtained constructivism learning models without media and direct learning models. in addition, the achievement of students who obtained a constructivist learning model without media was not as good as the achievement of students who obtained a direct learning model. (2) the achievement of students with high mathematical communication skills is better than students who have moderate and low mathematical communication skills. in addition, the achievement of students with moderate mathematical communication skills is better than students who have low mathematical communication skills. (3) in the category of high, moderate, and low mathematical communication skills, the mathematics learning achievement of students who have constructivism learning models assisted by constructivism media (smart books) are better than students who obtained the constructivism model of learning models without media and direct learning models. in addition, the achievement of students who obtained the treatment of constructivism learning models without media was as good as the achievements of students who obtained the direct learning model. (4) in the category of constructivism learning models assisted by constructivism media (smart books), constructivism learning models without media or direct learning models, the achievement of students with high mathematical communication skills is better than students with moderate and low mathematical communication abilities. in addition, the achievement of students who have moderate mathematical communication skills are also better than students who have low mathematical communication skills. based on the above conclusions, it is suggested that the mathematics teachers try to use an innovative learning media such as constructivism-based learning media (smart books) and apply it together with an appropriate learning model. this is so because based on the research, the use of instructional media provides effective results and helps students to understand mathematical concepts and improve students' mathematics learning achievements, especially on geometry material. in addition, the teacher should also pay attention to other factors in students, namely students 'mathematical communication skills, because in this study students' mathematical communication skills influence student achievement. it is also advised for other researchers or prospective researchers to continue or develop this research other learning models with the help of constructivism-based media or constructivism learning models with constructivist media with other materials such as spatial abilities, logical mathematical intelligence, and so on to be developed on other material and levels. acknowledgements the authors would like to thank for the support to the directorate of research and service society, directorate general strengthening research and development ministry of research, technology and higher education that has been support our research through beginner lecturer research scheme. volume 9, no 1, february 2020, pp. 1-14 13 references boddy, n., watson, k., & aubusson, p. 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(2021). supporting solving word problems involving ratio through the bar model. infinity, 10(1), 149-160. 1. introduction students’ poor academic achievement in mathematics has always been a concern and a worrying issue in brunei every year. these poor results are reflected in all examinations at both national and school levels, and one such example is the national examination for year 6 pupils, penilaian sekolah rendah (psr) before students enter the secondary level. in 2014, it was reported that only 62.29% of pupils managed to get grades a to c in mathematics for that year’s psr (jalil, 2015), whereas in 2015, the percentage increased to only 64.81% (hajar, 2015). although there was an improvement, the percentage was still considered low when compared to the other four subjects that the pupils sat for. there are several possible explanations for this low mathematics performance and one of them is the inability of the pupils to solve mathematics word problems, which is an integral component of such public examinations, and mathematics content specifically (saman, 2000; raimah, 2001; pungut & shahrill, 2014). https://doi.org/10.22460/infinity.v10i1.p149-160 said, & tengah, supporting solving word problems involving ratio through the bar model 150 brunei changed its old education system to a new one in 2009 to keep up with the fast changing world. the new system has identified problem solving skills as one of the essential skills for the 21st century in mathematics at both primary and secondary levels. some of the aims and focus under the new mathematics curriculum are the ability to interpret and communicate mathematics processes, mathematics reasoning and visualisation. the new national education system for the 21st century, spn21, suggests six modes of mathematics representations, namely, real life, verbal, concrete, symbolic representations, technology and diagram. a mathematics word problem is defined as a written description of some situation (greer, 1997). whereas, charles (2011) defined it as a real life situation in which both related known and unknown quantities given and described, a question is asked to determine value of unknown through one or a combination of operations to solve the problem. students who can form mathematical algorithms may not necessarily be able to solve word problems if they do not have the ability to apply mathematical concepts that they already learned in different situations given. according to saman (2000), a large number of year 6 pupils in the country find mathematics word problems to be difficult especially when multiplication and division are involved. while there are several factors that contribute to students’ poor performance in solving word problems, some of the common problems that are observed are such that they do not know when to apply certain mathematical concepts, students tend to misinterpret mathematical concepts (chin & clements, 2001; vaiyatvutjamai & clements, 2004) and they lack the ability to visualise the mathematical problem and subsequently cannot form the appropriate arithmetical algorithm to solve the problem. khalid and tengah (2007) stated that most students are unable to understand what is being asked when attempting word problem tasks. there are a considerable number of problem solving strategies developed by mathematicians to help students solve word problems. one effective strategy that can be taught to students is the famous polya’s four-step problem solving strategy: understand the problem, devise a plan, carry out the plan and look back at the solution (polya, 2004). liu and soo (2014) suggested several ways that students can devise a plan to solve a problem, such as work backwards, guess and check, look for a pattern, choose an operation, act it out, make a list, simplify the problem, draw a picture. the common strategy used by teachers in brunei is the use of keyword or phrases in attempting such word problem. examples of keyword include ‘altogether’, ‘more than’ and ‘total’ for addition, and ‘less’ and ‘reduced’ for subtraction. these at times might cause errors as students that uses keywords or keyphrases approached tend to ignore the whole context of the question being asked and just focuses on the keyword. for example: i have three marbles more than ali, and we have 9 marbles altogether. how many marbles do ali have? for such example, several students will look at ‘altoghter’ or ‘more than’ as addition process and add 9+3 marbles, where in context the question should be focusing on subtraction. in visual representation approach in mathematics, model(s) that reflect information is created to understand the problem as a whole (van garderen & montague, 2003). the use of visual representation to model a mathematical problem has been known to be useful and powerful to aid students in the problem solving process (pungut & shahrill, 2014; denis, 1991; piaget & inhelder, 1966), particularly in word problem. creation, interpretation, utilisation and reflection on pictorial representation enable students to develop greater understanding of the topic when they are able to form strong link between representations and abstract concepts (arcavi, 2003) and take into account the whole context of the problem. the ability to visualise mathematics concepts is what distinguishes competency between high and low achieving mathematics students (van garderen, 2006). volume 10, no 1, february 2021, pp. 149-160 151 an example of a tool for visual representation is the application of bar model. the bar model uses a series of bars or rectangles to model a word problem, and take into account of the whole context of the problem and actual question being asked, where the bars represent the quantities in the problem and the relationship between given and unknown quantities is made clear through diagrams created by the students. the bar model method was developed in 1983 in singapore by its ministry of education to address a national problem in the 1980s with the goal to raise mathematical competencies and improve problem solving abilities (kho, yeo & lim, 2009). therefore, it would be worth to determine if applying bar model should be considered as an alternative approach by brunei teachers when teaching their students. there are two main types of the bar model and they are the part-whole model and the comparison model (liu & soo, 2014). a variation of these two main types has resulted in other types of bar models. the use depends on the quantities and situation given in the word problem. the bar model can be used across many topics and some of them include whole numbers, fractions, ratio and percentage. figure 1. a part-whole (a) and a comparison (b) bar model, as illustrated by madani et al. (2018) as explained by liu and soo (2014), the comparison model can be used to relate or compare two quantities in the problem. the model shows the difference between the quantities and this helps to reduce students’ reliance on equating the terms ‘more than’ to addition and ‘fewer than’ to subtraction because depending solely on key words may not be helpful after all as ‘more than’ does not necessarily mean to add. figure 1 shows both partwhole and comparison bar model. for ratio problems, the comparison model is suitable to represent the quantities in the problem, hence the focus of this research intervention lessons. ratio is about comparing two or more quantities. word problems involving ratio require students to understand the relationship between the quantities contained in the problem and apply the correct arithmetic operations. however, students often fail to see the relationship and subsequently unable to form the correct arithmetical algorithm. with a bar model, the relationship between the quantities can be visually shown and this enables students to decide on the mathematical procedure to use (liu & soo, 2014). whilst bar diagram is useful in solving word problems, care must be taken when drawing the bars. ng and lee (2009) cautioned that every detail of the information in the text must be clearly, precisely and correctly translated to ensure correct solution of the word problem. the purpose of this research study was to examine the impact of using the bar model method on the academic achievement of students in solving word problems involving ratio through the use of bar model. the research question used to guide this study was: how does the bar model affect students’ performance in word problems involving ratio? said, & tengah, supporting solving word problems involving ratio through the bar model 152 2. method convenient sampling of thirty-three participants from two year 8 classes enrolled in an all-girls secondary school in the brunei-muara district were involved in this experimental research study. the participants have mixed mathematical abilities ranging from high to low. english language is their second language and their age ranged from 12 to 13, and all had already learned the topic ratio previously. the pre-test and post-test in this study contained the same questions. the ten questions were in the form of simple word problems on ratio and were designed to include different levels of difficulty from easy to challenging (see figure 2). the questions, adapted from past spe papers and spn21 mathematics year 8 textbook, include asking students to form ratio of two quantities, finding missing quantities given the ratio, finding the total quantities as well as finding difference in quantities. 1. there are 10 boys and 14 girls in a field trip. what is the ratio of boys to girls? 2. the total number of fruits in a basket is 30. there are 12 apples and the rest are oranges. what is the ratio of the number of apples to the number of oranges? 3. the ratio of green m&m’s to yellow is 2:5. if there are 20 yellow m&m’s, how many green m&m’s are there? 4. the ratio of the weight of wani’s cat to the weight of azam’s cat is 5:7. wani’s cat weighs 20 kg. how much does azam’s cat weigh? 5. the ratio of the number of students who wear glasses to the number of students who do not wear glasses is 2:3. if there are 25 students in the class, how many students do not wear glasses? 6. the ratio of red to green crayons in a box is 3:5. if there are 9 red crayons, what is the total number of crayons in the box? 7. the ratio of the number of dina's comics to the number of alif's comics is 5:2. if dina has 15 comics, how many comics do they have altogether? 8. farhan and azim get paid $49 for washing cars in a day. they split the money in the ratio 4:3. how much more does farhan receive? 9. the ratio of boys to girls at the football game is 5:3. there are 33 girls. how many more boys are there than girls? 10. a rope that is 18 metres long is cut into three strips in a ratio of 2:3:4. how long is the longest piece? figure 2. the 10 items in pre-test and post-tests prior to collecting data at the school of study, the validity and reliability of the test had been previously checked. a test-retest reliability method was administered to a group of twelve students not involved in the main study, produced correlation coefficient of 0.783. this positive correlation between the test and the retest shows that the test questions are reliable in producing stable and consistent results. both descriptive (mean, median, standard deviation) and inferential statistics were used in the analysis of the collected data to answer the above research question, namely through students’ overall scores and item-analysis of correct versus incorrect responses. for the pre-test and post-test, the sample paired t-test was used to analyse the results quantitatively to measure if there is a significant difference in students’ achievement before and after introducing the bar model method. volume 10, no 1, february 2021, pp. 149-160 153 the intervention lessons were delivered over the course of three lessons within a week and each lesson took one hour. the introduction lesson focused on introducing concept underlying bar model in this topic. it included testing whether students know how to evaluate the unknown quantities of given two bars, and constructing and partitioning bars of required quantities (see figure 3). here, the unitary method was the students’ preferred method to find the unknown quantities, where they calculated the value of 1 unit first by dividing appropriately and then once known, they multiplied it by the number of parts of the unknown quantities. similar approach was used in the deconstruction of the bar, where information were extracted from drawn bars. without the mention of the topic ratio, students received proper scaffolding when they transit from drawing bars to word problem. once the students were confident, this skill was transferred to represent different ratios using bars, which was covered in the end of the first and throughout the second lesson. figure 3. questions in lesson 1, focusing on understanding the concept and drawing of bar model in the second lesson, the students continue to represent ratio using bars as well as to solve word problems involving ratio using the bar model method. in this lesson, the importance of having accurate and correct size bars following the information in the word problem was emphasised. students learned to label the model correctly to represent the known and unknown quantities and finally utilised the model appropriately to solve the problem. the practice problems were varied, ranging from simple to slightly more challenging problems (see figure 4). said, & tengah, supporting solving word problems involving ratio through the bar model 154 1. hayati made 20 white chocolate cookies and 16 milk chocolate cookies. what is the ratio of milk chocolate cookies to white chocolate cookies? 2. at a summer camp the ratio of boys to girls was 5:4. if there were 45 boys, how many girls were there? 3. a classroom had 24 glue sticks. if the ratio of glue sticks to glue bottles was 4:3, how many glue bottles did the classroom have? 4. there were 220 girls who watched tennis. some of these girls wore caps. the ratio of the number of girls wearing caps to the number of girls not wearing caps was 4:7. how many girls did not wear caps? 5. bella gave a pocket money of $100 to her daughters, hani and emma, in the ratio 2:3. how much did emma receive? 6. kerrie ordered helmets and basketballs in the ratio 3:8. she ordered 64 basketballs. how many sports items did she order altogether? 7. the ratio of karen's cds to mary's cds is 5:6. if mary has 66 cds, how many cds do they have altogether? 8. azimah bought chicken and beef for a barbecue in the ratio 5:3. if she bought 6 kg of beef, how much chicken did she buy? 9. the ratio of the weight of meg’s cat to the weight of anne’s cat is 5:7. meg’s cat weighs 20 kg. how much more does anne’s cat weigh? 10. nina has a bag of 30 candies. the ratio of strawberry to orange to grape candies in the bag is 3:2:1. how many orange candies are there? figure 4. word problems in lesson 2 with and without prepared bar model in the final lesson, students worked on the provided worksheet to practise solving word problems using the bar model (see figure 5). they were encouraged to work in pairs and then by themselves to solve the given ratio problems. selected students were invited to explain their solutions to the whole class and this was used for class discussions. it was during this lesson as well that any errors demonstrated by the students were identified and corrected on the spot so that students could learn from their mistakes. volume 10, no 1, february 2021, pp. 149-160 155 1. a box of candy has 14 pieces total. if two of the pieces are cherry flavoured, what is the ratio of other flavours to cherry flavored pieces? 2. the ratio of girls to boys in a chess club was 5:4. there were 32 boys. how many girls were there in the club? 3. in a bag of red and green candies, the ratio of red candies to green candies is 3:4. if the bag contains 120 green candies, how many red candies are there? 4. the ratio of girls to boys in a swimming club was 1:3. there were 11 girls. how many total members were there in the club? 5. a teacher had 18 red pens. if the ratio of red pens to blue pens she owned was 3:5, how many pens did she have total? 6. a necklace is made using gold and silver beads in the ratio 3:2. if there are 100 beads in the necklace, how many are gold and silver beads? 7. jack and nina share a reward of $140 in a ratio of 2:5. how much of the total reward does jack get? 8. at an ice cream shop the ratio of chocolate ice cream sold to vanilla ice cream sold is 3:4. if there are 12 chocolate ice cream sold, how many more vanilla ice cream would be sold? 9. in one day a movie store rented out 9 comedies. if the ratio of comedies rented to action movies rented was 1:5, how many more action movies were rented? 10. a truck is carrying pear juice, cherry juice, and apple juice bottles in a ratio of 3:1:3. if there are 16 cherry juice bottles, then how many juice bottles in total are there? figure 5. word problems used in lesson 3 at different complexity 3. results and discussion as seen in figure 6, in the pre-test, only two students managed to achieve the maximum mark of 10 while there were eleven students scored zero. more than half of the students did not score past mark of 5, with only fifteen students obtaining marks 5 and above. these low scores indicated students’ poor performance and ability in solving word problems involving ratio to begin with. figure 6. bar chart of students' overall marks in the pre-test vs. post-test in the post-test, overall improvement can be seen in the number of students obtaining higher marks, with considerable reduction in the lower end. only two students obtained the lowest mark of 2 whereas twelve scored the maximum mark of 10. the bar chart also shows that a majority of the students successfully scored 5 marks and above, with just three getting 11 2 1 3 1 2 4 2 2 3 2 0 0 2 1 0 1 3 1 8 5 12 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 n o o f s tu d e n ts overall marks pre-test post-test said, & tengah, supporting solving word problems involving ratio through the bar model 156 mark 0-4. a total of twenty-five students obtained high marks of 8 and above, making up 76% of the sample. table 1. descriptive statistics of pre-test and post-test of the two classes test n minimum maximum mean std. dev class x pre-test 18 0 10 6.17 2.995 post-test 18 7 10 9.17 1.043 class y pre-test 15 0 6 1.13 1.995 post-test 15 2 10 6.73 2.738 overall pre-test 33 0 10 3.88 3.603 post-test 33 2 10 8.06 2.318 table 1 summarises the performance of each class as well as the whole group’s pretest and post-test. students in the study school were grouped into classes according to their abilities or more specifically, their primary year 6 psr results. class x was supposedly made up of students of high to average abilities, whereas, class y had students of low academic levels. the mean marks of the pre-test is reflective of the classes’ ability. the mean mark of pre-test of class x was 6.17, just slightly over the marginal mark of 5. the marks in this class ranged from 0 to max 10. as expected, the low mean mark of class y (m = 1.13) reflected the low ability students’ struggles in solving word problems involving ratio. the highest mark from this class y is just 6. both classes showed improvement in their class post-test result. the mean score increase in class x to 9.17 implied great achievement overall. the small range of marks between 7 and 10 marks being the minimum and the maximum respectively indicated all the students in class x scored in the high range. likewise, class y also showed great improvement, with post-test mean mark of 6.73. a majority of the students obtained good mark with only three students who did not manage to score past 5 marks. as a whole, the sample group produced mean mark of 3.88 for the pre-test, and increased to 8.06 in the post-test. before the paired sample t-test for the students’ marks was conducted, the four assumptions for the test were first inspected. firstly, the dependent variable, which is the students’ marks, was continuous as it measured from 0 to 10. due to the students’ scores spread almost evenly within this range, there was no outliers. thirdly, the same group of students was used for the pre-test and the post-test. lastly, through visual inspection on the bar graph of the differences between the paired marks, the data was approximately normally distributed. with these conditions fulfilled, a paired-sample t-test was carried out to determine significance of the mean difference. the paired sample t-test was used to test the following hypotheses: h0 = there is no significant difference between the pre-test and post-test results. h1 = there is a significant difference between the pre-test and post-test results. table 2. paired sample t-test of pre-test and post-test for overall sample (n=33) paired differences t df sig (2tailed) mean std. dev std. error mean 95% ci of difference lower upper prevs post test -4.182 2.973 0.518 -5.236 -3.128 -8.079 32 0.000 the t-test result revealed that there was a significant difference in the students’ mean marks (t = -8.079) at 𝑝 < 0.05 level (see table 2). hence, the null hypothesis, h0, was volume 10, no 1, february 2021, pp. 149-160 157 rejected indicating that there is a significant difference between the pre-test and post-test results. this implied that the lesson intervention of the bar model method had successfully enhanced the students’ performance in solving word problems involving ratio. this result is consistent with studies by gani et al. (2019), madani et al. (2018), mahoney (2012), and timah (2006), where the use of the bar model successfully improved the students’ problem solving performance in their respective topics. while the paired sample t-test of the overall sample has proven that the bar model method enhanced the sample students’ performance in the problem solving involving ratio, the analysis of post-test results particularly shows that students with lower academic abilities in class y could have benefited greatly from the bar model strategy, given the bigger increase in mean from pre-test to post-test of class y from 1.13 to 6.73, compared to class x with just mean increase of 3. however, due to the limitation of small number of students in each class (15 in class x and 18 in class y), t-test could not be used for significant difference test. from only 2 of them scoring 5 marks or more in the pre-test, this figure rose to 12 students in the post-test. a student from that class only obtained 1 mark in the pre-test but went on to obtain a full 10 marks in the post-test. the impact of the bar model method on the lower ability groups in solving word problems involving ratio is immense, as similarly found in goh (2009), where the bar model method especially helped the weaker students in her study. likewise, the bar model strategy helped all the mixed ability students in class x to do well in the ratio test with a mean mark of almost 10. furthermore, students performed significantly better in the post-test with thirty students scored 5 or more. this shows that the bar model is a problem solving heuristic that can cater to varying abilities of students but most significantly for the low abilities. a comparison of the number of correct responses against the number of incorrect responses in the pre-test and post-test is shown in figure 7. in the pre-test, there were clearly more incorrect responses than correct responses committed in almost all of the questions. except for q4 and q5, more than half of the students could not solve the problems posed. q2 and q8 have the highest number of incorrect responses (n = 27). q4 has the lowest number of students giving incorrect responses (n = 15), yet still considered high as almost half of the students failed to solve the problem. (a) (b) figure 7. comparison between the number of correct responses and incorrect responses in the pre-test (a) and post-test (b) looking at the post-test bars in figure 7(b), there is clearly a great improvement in the number of correct responses in each question after the students learned the bar model 14 6 15 18 17 15 15 6 11 11 19 27 18 15 16 18 18 27 22 22 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 1 0 n o o f s t u d e n t s item number correct incorrect responses 30 17 27 30 28 31 31 22 26 24 3 16 6 3 5 2 2 11 7 9 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 1 0 n o o f s t u d e n t s item number correct incorrect responses said, & tengah, supporting solving word problems involving ratio through the bar model 158 method. more than half of the students could solve all the problems correctly in the post-test as opposed to in the pre-test. questions 1, 6, 7 and 8 have the highest increase in correct responses by sixteen students. meanwhile, questions 2 and 5 have the smallest increase (eleven students) in the number of correct responses but this is still regarded as high. q2 which started with lowest correct responses in the pre-test ended up also being the lowest correct responses in post-test. overall, there is a positive indication of development in the students' problem-solving skills on ratio. the majority of the correct responses applied correct use of bar model in their answers in post-test, compared to keyword and unattempted questions in a pre-test. it suggests that teachers can consider different strategies for different topics such as: applying visual approach by using the graphic organizer in tackling word problem (sian et al., 2016); implementing thinking aloud pair and pólya in word problem approach (simpol et al., 2017); strengthening pattern discovery skills in the problem-solving task (tengah, 2011). the use of the bar model is also one of the strategies adopted in singapore classrooms taught as early as possible in elementary schools, thus making singaporean students one of the best math problem solvers as reported in timms and pisa. besides, previous empirical studies have shown the bar model's effectiveness in improving students' performance in word problems in singapore and brunei (gani et al., 2019; madani et al., 2018; mahoney, 2012; timah, 2006). all of this is a good justification for introducing the bar model to students in the country to solve word problems. this study particularly adds information to the existing body of knowledge regarding using the bar model in brunei. the positive results are evidence that the bar model strategy can be implemented into the local classrooms. it is also hoped that this study can encourage and convince more teachers to start using this type of diagram representation as part of their instructional approach as recommended by the new curriculum (cdd, 2006a; 2006b). furthermore, this bar model is very flexible and can be used across many topics such as whole numbers, fractions, and percentages. the bar model's simple nature, which does not consume time and money to teach, makes it a problem-solving heuristic that every mathematics teacher should practice as early as in primary levels across the country. early exposure would help provide a strong solid foundation in mathematics for young children, according to one of the intended learning outcomes of the new mathematics curriculum (cdd, 2006a; 2006b). moreover, the bar model's visual nature enables teachers to identify some of the difficulty’s students have with word problems. that information would be useful for the teaching of word problems (ng & lee, 2005). 4. conclusion this study was to examine the effects of using the bar model method on secondary students’ performance in solving word problems involving ratio. the results of the paired sample t-test revealed that there was a significant difference in the students’ mean marks between the pre-test and post-test results. furthermore, the results showed that students performed significantly better after they learned the bar model method as indicated by higher marks and more correct responses. weaker students particularly benefited from the introduction of the bar model as evident in the number of students scoring 5 marks or more in the post-test. hence for this study, the bar model successfully enhanced the lower secondary students’ performance in solving word problems involving ratio. it is evident from this empirical study that the bar model method is capable of helping students of different learning abilities and it can be an effective alternative strategy to solving word problems involving ratio. volume 10, no 1, february 2021, pp. 149-160 159 references arcavi, a. (2003). the role of visual representations in the learning of mathematics. educational studies in mathematics, 52(3), 215-241. https://doi.org/10.1023/a:1024312321077 cdd. (2006a). mathematics syllabus for lower primary school. gadong: curriculum department, ministry of education, brunei darussalam. cdd. (2006b). mathematics syllabus for upper primary school. gadong: curriculum department, ministry of education, brunei darussalam. charles, r. (2011). solving word problems: developing quantitative reasoning. retrieved from http://assets.pearsonschoolapps.com/playbook_assets/matmon110890charles_sw p_revise_ebook1.pdf chin, k. s., & clements, m. a. k. (2001). o-level students’ understanding of lower secondary school geometry. in energising science, mathematics, and technical education for all: proceedings of the sixth annual conference of the department of science and mathematics education, 213-222. denis, m. (1991). imagery and thinking. in imagery and cognition, 103-131. new york: springer. https://doi.org/10.1007/978-1-4684-6407-8_4 gani, m. a., tengah, k. a., & said, h. (2019). bar model as intervention in solving word problem involving percentage. international journal on emerging mathematics education, 3(1), 69-76. https://doi.org/10.12928/ijeme.v3i1.11093 goh, s. p. (2009). primary 5 pupils' difficulties in using the model method for solving complex relational word problems (doctoral dissertation). nanyang technological university, singapore. greer, b. (1997). modelling reality in mathematics classrooms: the case of word problems. learning and instruction, 7(4), 293-307. https://doi.org/10.1016/s09594752(97)00006-6 hajar, s. (2015). brunei sees improved results in psr exams. borneo bulletin. jalil, i. (2015). psr maths results still low: moe. the brunei times. khalid, m., & tengah, m. k. a. (2007). communication in mathematics: the role of language and its consequences for english as second language students. in third apec-tsukuba international conference innovation of classroom teaching and learning through lesson study iii—focusing on mathematical communication, criced, university of tsukuba, 1-8. kho, t. h., yeo, s. m., & lim, j. (2009). the singapore model method for learning mathematics. marshall cavendish education. liu, y. m., & soo, v. l. (2014). mathematical problem solving – the bar model method. scholastic education international. singapore. madani, n. a., tengah, k. a., & prahmana, r. c. i. (2018). using bar model to solve word problems on profit, loss and discount. journal of physics: conference series, 1097(1), 012103. https://doi.org/10.1088/1742-6596/1097/1/012103 https://doi.org/10.1023/a:1024312321077 http://assets.pearsonschoolapps.com/playbook_assets/matmon110890charles_swp_revise_ebook1.pdf http://assets.pearsonschoolapps.com/playbook_assets/matmon110890charles_swp_revise_ebook1.pdf https://doi.org/10.1007/978-1-4684-6407-8_4 https://doi.org/10.12928/ijeme.v3i1.11093 https://doi.org/10.1016/s0959-4752(97)00006-6 https://doi.org/10.1016/s0959-4752(97)00006-6 https://doi.org/10.1088/1742-6596/1097/1/012103 said, & tengah, supporting solving word problems involving ratio through the bar model 160 mahoney, k. (2012). effects of singapore's model method on elementary student problem solving performance: single subject research (doctoral dissertation). northeastern university. ng, s. f., & lee, k. (2005). how primary five pupils use the model method to solve word problems. the mathematics educator, 9(1), 60-83. ng, s. f., & lee, k. (2009). the model method: singapore children's tool for representing and solving algebraic word problems. journal for research in mathematics education, 40(3), 282-313. piaget, j. (1971). the psychology of intelligence. boston: routledge and kegan. piaget, j., & inhelder, b. (1966). mental imagery in the child. new york: routledge & kegan paul. polya, g. (2004). how to solve it: a new aspect of mathematical method (vol. 85). new jersey: princeton university press. pungut, m. h. a., & shahrill, m. (2014). students’ english language abilities in solving mathematics word problems. mathematics education trends and research, 2014, 111. raimah, m. (2001). an investigation of errors made by primary 6 pupils on word problems involving fractions. dissertation. universiti brunei darussalam. saman, a. (2000). investigating understanding by primary six pupils of word problems involving multiplication and division. dissertation. universiti brunei darussalam. sian, k. j., shahrill, m., yusof, n., ling, g. c. l., & roslan, r. (2016). graphic organizer in action: solving secondary mathematics word problems. journal on mathematics education, 7(2), 83-90. https://doi.org/10.22342/jme.7.2.3546.83-90 simpol, n. s. h., shahrill, m., li, h. c., & prahmana, r. c. i. (2017). implementing thinking aloud pair and pólya problem solving strategies in fractions. journal of physics: conference series, 943(1), 012013. https://doi.org/10.1088/17426596/943/1/012013 tengah, k. a. (2011). using simplified sudoku to promote and improve pattern discovery skills among school children. journal of mathematics education at teachers college, 2(1). https://doi.org/10.7916/jmetc.v2i1.710 timah, u. (2006). primary 5 pupils' performance on mathematical word problems using model-drawing/box-diagram strategy. dissertation. universiti brunei darussalam. vaiyatvutjamai, p., & clements, m. a. (2004). analysing errors made by middle-school students on six linear inequations tasks. globalisation trends in science, mathematics and technical education 2004, 173-182. van garderen, d. (2006). spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. journal of learning disabilities, 39(6), 496506. https://doi.org/10.1177/00222194060390060201 van garderen, d., & montague, m. (2003). visual‐spatial representation, mathematical problem solving, and students of varying abilities. learning disabilities research & practice, 18(4), 246-254. https://doi.org/10.1111/1540-5826.00079 https://doi.org/10.22342/jme.7.2.3546.83-90 https://doi.org/10.1088/1742-6596/943/1/012013 https://doi.org/10.1088/1742-6596/943/1/012013 https://doi.org/10.7916/jmetc.v2i1.710 https://doi.org/10.1177/00222194060390060201 https://doi.org/10.1111/1540-5826.00079 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p263-274 263 the common errors in the learning of the simultaneous equations pg. mohammad adib ridaddudin pg. johari1, masitah shahrill*2 1maktab sains paduka seri begawan sultan, ministry of education, brunei darussalam 2sultan hassanal bolkiah institute of education, universiti brunei darussalam, brunei darussalam article info abstract article history: received sep 12, 2020 revised sep 20, 2020 accepted sep 28, 2020 the purpose of this study is to understand the causes of common errors and misconceptions in the learning attainment of simultaneous equations, specifically on linear and non-linear equations with two unknowns. the participants consisted of 30 year 9 students in one of the elite government schools in brunei darussalam. further analyses of their work led to the categorisation of four factors derived from the recurring patterns and occurrences. these four factors are complicating the subject, wrong substitution of the subject, mathematical error and irrational error in solving the question. these factors usually cause participants to make errors or simply misconceptions that usually led them to errors in solving simultaneous equations. keywords: simultaneous equations, errors, misconceptions, secondary mathematics, brunei darussalam copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: masitah shahrill, sultan hassanal bolkiah institute of education, universiti brunei darussalam, tungku link road, gadong, bandar seri begawan, brunei darussalam. email: masitah.shahrill@ubd.edu.bn how to cite: pg. johari, p. m. a. r., & shahrill, m. (2020). the common errors in the learning of the simultaneous equations. infinity, 9(2), 263-274. 1. introduction simultaneous equation is often perceived as a difficult and demanding topic to deal with requiring a lot of algebraic processes to find the solution (ugboduma, 2012). the nature of it being heterogeneous and often vigorous is why most participants have little to no interest in studying or even attempting the question during their test or examination (ugboduma, 2006). this is particularly true in brunei darussalam (hereinafter, referred to as brunei) where rote memorisation has normally been the common way of teaching and learning mathematics (khalid, 2006), only to be used for passing certain tests or examinations (shahrill, 2009, 2018; salam & shahrill, 2014; shahrill & clarke, 2014, 2019; zakir, 2018). this prevents participants to utilise and relate any lessons learned from the class to the real-life situation. moreover, the lack of understanding in learning mathematics due to rote memorisation usually led participants to forget most of the knowledge taught after going through their said tests or examinations, which usually daunts them once they have to go through it again, due to the repeating nature of mathematics (matzin et al., 2013; https://doi.org/10.22460/infinity.v9i2.p263-274 pg. johari & shahrill, the common errors in the learning of … 264 shahrill et al., 2013). this is why for most students, they have negative attitude towards mathematics rendering it as one of the most challenging subjects for students in brunei generally (chua, et al., 2016; khoo et al., 2016). hence, an intervention to alleviate this negative trend is required. although there are some literature studies that investigate the matter of simultaneous equations (ugboduma, 2006, 2012; yunus et al., 2016; nordin et al., 2017), this study in particular focuses on linear and non-linear equations in two unknowns. this makes the investigation of this material to be more significant, at least in the opinion of the researchers. this is because simultaneous equations is an integral part of algebra which is needed in most mathematical topics or even other learning area of the 21st century such as computer, sciences or even engineering to name a few. nevertheless, simultaneous equations are usually one of the challenging topics to be taught in school as participants usually struggle to understand the concepts and just prefer to memorise steps and methods for the sake of getting through tests or exams. accordingly, this really creates a question whether the current method of teaching is ineffective and should a different method of teaching be required as an alternative of teaching simultaneous equations, particularly of linear and non-linear equations in two unknowns in brunei. yunus et al. (2016) also pointed out that most teachers in brunei teaches simultaneous equation by telling which only provide participants with instrumental understanding in applying the rules of algebra in solving simultaneous equations, neglecting the relational understanding which is more helpful in participants’ understanding when also present. yunus et al. (2016) then further mentioned that, because of this, participants usually interpret teacher’s instruction wrongly due to failure in understanding participants’ thought processing mechanism. therefore, a learner-centred approach is recommended in order to minimise any misconceptions that might arise. this is in-line with the thoughts from ugboduma (2006, 2012) that mentioned good methodologies are required to help stimulate participants in enhancing their understanding of simultaneous equations. he further stated that a carefully designed methodology that is adopted by an adept teacher is a key factor for participants to improve their learning. jaggi (2006) clearly impart that a statement of equality is defined as an equation. we call it as an identity when the statement of equality is true for all the unknown values involved, denoted by the symbol ≡, and we call it as conditional equation using the symbol =, when the statement of equality is only true for certain values of the unknown qualities. häggström (2008) defined an act of equaling by which a state is being equal as an equation. this formal statement of equivalence in terms of mathematical logical expressions is often denoted by the symbol of equal sign, =. a mathematical statement that has two same values is an equation. for example, 2+1=4−1. häggström (2008) later explained that when two events are done, occurring or happening at the same time, it is then called simultaneous. therefore as stated by ugboduma (2012), if we have two or more equations that are true at one end, satisfying the same values of involved unknowns, then we can call it as simultaneous equations. for a straight-line equation that has two variables, the number of solutions will be infinite. if we denote the first variable as 𝑥 and the other variable as 𝑦, then any one of the solutions for 𝑥 can be substituted into the straight-line equation giving its corresponding 𝑦 value. however, if two of such equations are simultaneously calculated together then there might be only one set of solution of 𝑥 and 𝑦 that satisfy both equation simultaneously (ugboduma, 2012). for simultaneous equations of linear and non-linear equations in two unknowns, this amount of solution that can satisfy both equations simultaneously increase depending on the degree of the non-linear equations. for instance, if the non-linear part of volume 9, no 2, september 2020, pp. 263-274 265 the equation is a quadratic equation, then the solutions should come in two sets or one repeating solution. as mentioned by yahya and shahrill (2015), it should be easier to improve participants’ understanding in their future endeavours in solving algebraic problems, which is crucial for simultaneous equations, if the reasons of their workings can be identified. consequently the purpose of this present study is to understand the causes of common errors and misconceptions made by participants in their attainment of simultaneous equations, particularly of linear and non-linear equations in two unknowns. this is so that an alternative method of teaching can be proposed to minimise these misconceptions and errors as much as possible by way of analysing and thinking. another reason is to investigate the causes of common errors and misconceptions that participants keep on committing in attaining the learning of simultaneous equations of linear and non-linear equations in two unknowns, especially for mid to low level ability participants. although there are a lot of studies that cover on types common errors and misconceptions (sarwadi & shahrill, 2014), it is hoped that in identifying the causes of it may help participants in preventing in committing those common errors and misconceptions so participants can have a better understanding, attitude and mindset in the process of learning the topic. importantly, with the formation of this study, we hope that further contribution can be made on the literature concerning how simultaneous equation is taught in brunei. the authors also feel the necessity of the study since upon reviewing literature, particularly in simultaneous equations of linear and non-linear equations in two unknowns, almost none surfaced. its instant existence in literature can be used as a doorway to pave for future studies in providing an alternative way of teaching simultaneous equations in mathematics lessons, particularly of linear and non-linear equations with two unknowns. as such, this present study is guided by the research question “what are the common errors and misconceptions made by participants in their learning of the simultaneous equations?” 2. method a total of 30 participants participated for this study taken from two year 9 classes in one of the elite government schools in brunei. the level of ability of both classes ranges from medium to low ability, mostly being medium. both classes have the required algebraic and arithmetic skills to do simultaneous equations of linear and non-linear equations with two unknowns, such as linear equation manipulation and solving quadratic equations. a test was administered to the participants that contained three item questions chosen from a pool of questions (refer to table 1) validated by experienced mathematics teachers. the questions chosen should test participants in various ways on solving simultaneous equation of linear and non-linear equations in two unknowns, such as choosing a proper subject to be used for the substitution method or how they can avoid complications of simultaneous equations by simplifying equations further before solving. the validity was assessed through judgmental methods collected from comments and opinions of experienced mathematics teachers from the school mathematics department. it was also assessed to a specification method using the first four levels of blooms’ taxonomy namely remembering, understanding, applying and analysing. pg. johari & shahrill, the common errors in the learning of … 266 table 1. list of questions for the test with item number item number questions 1 𝑦 = 𝑥𝑦 + 𝑥2 − 9 𝑦 = 3𝑥 − 1 2 4𝑥 + 𝑦 = −8 𝑥2 + 𝑥 − 𝑦 = 2 3 𝑥 3 − 𝑦 2 + 3 = 0 3 𝑥 + 2 𝑦 − 1 2 = 0 3. results and discussion 3.1. results the nature of item 1 (𝑦 = 𝑥𝑦 + 𝑥2 − 9, 𝑦 = 3𝑥 − 1) consisting of one simple linear equation and a quadratic equation is quite straightforward in relative to the other items. for the linear equation, the variable 𝑦 has already been arranged as the subject. both equations are not in a fraction form making it easier for algebraic manipulations. ideally this should be a straightforward task of substitution, expansions, simplifications and quadratic equation solving, skills that all participants already acquired. there are two common errors that were made most by participants for this item. the first one is where participants try to make 𝑥 as the subject from the linear equation. while this is actually not a form of error in any ways, making 𝑥 as the subject in this case will give a fractional subject of 𝑥 = 𝑦+1 3 which usually will result in an error due to complication that it will produce. typically, a lot of errors were usually made when the subject is made into a complicated fraction form (low et al., 2020). figure 1 exhibits a sample of a student 1’s work dealing with said fractional subject. this unnecessary step causes the question to become more complicated in a form of fractions, expanding fractions, and fractional algebra manipulation, to name a few. these unnecessary complications usually increase the risk of participants being careless as shown by this participant’s work, where he did not expand ( 𝑦2+1 3 ) × 3 3 correctly which then leads him to get the wrong final quadratic equation. one can assume that this is just an overlooked error made by the participant since the other mathematical part of his workings was done quite well. one mark was given for his correct although unnecessary substitution of subject 𝑥, and another one mark is given for his valid attempt in solving the final quadratic equations. volume 9, no 2, september 2020, pp. 263-274 267 figure 1. sample of student 1’s work for item 1 secondly, there are quite a number of participants who made an error of failing to substitute their subject 𝑦 = 3𝑥 − 1 into both side of the quadratic equation 𝑦 = 𝑥𝑦 + 𝑥2 − 9. this will make their equation impossible to solve since both variable 𝑥 and 𝑦 still exist, defeating the purpose of substitution, which is to eliminate one out of the two variables. from the sample work by one participant shown in the figure 2, the participant only substituted 𝑦 = 3𝑥 − 1 into the right-hand side of 𝑦 = 𝑥𝑦 + 𝑥2 − 9 which in the end gives her the final equation of 𝑦 = 4𝑥2 − 𝑥 + 9. student 2 then forces her way through in solving the final quadratic equation even though both variables still exist. this then resulted a wrong answer with no marks, since there were no opportunities for the marker to give one throughout the workings. pg. johari & shahrill, the common errors in the learning of … 268 figure 2. sample of student 2’s work for item 1 for item 2 (4𝑥 + 𝑦 = −8, 𝑥2 + 𝑥 − 𝑦 = 2), almost half of the participants managed to get full marks, reflecting the easier nature of the question. the common error made by participants, who mostly scored 2 marks for this item, is very similar to the error made in item 1. it is using the fractional subject for substitution, which as mentioned before is not an actual error, but usually leads to mathematical errors since it complicates workings. figure 3 below shows another working of a participant where he makes the variable 𝑥 as the subject, i.e. 𝑥 = −8−𝑦 4 . this subject is not only fractional in nature but also contains a lot of negatives sign, which usually can cause carelessness that leads to complications. however, the error made by student 3 is the expansion of ( −8−𝑦 4 ) 2 , where the denominator 4 is not expanded. this can be due to a simple misstep or lack of indices skills or knowledge. one can assume that these two factors can be easily eliminated if variable 𝑦 was made as the subject, since the participant will then have a fraction-less subject leading to a straightforward expansion. this error then leads to a wrong solution. student 3 was awarded with 2 marks for a correct substitution and a valid attempt on solving the final quadratic equation. figure 3. sample of student 3’s work for item 2 volume 9, no 2, september 2020, pp. 263-274 269 for item 3 ( 𝑥 3 − 𝑦 2 + 3 = 0, 3 𝑥 + 2 𝑦 − 1 2 = 0), although unanimously, all of the participants were unable to score more than 1 mark, some attempts can be seen to have the correct idea in generally solving the simultaneous equations. however, most participants lack the skills in manipulating algebraic fractions that led to errors that hindered them to get the required final quadratic equations, resulting in the severe loss of marks. this also caused a lot of participants to quit trying after their working seems to get very complicated. the sample work of student 4 in figure 4 reflects on this. after choosing the subject 𝑥 = 3𝑦 2 − 9, a correct substitution into the second equation yield her 1 mark. she then proceeds to simplify the equations by trying to get rid of the fraction. in doing so, she made an error by dividing the whole equation with 2 instead of multiplying. as the equation goes peculiar, she then stops trying. it can also be observed that she failed to see 12𝑦 2 as 6𝑦, which can then make her equation simpler. one can perceive from this that lack of critical thinking was present when attempting the question. figure 4. sample of student 4’s work for item 3 then some attempts, especially from participants who managed to score 1 mark, were quite decent relative to the challenging nature of the question. figure 5 shows an example of this. student 5 can be seen to have a very good algebraic manipulation skills but made an error in expanding (2 − 1 2 𝑦)(−18 + 3𝑦) which then made him lose 4 marks. the 1 mark was given for a valid attempt to find the solution from his wrong working. errors were generally made when the subject is made into a complicated fraction form, which seems to be the case here. once again one can assume that student 5 may obtain more than 1 mark if the error can be avoided by getting the fraction in the first instance. item 3 can be categorised as a challenging due to the fact that both are in a fraction state of form. however, the difficulty can be lowered if participants can change the fraction nature to whole number by multiplying it with the lcm of the denominator. from there the question will then be on par as items 1 and 2. pg. johari & shahrill, the common errors in the learning of … 270 figure 5. sample of student 5’s work for item 3 to achieve in-depth insights of the students’ work for further analysis, the following four categories of factors were derived from the recurring patterns and occurrences, which affects the students’ work. these factors usually cause participants to make errors or simply misconceptions that usually led them to errors. these four factors are: complicating the subject (cs), wrong substitution of the subject (ws), mathematical error (me) and irrational error in solving the question (i). briefly, (cs) is a factor when a student complicates a simple subject that then may lead to producing errors. for example, in item 1, instead of using the simple subject 𝑦 = 3𝑥 − 1, the student might complicate it by using 𝑥 = 𝑦+1 3 instead. this may cause errors further in their workings. (ws) is a factor when participant error in substituting their subject into another equation. this can be either literally substituting their subject wrongly or only substituting their subject partially, which this factor will focus on solely. for example, substituting 𝑦 = 3𝑥 − 1 into the right-hand side of 𝑦 = 𝑥𝑦 + 𝑥2 − 9 only, will not make the elimination of variable 𝑦 complete, since there will still be variable 𝑦 on the left-side of the equation. hence, making it impossible to solve the simultaneous equation. (me) is self-explanatory where participants made simple mathematical errors such as expanding, rearranging, changing signs or algebraic manipulations, to name a few. it can be due to carelessness of the participant or lack of mathematical skills. (i) is when participants have no understanding about the question and in solving it. usually this can be seen when participant tend to give unreasonable solution or answering it as a different topic such as solving both equations from the simultaneous equations volume 9, no 2, september 2020, pp. 263-274 271 independently or solving it as another topic, for example using 𝑏2 − 4𝑎𝑐 from the topic discriminant of intersections. figure 6. factors contributing to participants’ marks on the test the bar graph from figure 6 shows that for item 1, a large number of 12 participants changed their subject from 𝑦 = 3𝑥 − 1 into 𝑥 = 𝑦+1 3 . out of these 12 participants who committed cs, 9 of them yield 0 mark. this can tell us that indeed complicating the initial subject usually leads to errors and an effort to prevent participant from doing cs might improve their marks. then 6 participants committed the ws where most of them failed to substitute the subject properly. only 2 of these participants scored 0 mark while others had a good attempt acquiring 1 or 2 marks. a small number of irrational errors tell us that most participants have the general idea on solving the question even though the topic has not been covered with them. for item 2, there is still a moderate fraction of participants committing the cs, although only 2 out of these 8 participants scored 0 marks indicating a good attempt. this might be due to the simpler nature of the algebraic equation. no ws was recorded since the equation was designed for a one-sided substitution only, unlike item 1. most participants were able to score well on this question, but there is an interesting case where a participant who scored full mark on item 1 and item 2 committed a cs on item 2 but not on item 1. the participant choose variable 𝑥 as the subject regardless how difficult it can be. this can tell us that although his mathematical skills are very high, a critical thinking might be lacking. one can assume that the reason 𝑥 is used as the subject on both occasions might be because he is used to it from his previous lower level education, where 𝑥 is always used as the starting subjects in classroom or exams. finally, item 3 shows that most participants committed the me, understandably due to the complicated nature of the fraction form. this though can be avoided since the question is designed in such a way that the fraction can be get rid of and changed into a much simpler fraction-less equation by a simple algebraic manipulation, which all of the participants should already possess the skill to by now. 3.2. discussion based on the in-depth analysis of the repeating pattern found in the participants’ work, four main factors were detected in affecting participants’ test as mentioned earlier. the three major factors were complicating the subject needed for substitution method (cs), 12 6 7 3 8 0 8 2 0 0 18 6 0 5 10 15 20 cs ws me i item 1 item 2 item 3 pg. johari & shahrill, the common errors in the learning of … 272 making error while substituting their subject into the other equation (ws) and simple mathematical error (me). the first factor, while mathematically correct, was committed by a total of 20 participants for both items 1 and 2 of the test, which mostly led participants in making mathematical error for their subsequent workings, scoring them a very low mark in average. the second factor was mostly found in item 1, where six participants failed to substitute their subject of 𝑦 = 3𝑥 − 1 to the both side of 𝑦 = 𝑥𝑦 + 𝑥2 − 9 rendering their following workings wrong. both of these factors were perceived by the researchers as misconceptions believed due to the lack of understanding on the meaning of the mathematical process in solving the simultaneous equations, which may have resulted them to be rigidly stuck in their sole method, regardless of how difficult it was for them. the third factor comprises the highest frequency count out of the other factors (33 total for all items), where participants committed the mathematical errors. although this result is understandable due to the fact that most participants have the mathematical ability of lower to medium, the mathematical part of the question was designed and allowed to be simpler if certain mathematical skills were to be applied such as simplification, changing fractions to whole numbers or changing negative coefficient of quadratic equations to positive before solving it. failure in doing it usually leads to participants having to deal with complicated equations or forcing them to commit errors due to carelessness. these three factors are believed to contribute to the poor results obtained by almost all participants during the test, where 28 out of the 30 them only managed to score marks of below average out of the total 15 marks, where 2 marks being the mode of the result. consequently, rohmah and sutiarso (2018) mentioned that a weak prior knowledge is one of the major problems in solving simultaneous equations. however, the participants of both classes, who are mostly of medium ability participants, managed to score above average marks if not excellent, even though they were not taught on how to solve the simultaneous equations. 4. conclusion simultaneous equations, especially linear equation versus non-linear equation, with its vigorous and heterogeneous nature always intimidate participants in learning it wholeheartedly. they perceive it as a subject that is very difficult to follow rather than something that is yet to be fully understood. this incomplete understanding usually leads them to make misconceptions and common errors along the line that further hinder their study. one of the major factors that resulted from this study was misconception in making only 𝑥 as the main subject regardless how difficult it can render their subsequent workings, although other variable as subject can offer much simpler workings. another factor is failing to use their understanding in simplifying equations to achieve simpler mathematical workings in avoiding complications and careless mistakes. it is imperative that we understand the causes of common errors and misconceptions made by participants in their attainment of simultaneous equations, particularly of linear and non-linear equations in two unknowns. this way we may be able to minimise these misconceptions and errors as much as possible and to analyse and reason on every steps taken in calculating that leads to the attainable correct answer. volume 9, no 2, september 2020, pp. 263-274 273 references chua, g. l. l., shahrill, m., & tan, a. 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(2015). the strategies used in solving algebra by secondary school repeating students. procedia – social behavioural and sciences, 186, 1192-1200. https://doi.org/10.1016/j.sbspro.2015.04.168 yunus, d. h. r. p. h., shahrill, m., & abdullah, n. a., & tan, a. (2016). teaching by telling: investigating the teaching and learning of solving simultaneous linear equations. advanced science letters, 22(5/6), 1551-1555. https://doi.org/10.1166/asl.2016.6676 zakir, n. (2018). the impact of educational change processes in brunei preschools: an interpretive study. unpublished doctoral dissertation, university of sheffield, united kingdom. https://doi.org/10.34044/j.kjss.2019.40.2.06 https://doi.org/10.5539/ies.v6n10p39 https://www.globalacademicgroup.com/journals/knowledge%20review/ugboduma.pdf https://www.globalacademicgroup.com/journals/knowledge%20review/ugboduma.pdf https://doi.org/10.4314/gjedr.v11i2.8 https://doi.org/10.1016/j.sbspro.2015.04.168 https://doi.org/10.1166/asl.2016.6676 http://etheses.whiterose.ac.uk/21197/ http://etheses.whiterose.ac.uk/21197/ http://etheses.whiterose.ac.uk/21197/ sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p53-68 53 mathematical trauma students’ junior high school based on grade and gender ayu faradillah, leha febriani* universitas muhammadiyah prof. dr. hamka, indonesia article info abstract article history: received sep 27, 2020 revised nov 23, 2020 accepted jan 5, 2021 mathematical trauma is a mental condition of students caused by experiences that make it difficult or emotionally with mathematics. this study analyzes students’ trauma of mathematics based on grade and gender by using survey research methods and data processing using winsteps. the subjects of the research were 204 students in every grade in junior high school. the questionnaire consisted of 20 statements. this instrument was developed measuring mathematical trauma. it consisted of three indicators: difficult to adapt to the environment, saturated with the learning system that is done, and difficult to get along and organize themselves. based on the rasch analysis, for the item, two items (i10 and i7) are a misfit, and the rest items are fit. the items were suitable for 147 respondents (72.06%). based on the wright maps table on winsteps, the percentage of male students who are very traumatized by mathematics is more than female students. meanwhile, based on grade, students from grade 8 are more very traumatized by mathematics than students from grade 7 and grade 9. furthermore, based on a questionnaire, students prefer to class condition that made very traumatized by mathematics. keywords: mathematical trauma, mathematics, rasch model, trauma copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: leha febriani, department of mathematics education, universitas muhammadiyah prof. dr. hamka, jl. tanah merdeka no. 20, ciracas, jakarta timur, dki jakarta 13830, indonesia email: lehafebri21@gmail.com how to cite: faradillah, a., & febriani, l. (2021). mathematical trauma students’ junior high school based on grade and gender. infinity, 10(1), 53-68. 1. introduction the trauma is an emotional response because of anxiety from bad experiences that have occurred and give a negative impact on individual personality and social environment (clayton, 2020; luyten et al., 2020; williams et al., 2018). in the us, 40% of students have experienced of traumatic stress in their lives according to the national child traumatic stress network (brunzell, stokes, & waters, 2016). experiences that make trauma easier to remember, if always remembered it, it will have an impact on daily life that causes stress. a negative impact on emotional that allows increased problems in schools if students are have high levels of stress and trauma (mendelson, tandon, o’brennan, leaf, & ialongo, 2015). school is a potentially critical place to provide services to overcome the trauma experienced by children (chafouleas, koriakin, roundfield, & overstreet, 2019). trauma is related to students' understanding and occurs when students has a very difficult experience or https://doi.org/10.22460/infinity.v10i1.p53-68 faradillah & febriani, mathematical trauma students’ junior high school … 54 emotional or mathematical disturbances, by allowing them to have a long-term influence on life, including the lives of students as students and users of mathematics (lange & meaney, 2011). in fact, there are many students who do not like mathematics, the biggest factor is because students feel mathematics is a difficult subject, therefore many students avoid mathematics. there are three indicators of learning trauma, which are difficult to adapt to the environment, saturated with the learning system that is done, and difficult to get along and manage themselves (romli, 2017). one potential trauma experienced by students is when math problems that require completion steps are made homework, especially if the student is having difficulty with mathematics (lange & meaney, 2011). students who have problems with number symbols usually have difficulties with mathematics (jordan & levine, 2009). mathematics is closely related to numbers, so if students have difficulties with numbers, it will also have an impact on their mathematical abilities. mathematics is considered as the basis of science and technology. it is a very important subject in high school, therefore every country applies mathematics to support the development of science and technology (faradillah, 2018). mathematics has a very important role in daily life, because there are many activities that involve mathematics. mathematics has an important role in the process of developing human thought because mathematics and reasoning are inseparable (crosby et al., 2018). measure mathematical trauma in some research are (crosby et al., 2018; lange & meaney, 2011; mendelson et al., 2015; proulx, 2019). the results of the above research are school discipline practices can greatly influence adolescent development. students who have trauma can still have good social relationships and success if the school environment provides to support these students (crosby et al., 2018). some of the causes of students experiencing mathematical trauma arising from homework, namely if procedural mathematics is made homework, if students cannot do the calculations they need, and if schools only provide mathematics learning limited just to practicing multiplication or calculation (lange & meaney, 2011). which can influence emotions and increase problems in the school are when adolescents who have high levels of stress and trauma (mendelson et al., 2015). mathematical activities carried out to find exploratory relationships between representations which then make a conclusion. the mental state of students towards mathematics in ongoing mathematics activities deserves attention for the scope of mathematics education (proulx, 2019). mathematics is one of the subjects that is considered difficult because of its abstract concept so that many students in secondary school consider mathematics to be a difficult, boring, and uninterest subject. this has become one of the triggers for some students get low scores in mathematics in their school. in school, challenge for students is learning mathematics (daud et al., 2020). mathematics ability of male is higher than female in grade 7 and grade 9 (petersen & hyde, 2017). several topics can be related to gender to be research material because it is interesting problems in educational studies. male students have a higher psychological interest in mathematics than female students (chen, yang, & hsiao, 2016). men have higher self-concept than women in some other research results (peteros et al., 2019). no one has discussed based on research related to trauma mathematics using the rasch model for mathematical trauma of students in junior high school based on grade and gender. this research is aimed to analyzing item difficulties mathematical trauma students’ junior high school based on grade and gender. finally, new studies are produced to enrich innovation in the world of mathematics education which can then be utilized by all groups. volume 10, no 1, february 2021, pp. 53-68 55 2. method this study used survey method for data collection. the instrument consisted of 20 statements (table 1). it was validated by two validators, including lecturers and teacher, where it had gone through a revision process and was declared eligible for testing by both validators. subject of the research was 204 students of every grade in junior high school. this instrument was developed measuring mathematical trauma. mathematical trauma divided into three indicators, including difficulty to adapt to the environment, saturated with the learning system that is done, and difficult to get along and organize themselves. data collected from the results of questionnaire distribution were evaluated by rasch mode. the answer choices are guided by a likert scale with four answer options in the form of "strongly agree (sa)", "agree (a)", "disagree (d)", and "strongly disagree (sd)”. table 1. instrument of survey no statements answer sa a d sd difficult to adapt to the environment. i1 if the class conditions are quiet and comfortable, then i can study with focus. i2 the sound of vehicles outside the school distracts concentration while studying in class. i3 noise from other classes disturbs concentration while studying in class. i4 the classroom atmosphere became tense when i started math class which made me nervous. i5 it is easier for me to understand math when studying outside school hours. saturated with the learning system that is done. i6 the teacher appoints the same student every time he gives a math question/problem. i7 i like it when teachers use math software when teaching in class. i8 i feel sleepy while doing math problems. i9 i am bored when the teacher teaches mathematics with the lecture method. i10 i am excited when teachers do learning with a group discussion system. i11 i like it when i have an assignment to make math teaching aids. i12 i like it when teachers use teaching aids when teaching mathematics. faradillah & febriani, mathematical trauma students’ junior high school … 56 no statements answer sa a d sd difficult to get along and organize themselves. i13 i am not awkward to ask friends/teachers if i don't understand the math material explained by the teacher. i14 i did my math exam with honestly. i15 i did my math exam with confidence. i16 i was worried when math class started. i17 i'm afraid of the math teacher. i18 i did discussion with friends to solve difficult math problems. i19 i am disappointed when i get unsatisfactory math scores. i20 i get scared when asked to do math problems on the whiteboard. the rasch model made it possible to change ordinal data from questionnaires to interval data (nguyen & ng, 2014; park & liu, 2019; setiawan, panduwangi, & sumintono, 2018). the rasch model shows diagnostic information for improving the quality of instruments and benefits to scaling and interpretation (willse, 2017). some of the advantages of the rasch model include being able to predict missing dates and data bias, measuring based on the logit scale, knowing student distribution, converting ordinal data into intervals, and showing wright maps of variables (ölmez & ölmez, 2019; soeharto & rosmaiyadi, 2018). not only using the rasch model for evaluation but also to determine the validity and reliability of the mathematical trauma questionnaire by using winsteps software as well as determining the validity and reliability of the data for the cronbach’s alpha (α) reliability evaluation (osman et al., 2016). this study categorized the subject’s demographic data into two parts, namely grade and gender. table 2. reliability in rasch analysis (sumintono & widhiarso, 2014) statistics fit indices interpretation cronbach’s alpha (kr-20) <0.5 low 0.5-0.6 moderate 0.6-0.7 good 0.7-0.8 high >0.8 very high item and person reliability <0.67 low 0.67-0.80 sufficient 0.81-0.90 good 0.91-0.94 very good >0.94 excellent volume 10, no 1, february 2021, pp. 53-68 57 statistics fit indices interpretation item and person separation high separation value indicates that the instruments has a good quality since it can identify the group of item and respondent. according to sumintono and widhiarso (2014) on table 2, for establishing the reliability from the rasch model consist by three fit indices criteria which are cronbach’s alpha, item and person reliability, and item and person separation. table 3. the summary of instrument statistics mean separation reliability cronbach’s α person 0.04 1.05 0.53 0.53 item 0.21 8.84 0.99 table 3 shows a summary of statistical instruments, including reliability of person and items. the result shows that measurement is reliable. the results of the analysis are person-output data that shows the suitability of the respondent and item-output data shows the suitability of the instrument. (ölmez & ölmez, 2019). the separation coefficient is a very important addition to evaluating the function of a measurement instrument. coefficient of the cronbach’s α has a value of 0.59, it means moderate. the reliability results of the items with a score of 0.99 show that the items used have a very good level of reliability and the 20 items measured have a separation with a score of 8.84 (rounded to 9) which means it is divided into 9 groups of items from which can be easily considered to the most difficult to considered (setiawan et al., 2018). table 4. fit indices for item fit and person fit (sumintono & widhiarso, 2014) statistics fit indices outfit mean square values (mnsq) 0.5 – 1.5 outfit z-stardarized values (zstd) -2.0 – +2.0 point measure correlation (ptmea-corr) 0.4 – 0.85 table 4 show three criteria can used for assessing the item fit, which are outfit mean square values (mnsq), outfit z-standarized values (zstd), and point mesure correlation (ptmea-corr). item fit can inform that the item is functioning normally to supposed measurements, meanwhile if the item shows misfit it is indicated that the respondents had a misconception to the item. three criteria above can also used for assessing the person fit (sumintono & widhiarso, 2014). so, researcher can find out the items and persons are fit or misfit. the mnsq shows a measure of randomness which is the amount of distortion in the measurement system. expected values are between 0.5 1.5; if the value is less than that value indicates the data overfit the model; while a greater value indicates the data underfit the model. the zstd is the t-test for the hypothesis of the suitability of the data to the model, the result is a z-value which is the unit deviation (sumintono & widhiarso, 2014). faradillah & febriani, mathematical trauma students’ junior high school … 58 table 5. the value of person cronbach’s alpha (kr-20), person reliability, item reliability, person separation, and item separation statistics value cronbach’s alpha (kr-20) 0.59 person reliability 0.53 item reliability 0.99 person separation 1.05 item separation 8.84 table 5 shows the value for cronbach’s alpha (kr-20), person reliability, item reliability, person separation, and item separation based on the rasch analysis in winsteps. when the value of person reliability is less than 0.67, it is “low” (table 2). based on table 5, the value of person reliability is 0.53 with the person separation value of 1.05. according to table 1, an item reliability which is higher than 0.94 is interpreted as “excellent”. in this study, the value for item reliability is 0.99 with an item separation value of 8.84. moreover, the instrument has a “moderate” reliability if the value of the cronbach’s alpha (kr-20) between 0.5 – 0.6 (table 2) and in table 5, the value of cronbach’s alpha (kr-20) is 0.59. 3. results and discussion 3.1. results 3.1.1. item fit item fit can inform that the item is functioning normally to supposed measurements, meanwhile if the item shows misfit it is indicated that the respondents had a misconception to the item. any item that fails to fulfill these three criteria (outfit mnsq, outfit zstd, and ptmea-corr) needs to be improved or modified to ensure the quality and suitability of the item (saidi & siew, 2019). researcher can find out the items are fit or misfit. table 6. misfit order of the items item measure outfit mnsq (0.5 1.5) outfit zstd (-2.0 +2.0) ptmea-corr (0.4 0.85) i10 -1.15 1.73 6.31 0.05 i7 -0.08 1.58 5.14 0.13 i19 -1.26 1.28 2.68 0.20 i9 1.00 1.23 2.42 0.31 i17 0.08 1.15 1.52 0.50 i6 0.21 1.14 1.42 0.30 i18 -.1.07 1.09 0.94 0.19 i1 -2.04 0.95 -0.44 0.32 i5 0.56 0.97 -0.34 0.35 i20 0.99 0.97 -0.31 0.40 i8 0.77 0.92 -0.88 0.64 i16 0.40 0.91 -0.95 0.57 i13 -0.80 0.83 -1.81 0.36 volume 10, no 1, february 2021, pp. 53-68 59 item measure outfit mnsq (0.5 1.5) outfit zstd (-2.0 +2.0) ptmea-corr (0.4 0.85) i15 -0.41 0.83 -1.80 0.46 i3 1.45 0.83 -1.98 0.06 i11 0.22 0.83 -1.82 0.27 i14 -0.34 0.80 -2.25 0.42 i4 0.86 0.79 -2.45 0.59 i12 -0.75 0.77 -2.39 0.31 i2 1.34 0.69 -3.94 0.20 table 6 shows based on the value of outfit mnsq, outfit zstd, and ptmeacorr for misfit order of the items. there are some bold numbers, its indicate that items failed to fulfill the criteria suggested by (sumintono & widhiarso, 2014). it is known that items i10 and i7 are misfit because its failed to fulfill the three criteria suggested by (sumintono & widhiarso, 2014), items i10 and i7 should be changed. meanwhile, there are five items (i17, i20, i8, i16, and i15) that fulfilled the three criteria suggested by (sumintono & widhiarso, 2014), while the rest fulfilled at least one of the three criteria and should be retained. 3.1.2. person fit the information of the person fit also provided by the rasch analysis. person fit can identify by rasch model based on the unusual response pattern. for example, the unusual patterns that are detected by rasch analysis suggests that the student may not seriously when answering the items (boone, 2016). according to saidi and siew (2019), ‘measure’, outfit mnsq, and outfit zstd are the criteria for assessing person fit or misfit. table 7. misfit order of the person person total score measure outfit mnsq (0.5 – 1.5) outfit zstd (-2.0 +2.0) 160pc 59 1.77 4.28 6.21 138pc 61 1.98 3.35 4.89 047la 59 1.77 3.08 4.56 150pc 46 0.56 2.86 4.33 049pa 63 2.21 2.55 3.57 055la 57 1.57 2.40 3.47 176lc 60 1.88 2.44 3.47 028pa 40 0.04 2.33 3.35 032la 62 2.09 2.21 2.99 144lc 58 1.67 2.07 2.82 034la 57 1.57 2.19 3.06 003pa 70 3.14 2.17 2.44 016pa 49 0.82 2.16 3.05 142pc 59 1.77 2.09 2.83 058pa 57 1.57 1.99 2.65 faradillah & febriani, mathematical trauma students’ junior high school … 60 person total score measure outfit mnsq (0.5 – 1.5) outfit zstd (-2.0 +2.0) 123lb 61 1.98 1.94 2.48 013la 59 1.77 1.97 2.59 099lb 49 0.82 1.89 2.47 115lb 58 1.67 1.83 2.29 171pc 58 1.67 1.73 2.09 077pb 52 1.10 1.75 2.15 048la 56 1.47 1.72 2.07 004pa 55 1.38 0.48 -2.08 021la 52 1.10 0.48 -2.07 181pc 55 1.38 0.48 -2.06 074lb 59 1.77 0.48 -2.01 157pc 54 1.28 0.47 -2.11 103pb 53 1.19 0.47 -2.16 186lc 55 1.38 0.46 -2.16 194lc 55 1.38 0.46 -2.16 196pc 49 0.82 0.47 -2.14 010pa 51 1.00 0.44 -2.28 117lb 60 1.88 0.45 -2.19 185lc 54 1.28 0.45 -2.22 125pb 63 2.21 0.44 -2.15 017pa 57 1.57 0.41 -2.47 064pa 57 1.57 0.40 -2.49 012la 53 1.19 0.39 -2.59 024pa 53 1.19 0.39 -2.59 029pa 54 1.28 0.39 -2.62 137pc 58 1.67 0.39 -2.57 197pc 51 1.00 0.39 -2.63 081lb 58 1.67 0.38 -2.64 070lb 59 1.77 0.37 -2.64 147pc 59 1.77 0.37 -2.67 156pc 59 1.77 0.37 -2.67 066pb 58 1.67 0.36 -2.75 135lc 58 1.67 0.36 -2.75 167pc 57 1.57 0.35 -2.80 091lb 57 1.57 0.32 -3.03 108pb 59 1.77 0.31 -3.09 107lb 59 1.77 0.26 -3.42 067lb 60 1.88 0.25 -3.46 volume 10, no 1, february 2021, pp. 53-68 61 person total score measure outfit mnsq (0.5 – 1.5) outfit zstd (-2.0 +2.0) 068lb 59 1.77 0.25 -3.55 072lb 58 1.67 0.25 -3.58 026pa 54 1.28 0.24 -3.70 073pb 56 1.47 0.24 -3.65 table 7 presented the respondents (which is the students in this case), whose response was mos misfit with the rasch model analysis, it means their response was different from the range given by the rasch model. the respondents on the table 6 were coded accordingly, the 160 was the number of respondents, p refers to the female while c was the grade 9. according to table 7, there are 22 respondents’ scored an outfit zstd value greater than 2.0, it means the items was unpredictable, while 35 respondents’ scored less than -2.0, it means the items was too predictable (sumintono & widhiarso, 2014). meanwhile, the rest respondents have an outfit zstd value within the acceptable range (-2.0 – 2.0). in this study, the items were suitable for 147 respondents (72.06%) and the analysis conducted on those respondents showed quality findings for the assessment using the rasch analysis. 3.1.3. the wright maps the analysis in this study was seen from two aspects, namely grade and gender. the data obtained were analysed using the rasch model to determine student responses based on distributed questionnaires. the rasch model presents diagnostic information for the purpose of evaluating and improving instruments (willse, 2017). data were collected on the scale from 204 students from every grade in junior high school. figure 1. the wright maps of mathematical trauma based on grade faradillah & febriani, mathematical trauma students’ junior high school … 62 figure 1 students from grade 8 who were very traumatized compared to grade 7 and grade 9, which is grade 8 (5.88%), grade 9 (4.17%), and grade 7 (3.125%). students in grade 7 have the lowest math anxiety (luo, wang, & luo, 2009) so they have the lowest percentage students who very traumatized. meanwhile, abstract reasoning ability of students from grade 8 was low and should reach extended level (kusmaryono et al., 2018) so that it makes students very traumatized by mathematics. the percentage result was calculated of every grade, not from all of the respondents. figure 2. the wright maps of mathematical trauma based on gender figure 2 male students who were very traumatized compared to female students, which is male (5.43%) and female (3.57%). work on mathematics for students who have good mathematics ability will easier. in mathematics, female students are better than male students (purwasih, anita, & afrilianto, 2019), so male students more trauma with mathematics than female because mathematics ability of male students are low than female students. male students have more bad experiences with mathematics so the level of trauma is higher than female students. the percentage result was calculated of every gender, not from all of the respondents. table 8. students’ quantity based on level of trauma level of trauma quantity very traumatized 9 traumatized 191 not trauma 4 volume 10, no 1, february 2021, pp. 53-68 63 table 8 shows from 204 students as subject, there are 9 students who were very traumatized, 191 students who were traumatized, and 4 students who were not traumatized. in percentage there are 4.42% for students who were very traumatized, 93.63% for students who were traumatized, and 1.96% for students who very traumatized. figure 3. the wright maps of mathematical trauma questionnaire figure 3 shows based on the results of wright maps on winsteps. data were consisted of 20 statements from questionnaire and analyzed through winsteps. the item that make students very traumatized by mathematics students from grade 8 who were very traumatized was “if the classroom conditions are calm and comfortable, then i can study with focus”, which is 5% of very traumatized, 95% traumatized and 0% not traumatized. students will be more motivated to learn if the classroom conditions are conducive and comfortable (mawardi & supadi, 2018). keep the classroom climate in mind when studying is likely to increase positive outcomes for students (barr, 2016). class condition is one of the factors in student success in learning. 3.2. discussion based on questionnaire result, 8 from 9 students who very traumatized (table 8) by mathematics strongly agree with statement “i'm afraid of the math teacher” in the instrument research. afraid is one of the characteristics of anxiety. anxiety is an emotional response faradillah & febriani, mathematical trauma students’ junior high school … 64 when experiencing frustration or conflict, so that feeling will arise through various emotions such as afraid, surprise, weakness, guilt, and feeling threatened (aminullah, 2013; hidayat & ayudia, 2019; prahmana et al., 2019). according to prawirohusodo (anita, 2014), in behavioral theory, frustration and trauma that are continuous and not handled will cause anxiety for students. if this is allowed, it will affect the psychological and emotional condition of students when studying or interacting with subjects which are a source of anxiety. mathematics anxiety cannot be seen as a normal thing, because the inability of students to adapt to lessons causes students difficulties with mathematics which ultimately lead to low learning outcomes and achievement in mathematics. there are several ways to reduce math anxiety for students according to freeman (syafri, 2017), namely: (1) overcoming negative self-impressions of mathematics. (2) frequently asking questions when feeling difficulties with mathematics (3) must be brave in trying to understand mathematics because mathematics is new knowledge. (4) not limited to one text book. (5) creating a state of relaxation, comfort, and pleasure when learning mathematics. as for teachers, woodard (zakaria and nordin, 2008) suggests several techniques that can be used to reduce math anxiety, including: (1) creating a calm and relaxed mathematics learning environment. (2) using cooperative groups. (3) teaching without haste. (4) provide additional learning, so that no student is left behind academically. overall, according to the analysis from the rasch model, the instruments has moderate cronbach’s alpha (kr-20), low person reliability, and excellent item reliability. in terms of validity, there are two items (i10 and i7) that must be changed. meanwhile, there are five items (i17, i20, i8, i16, and i15) that fulfilled the three criteria suggested by (sumintono & widhiarso, 2014), while the rest should be retained because the items fulfilled at least one item of the suitability criteria for outfit mnsq, outfit zstd, and ptmeacorr. an item may misfit because it is difficult for the students but is unexpectedly answered correctly by poor-performing students or because it is an easy item that is unexpectedly answered incorrectly by high-performing students (boone, 2016). for person fit, the items were suitable for 147 respondents (72.06%) and misfit for 57 respondents (27.94%). from 52 respondents are misfit, there are 22 respondents’ scored an outfit zstd value greater than 2.0, it means the items was unpredictable, while 35 respondents’ scored less than -2.0, it means the items was too predictable (sumintono & widhiarso, 2014). 4. conclusion according to analysis result, concluded that from 204 students consisting of every grade in junior high school as subjects, male students and students from grade 8 more traumatized by mathematics caused by class condition that not conducive. moreover, based on the rasch analysis, the items i10 and i7 are misfit because its failed to fulfill the three criteria, there are mean square values (mnsq), outfit z-standarized values (zstd), and point mesure correlation (ptmea-corr), items i10 and i7 should be changed. meanwhile, there are five items (i17, i20, i8, i16, and i15) that fulfilled the criteria, while the rest fulfilled at least one of the criteria and should be retained. the items were suitable for 147 respondents (72.06%) and the analysis conducted on those respondents showed quality findings for the assessment using the rasch analysis. furthermore, depend on the wright maps table on winsteps, the percentage of male students who very traumatized by mathematics is more than female students. meanwhile, based on grade, students from grade 8 are more very traumatized by mathematics than students from grade 7 and grade 9. volume 10, no 1, february 2021, pp. 53-68 65 therefore, based on questionnaire, students prefer to class condition that make very traumatized by mathematics. references aminullah, m. a. 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(2008). the effects of mathematics anxiety on matriculation students as related to motivation and achievement. eurasia journal of mathematics, science and technology education, 4(1), 27-30. https://doi.org/10.12973/ejmste/75303 https://doi.org/10.1016/j.jmathb.2019.100725 https://doi.org/10.1088/1742-6596/1315/1/012073 https://doi.org/10.29333/iejme/5755 https://doi.org/10.1108/ijse-07-2017-0294 https://psycnet.apa.org/doi/10.1037/pri0000076 https://doi.org/10.1080/07481756.2017.1362656 https://doi.org/10.12973/ejmste/75303 faradillah & febriani, mathematical trauma students’ junior high school … 68 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p247-262 247 infinity the role of scaffolding in the deconstructing of thinking structure: a case study of pseudo-thinking process imam kusmaryono*, nila ubaidah, mochamad abdul basir universitas islam sultan agung, indonesia article info abstract article history: received jun 17, 2020 revised sep 21, 2020 accepted sep 22, 2020 this study aimed to (1) analyze the role of scaffolding in deconstructing pseudo-thinking structure, and (2) analyze the development of students' thinking structures after receiving scaffolding. the study was framed with a qualitative methodology by involving case study design. this research was conducted at state junior high school 31 in semarang city, indonesia. data was collected through tests, observation, and interview methods. result of the study indicated that (1) scaffolding has changed the pseudo thinking process into a real thought process, and (2) scaffolding could develop students’ thinking structure into a more complex (abstract) level. their thinking structure was initially on the stage of comparative thinking structure before receiving scaffolding assistance and after receiving scaffolding, to developed into the stage of logical reasoning thinking structure. in other words, scaffolding could become a useful strategy to help students go through different zone of proximal development (zpd). keywords: pseudo thinking, scaffolding, thinking structure, zone of proximal development copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: imam kusmaryono, departement of mathematics education, universitas islam sultan agung, jl. kaligawe raya km. 4, semarang, center java 50112, indonesia. email: kusmaryono@unissula.ac.id how to cite: kusmaryono, i., ubaidah, n., & basir, m. a. (2020). the role of scaffolding in the deconstruction of thinking structure: a case study of pseudo-thinking process. infinity, 9(2), 247-262. 1. introduction develop students’ ability in problem-solving is one of the teacher’s roles (keiler, 2018; simamora, saragih, & hasratuddin, 2018). in problem-solving, the procedure of mathematics problem solving is a cognitive process based on what had been known (ekawati, kohar, imah, amin, & fiangga, 2019; samsonovich, kitsantas, o’brien, & de jong, 2015). in this case, students should use cognitive strategy to determine how they learn, reprocess information, to use what had been learned. they must think to obtain a suitable problem-solving strategy, so they could achieve the cognitive purpose, that is, to solve problems (evans & swan, 2014; novita, widada, & haji, 2018; susanti, 2018). students in the activity of thinking to solve mathematical problems can occur the possibility of answers obtained is true or false. incorrect answers are not necessarily caused https://doi.org/10.22460/infinity.v9i2.p247-262 mailto:kusmaryono@unissula.ac.id kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 248 by thinking processes that are also wrong (herna, nusantara, subanji, & mulyati, 2016; subanji & nusantara, 2016). this wrong answer does not mean that the subject (student) cannot solve it. when students solve problems, they often give "false" answers relatively quickly, spontaneously, and do not check or reflect the results of their work. this happens because of the low ability to think reflective in solving problems, so their thought processes tend to produce wrong answers (choy, yim, & tan, 2017; deringöl, 2019). this process of thinking is still "raw" rather than the actual thought process so that the process of pseudo thinking occurs (herna et al., 2016). the pseudo-thinking process is a thinking process in which students thought as if they had solved a problem, but they just imitated what the teacher or someone else did (subanji & nusantara, 2016; vinner, 1997). the students did not understand what they just did. they often provide answers spontaneously without any check or control on the thinking process (thanheiser, 2010; vinner,1997). pseudo-thinking process has become interesting as students experienced it "unreal", just a pseudo-thinking. the pseudo-thinking process could be discovered in two forms i.e., (1) the answer is correct, but the student could not justify it. this is called as "true" pseudo (caglayan & olive, 2010; herna et al., 2016; thanheiser, 2010) and (2) the answer is incorrect, but the student actually could solve it well. this is called as "false" pseudo (subanji, 2013; vinner, 1997). both pseudo-thinking processes could be fixed after being reflected with or without scaffolding (wibawa, nusantara, subanji, & parta, 2018). this pseudo-thinking process could still be fixed as it is not the real thinking process as incapability. furthermore, through the reflection process, the teacher should believe that the students actually have the potentials to develop with assistance from teachers and or more capable peers. according to vygotsky, each student had the zone of proximal development (zpd), usually mentioned as a distance between actual development level and higher potential development level. vygotsky suggested that students could achieve maximal area if they received enough assistance. if students learned without assistance, they would still be in the actual area and could not develop to higher potential development level (chairani, 2015). given that the pseudo thinking process is not a real thought process, it can still be improved through the process of reflection. at the time of reflection, the teacher provides scaffolding with the aim of improving students' thinking structures. the improvement of this thinking structure is based on the belief that when students are in the zone of proximal development (zpd), they have the potential to develop optimally. cognitive structure is a basic mental process used by an individual to understand information (garner, 2007). then it was also called a mental structure, or thinking pattern (kusumadewi, kusmaryono, jamallullail, & saputro, 2019). a students thinking pattern would develop based on the cognitive development stage (mascolo, 2015). however, some students met obstacles in cognitive development, so that it is the time for the student to get assistance to face learning barriers. after the scaffolding was given, the student’s cognitive structure became more developed and more complex. scaffolding was learning assistance provided by teachers to students who had barriers to learning. the learning would be more effective if teachers helped students to develop cognitive structures that would equip them for individual learning (kusumadewi et al., 2019). this research focuses on two objectives. the first objective is to analyze the role of scaffolding in deconstructing pseudo-thinking structures. the second objective is to analyze the development of students' thinking structures after receiving scaffolding. volume 9, no 2, september 2020, pp. 247-262 249 infinity 2. method 2.1 research design this research was conducted in qualitative methodology by involving case study design. this research was conducted at state junior high school 31 in semarang city, indonesia. setting of the study involved kind of naturalistic investigation because the scaffolding was studied on natural situation without manipulating any variables (kalu & bwalya, 2017). 2.2 participants this study involved 36 students in grade 8, a teacher, and a supervisor. the students had attended mathematics learnings with a group investigation approach and attended problem-solving test. based on the test results, students whose scores did not reach the specified criteria and had the most error answers, it is reasonable to suspect that these students experienced a pseudo thinking process, then considered as data sources. researchers act as supervisors of learning in the classroom. where as the instructor is a senior teacher who has more than 10 years of teaching experience in school. 2.3 instruments the data collection method included tests, observation, and interviews. the instruments used in this study were written test questions and a list of interview questions. the interaction in scaffolding between students and teachers is observed by supervisors during the mathematics learning process. the deep interview was conducted on students by purposive snowball technique in order to collect complete information. an instrument of the written test was a test of problem-solving as follows. problem: a water storage tank had 10 meters length, 5 meters in width, and 4 meters in height. the tank is full of water and will be distributed to 40 houses. each house receives 500 liters of water every day. if the tank is empty, the company will refill it again until it is full. question: how many times does the company should refill the water storage tank in one month to fulfill the need of water for 40 houses? 2.4 procedure the subject of the study was focused on students who were in the lower group, namely students who received low test scores or did not reach the minimum completeness criteria of 70.0. the study was conducted by tracking the response of students' answers in solving problems. then so that students can correct their mistakes students are given the opportunity to reflect with the help of scaffolding. the author conducted this research by exploring the response of student answers in solving problems. then, in order to correct the mistakes, the students are given a chance to do reflection with scaffolding assistance. scaffolding is carried out through the steps: (1) giving questions; (2) problems to solve for students; (3) asking students to express what they knew; (4) giving a chance for students to review their work; (5) asking students to describe the plan of problem-solving; (6) asking students to combine their ideas; (7) asking students to share with others; (8) giving question and keywords to students; (9) if the students need further information, teachers guide them to go back to step 4, dan restart until it is finished kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 250 (buli, basizew, & abdisa, 2017; san martín, 2018; van de pol, volman, & beishuizen, 2010). 2.5 data analysis, and validation this research is natural, researchers as data collection tools (human instruments). the data is analyzed inductively and is a descriptive one. data analysis was described as an interactive cycle through stages of data collection, data reduction, data display, and drawing conclusions (miles & huberman, 2016; moleong, 2007). to ensure data validity, the authors used the triangulation of theory and data source (moleong, 2007). 3 results and discussion 3.1 results after students complete formative tests at the end of learning, all student answers are analyzed and presented in table 1. table 1. description of the test score statistics for each group group students test score range mean std. deviation top 10 95 100 98.5 2.415 middle 17 70 90 74.4 11.023 bottom 9 40 65 45.6 5.270 total 36 based on the data in table 1, there were 9 students in the lower group whose test scores did not reach the minimal completeness criteria of 70.0. so that the students' answers were analyzed for errors, and the following results were obtained (see table 2). table 2. error analysis of student answers no. subject analysis of student answers 1 (s.01) incorrect concept 2 (s.05) incorrect concept 3 (s.17) incorrect concept 4 (s.18) incorrect concept 5 (s.20) incorrect concept 6 (s.22) incorrect concept 7 (s.06) incorrect analysis 8 (s.29) incorrect analysis 9 (s.31) incorrect analysis the following is the result of students' answer response to represent 2 cases happened i.e., the case of incorrect concept (s.01) and incorrect analysis (s.06). student (s.01) was volume 9, no 2, september 2020, pp. 247-262 251 infinity supposed to experience a "false" pseudo-thinking process, and the student (s.06) was supposed to experience a "true" pseudo-thinking process. student’s answer response to mathematics problem that was tested in this research was taken as a sample to discuss in figure 1. figure 1. student answer (s.01) before reflection considering the student answer (s.01) in figure 1, finally, it was found that the company should refill the tank 10 times a month. the answers given by students are not right. to find out the student's thought process when solving a problem, the researcher conducted the interview as follows. ………………………. researcher : why did you write down 1 liter = 1 dm3, and 1 m3 = 1000 liter? subject (s.01) : i have been thinking that i would convert the volume of the cube in liter measurement. researcher : did you understand how to solve question number 1? subject (s.01) : i do, and i have some ideas to do. researcher : was your answer saying that the company should refill the tank 10 times a month, correct? subject (s.01) : i can not explain, but i hope it is correct. researcher : the calculation you made had not finished yet. it still had to be continued to the next step. subject (s.01) : oh, i see. (the subject looked disappointed). researcher : did you conduct a recheck? subject (s.01) : i did not recheck it. ………………………. kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 252 based on students' answers in the interview passage indicated that students experience pseudo thinking processes. this pseudo thinking process occurs because the answers are spontaneous, and students do not check their work in the form of reflection. the structure of student thinking (s.01) in solving problems before reflection (see figure 2). thinking scheme code information a. problems should be defined: amount of tank refill in a month. b. providing information of the known data c. questioning : volume of water storage tank d. converting cubic measurement into liter e. calculating the need of water for 1 day f. water adequacy in the water stage tank g. prediction of time the water will be used up in 10 days h. the work is done: the subject was unsure of what had been done. rf. recheck of reflection should be conducted figure 2. the structure of student thinking (s.01) before reflection based on the student (s.01) thinking structure as displayed in figure 2, it could be said that the student was on a comparative thinking structure. it is basic to learn and as a requirement for other more complex cognitive structures (garner, 2007). student mathematical ability on this level was processing information by identifying how the data were similar or different, including recognition, memorization, constant conservation, classification, spatial orientation, temporal orientation, and metaphoric thinking. thinking structure on this stage could still be developed with scaffolding assistance from the teacher. the scaffolding was given to the students to reflect on what had been done and direct their initial knowledge to solve the problems they faced (maharani & subanji, 2018). the role of the teacher in learning as resource persons and facilitators who provide assistance as needed (scaffolding) in order to facilitate the construction process of knowledge that is built by students. at a certain level of cognition, teachers provide scaffolding assistance by guiding them or providing key instructions, cues, questions, and justifications so that students will be easier to move or develop into higher thought processes. to correct the mistakes, the students were given a chance to do a reflection. they rechecked the steps of problem-solving. while students were having difficulties, the teacher gave the scaffolding. the work of students after reflection with the help of scaffolding is shown in figure 3. h rf ? g f e c d b a volume 9, no 2, september 2020, pp. 247-262 253 infinity figure 3. student answer (s.01) after reflection to understand the student (s.01) thinking structure, consider the following interview between the researcher and the student. ………………………. researcher : please check whether the result you obtained had answered the question? subject (s.01) : i had rechecked and i found that my answer did not solve to the problem questioned. researcher : was there any mistake you did while working on it? subject (s.01) : i did a mistake. researcher : in which part did you find the mistake? subject (s.01) : the answer should not be 10. i would recheck it. ………………………. to help the student, the scaffolding was implemented through words to track or keywords. by the keywords, the student could find the solution. ………………………. researcher : please calculate the need for water in a month. subject (s.01) : the need for water in a month is 500 liters x 40 x 30 = 600.000 liters. researcher : then, make an equation (relation) with the volume of the water tank. subject (s.01) : the volume of the water tank as 200,000 liters will be used up in 10 days (1-time refill). so, in a month should be 600.000 liters divided by 200.000 liters equals to 3 or (3 times refill). researcher : have you found your answer? kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 254 subject (s.01) : the answer is 3 times a refill in a month. researcher : what conclusion did you obtain? subject (s.01) : the need for water for 40 houses in a month equals to 3 times of water tank volume. researcher : are you sure? how do you prove it? subject (s.01) : i’m sure it’s correct. subject (s..01) : (volume of water tank) x (amount of water refill) = need of water in a month. it was (10 m x 5 m x 4 m) x 3 = 40 x 500 liters x 30. obtained 600 m3 = 6.000 liter. ………………………. based on the result of the student’s work and the interview, the following is the thinking structure after reflection (see figure 4). thinking scheme code information rf. rechecking or reflection. a. problems should be found: number of time to refill water in a month b. displaying information of the data known c. questioning : volume water storage tank d. calculate volume of water tank e. converting cubic measurement into liters f. calculating the need of water in one day g. calculating the need of water in one month h. forming an equation, the need of water in a month equals to numbers of water refill in the tank j. result of number of water refill in a month end done. no doubt. figure 4. the structure of student thinking (s.01) after reflection the response of student answer (s.06) on mathematics questions tested in this research was chosen as sample to discuss and present in figure 5. j end h d c b a f g rf. e volume 9, no 2, september 2020, pp. 247-262 255 infinity student: (s.06) figure 5. student answer (s.06) before reflection based on the result of the analysis on the student (s.06) response answer, it indicated that the final result was correct. however, there were some solving steps missing that were not written by the student (s.06). furthermore, an interview was conducted to determine how student's (s.06) thinking process was, as presented below. ………………………. researcher : what is the conclusion (answer) of this problem? subject (s.06) : the company should fill the water tank 3 times a month. researcher : are you sure about your answer? subject (s.06) : probably. researcher : did you re-check it? subject (s.06) : i did not. researcher : could you prove that your answer is correct? subject (s.06) : sorry, i have no idea. i could not explain it. ………………………. the student's thinking structure (s.06) could be described in the following scheme (see figure 6). kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 256 thinking scheme code information a. problems should be defined: amount of tank refill in a month. b. providing information of the known data c. question: lots of water needed for a month. d. calculate the volume of a water tank e. converting cubic measurement into liter f. calculating the need of water for 1 day g. adequacy of water in the tank h. prediction of time the water will be used up in 10 days j. the work is complete, but the subject cannot prove the argument rf. re-examination or reflection is required figure 6. the structure of student thinking (s.06) before reflection after that, the student (s.06) got a chance to do reflection in order to correct the answer with scaffolding assistance from teacher, through the following steps. ………………………. researcher : calculate the need for water in a month! subject (s.06) : need of water = 40 x 500 liters x 30 = 600.000 liters researcher : what is the volume of the water tank? subject (s.06) : volume of water tank = 10 m x 5 m x 4 m = 200 m3 = 200.000 liters. researcher : explain the correlation between the need for water and the volume of the water tank! subject (s.06) : the need for water in a month should be equal to the volume of water in the tank. researcher : create the equation for the correlation of both variables! subject (s.06) : (volume of the tank) x (number of water refill) = need of water in a month. (10 m x 5 m x 4 m) x k = 40 x 500x 30. 200.000 liters x k = 600.000 liters. k = 600.000 : 200.000 k = 3 a rf. ? b d f e g h j c volume 9, no 2, september 2020, pp. 247-262 257 infinity researcher : what is your conclusion? subject (s.06) : the number of water refill into the tank is 3 times a month. researcher : please re-check your answer! subject (s.06) : (volume of the tank) x (number of water refill) = need of water in a month. (10 m x 5 m x 4 m) x 3 = 40 x 500 liters x 30 600 m3 = 600.000 liters. yes, the result is similar. researcher : are you sure? subject (s.06) : pretty sure. ………………………. the following is a student's (s.06) answer response after the reflection process with scaffolding assistance as presented in figure 7. figure 7. student answer (s.06) after reflection after reflection with scaffolding, the student could find the solution and provide an explanation for the answer obtained correctly. through scaffolding, the student (s.06) could fix the thinking structure as presented in figure 8. kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 258 thinking scheme code information rf. reflection a. problems should be defined: amount of tank refill in a month. b. displaying information of the data known c. question: lots of water needed for a month. d. calculate volume of water tank e. converting cubic measurement into liters k. calculating the need of water in one month m. forming an equation, the need of water in a month equals to numbers of water refill in the tank n.. resolve the equation end. work completed: subject confident of the results of the solution figure 8. the structure of student thinking (s.06) after reflection 3.2 discussion 3.2.1. before scaffolding considering the result of analysis on students' answer responses (s.01 and s.06) and interview toward the students on how they solved mathematics problems, it could be said that the students experienced a pseudo-thinking process before the reflection process. the student (s.01) experienced a "false" pseudo-thinking process, while the student (s.06) experienced a "true" pseudo-thinking process. based on the interview, the student (s.01) was detected to have a "false" pseudothinking process. the student did not understand what he has done. the concept of problemsolving was not complete yet. the solution obtained did not receive treatment of re-checking, but it had been considered as a conclusion, so the answer was incorrect. specifically, the student (s.01) was said to experience the so-called “pseudo-conceptual” thinking process, as the main process focused on the inappropriate concept, reasoning, correlation between concepts, and implementation (vinner, 1997). the student's answer (s.06) in the interview was a manifestation of "true" pseudothinking process behavior (caglayan & olive, 2010; herna et al., 2016; thanheiser, 2010). although the answer was correct, the student (s.06) was unsure and could not provide explanation (proof) for the answer. in addition, the student did not conduct reflection. according to experts, a "true" pseudo-thinking process happens when the answer provided by a student is correct, but the student (s.06) could not provide justification (caglayan & olive, 2010; herna et al., 2016; a j b d e n c k rf m end volume 9, no 2, september 2020, pp. 247-262 259 infinity thanheiser, 2010). specifically, the student (s.06) was said to experience the so-called “pseudo-analytical” thinking process, as the main process is a problem-solving process (vinner, 1997). considering the student's thinking structure (s.01) before reflection (figure 2), it could be said that both of the students were on a comparative thinking structure. it is basic to learn and as a requirement for other more complex cognitive structures. student's mathematical ability on this level is to process information by identifying how data is similar or different. it includes recognition, memorization, constant conservation, classification, spatial orientation, temporal orientation, and metaphorical thinking. on the other hand, a student's thinking structure (s.06) before reflection (figure 6) was on a symbolic representation thinking structure. the symbolic representation structure changes information into a coding system that could be accepted in general (culture) including verbal and nonverbal language; mathematics; graphics (2-dimension figures, painting, logo); construction; simulation and multimedia (garner, 2007). both comparative and symbolic representation thinking structures could still be developed to construct more complex mathematics knowledge structures with scaffolding assistance from the teacher. the role of the teacher is to provide assistance (scaffolding) as needed in order to ease the process of knowledge construction that is built by students themselves (hmelo-silver, 2004; darling-hammond et al., 2019). 3.2.2. after scaffolding scaffolding describes processes to support students for (puntambekar & hubscher, 2005; van de pol et al., 2010). scaffolding is given to students in order to reflect the learning outcome and direct the initial knowledge in solving problems (maharani & subanji, 2018). learning would be more effective if the teacher helps students (to provide scaffolding) to develop a cognitive structure for them to study independently (kusumadewi et al., 2019). after reflection with scaffolding assistance, the pseudo-thinking process never happened anymore. students' answers after reflection indicated that they succeeded to correct mistakes and turn them into correct answers. it was in line with the interview result in which the students could provide explanations or proof for the answer with proper reason. the answers had been re-checked it’s solving steps, the students became sure with no doubt, and the thinking process became true. on certain cognition levels, teachers provide scaffolding assistance by providing students guidance, key instruction, signs, questions, and corrections, so the students could move or develop more easily into higher thinking processes (van de pol et al., 2010). after having scaffolding, the students succeeded to do reflection and correct the previous mistakes. moreover, scaffolding had developed students' thinking structure (s.01 and s.06). before reflection, their thinking structure was on comparative and symbolic representation structure. after the reflection, the thinking structure developed into a logical reasoning structure (figure 4 and figure 8). on the logical reasoning structure, the students (s.01 and s.06) had used an abstract thinking strategy to systematically process and produce information. they could conduct analysis on problems, causal relation, and evaluation in problem-solving (garner, 2007). the provision of scaffolding to students is based on vygotsky's theory (the zone of proximal development) has proven to be effective. that it can reach the level of potential development that students can actually do like problem-solving abilities under adult guidance or through collaboration with other more capable students. according to van de pol et al. (2010), scaffolding could become a useful strategy to help students move through different zpd. the scaffolding involved support from teachers kusmaryono, ubaidah, & basir, the role of scaffolding in the deconstruction of thinking … 260 to students while working on a task that can’t be done alone (van de pol et al, 2010). however, it was a different mental process, constructing a new mental structure to adjust and model the model they have learned with an emphasis on the relationships among thinking objects (navaneedhan & kamalanabhan, 2017). the success of scaffolding in this research was because the teacher and students were in the correct place. the teacher as a facilitator had applied steps of scaffolding correctly and effectively. the teacher showed an attitude of appreciating student ideas and then directed them to a decision or choice that they needed to develop based on the basic rule. the final decision was brought back to the students to decide how they used the teachers’ suggestions in redesigning their work. it leads to boost their confidence and activities to encourage them to use reasoning, communicating, and connecting ability between knowledge and experience they had before. scaffolding concept could be used as an analytical tool to help students obtaining much more understanding in learning. 4 conclusion in the scaffolding process, effective learning interactions have occurred between the teacher and students, so the conclusions of this study are (1) scaffolding changes the pseudo thinking process into a real thought process, and (2) scaffolding could help develop students' thinking structures from simple ones to more complex (abstract) level. the development of students' thinking structures is at the stage of comparative thought structures before scaffolding is given, and then developed into logical reasoning thinking structures after receiving scaffolding. in other words, scaffolding can be a useful strategy to help students move through different zona of proximal development (zpd). acknowledgements special thanks to students and teachers at state junior school 31 semarang, indonesia. the head of sultan agung islamic university, which provides funding to conduct research. references garner, b. k. 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(2018). defragmentation of student’s thinking structures in solving mathematical problems based on cra framework. journal of physics: conf. series, 1028(1), 012150. https://doi.org/10.1088/1742-6596/1028/1/012150 https://ejournal.unib.ac.id/index.php/jpmr/article/view/6286 https://ejournal.unib.ac.id/index.php/jpmr/article/view/6286 https://ejournal.unib.ac.id/index.php/jpmr/article/view/6286 https://ejournal.unib.ac.id/index.php/jpmr/article/view/6286 https://doi.org/10.1207/s15326985ep4001_1 https://doi.org/10.1016/j.procs.2015.12.218 http://dx.doi.org/10.15446/profile.v20n1.63032 https://doi.org/10.12973/iejme/3966 https://www.researchgate.net/profile/subanji_subanji/publication/299364474_proses_berpikir_pseudo_siswa_dalam_menyelesaikan_masalah_proporsi/links/56f2046e08ae1cb29a3d206e.pdf https://www.researchgate.net/profile/subanji_subanji/publication/299364474_proses_berpikir_pseudo_siswa_dalam_menyelesaikan_masalah_proporsi/links/56f2046e08ae1cb29a3d206e.pdf https://doi.org/10.5539/ies.v9n2p17 http://www.journal.unipdu.ac.id/index.php/jmpm/article/view/998 http://www.journal.unipdu.ac.id/index.php/jmpm/article/view/998 http://www.journal.unipdu.ac.id/index.php/jmpm/article/view/998 https://doi.org/10.1007/s10649-010-9252-7 https://doi.org/10.1007/s10648-010-9127-6 https://doi.org/10.1023/a:1002998529016 https://doi.org/10.1088/1742-6596/1028/1/012150 after students complete formative tests at the end of learning, all student answers are analyzed and presented in table 1. table 1. description of the test score statistics for each group table 2. error analysis of student answers sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p259-270 259 mathematical literacy ability and metacognitive characteristics of mathematics pre-service teacher christina m. laamena*, theresia laurens universitas pattimura, indonesia article info abstract article history: received oct 1, 2020 revised feb 14, 2021 accepted july 4, 2021 this study aims to determine the characteristics of students' metacognition in solving mathematical literacy problems. the metacognitive traits explored are related to awareness in planning, monitoring, and evaluating the design of the thinking process used. the research method used is a mixed-method (sequential explanatory), which uses quantitative research results to conduct qualitative research. the research subjects were 80 early semester students who took the literacy test and chose six respondents representing the upper, middle, and lower groups, with two people in each group to be interviewed. the results showed that the mathematical literacy skills of pre-service teachers were at a low level. metacognitive characteristics that appear in the low group are (1) realizing that the solution of strategy is not right but not improved; (2) planning to develop a settlement strategy, but are not sure, (3) do not carry out the re-check process, and (4) do not believe what is being thought and do not understand the concept. metacognitive traits in the middle group are (1) aware of what they are thinking, (2) consciously plan various strategies to improve thinking accuracy, but do not always use these strategies, (3) tend to monitor the thinking process, and (4) show tendency to master the basic mathematical concepts of the problems at hand. the characteristics of metacognition in the high group during problem-solving are (1) using various strategies to demonstrate or improve the accuracy of thinking (sketching, drawing), (2) analyzing the problem before solving it, and (3) understanding and mastering the mathematical concepts that underlie the problem which is given. keywords: mathematical literacy, metacognition copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: christina m. laamena, department of mathematics education, universitas pattimura jl. ir. m. putuhena, poka, tlk. ambon, maluku 97233, indonesia email: christinmath18@gmail.com how to cite: laamena, c. m., & laurens, t. (2021). mathematical literacy ability and metacognitive characteristics of mathematics pre-service teacher. infinity, 10(2), 259-270. 1. introduction mathematics has an important role in life, but in practice some students still consider mathematics to be a difficult subject (angateeah, 2017; khiat, 2010; laurens, 2010; sholihah & afriansyah, 2017). this is because mathematics is always introduced as an abstract discipline without relating it to everyday realities (fitriani et al., 2018; kadarisma et al., 2020; kariadinata, 2021; novriani & surya, 2017). in addition, students are also less https://doi.org/10.22460/infinity.v10i2.p259-270 laamena & laurens, mathematical literacy ability and metacognitive characteristics … 260 taught with mathematical skills, as a result they are not skilled at solving problems even according to tambychik and meerah (2010) and heong (2005) there are phases in problem solving that are not implemented properly. problems in mathematics are interpreted differently by the individuals who face them. according to schoenfeld (1987), mathematics problems for students are tasks that: (1) attract students' attention and challenge them to solve them, (2) it is not easy to find solutions. the problem as a situation that requires resolution and for which the individual sees no apparent or obvious means or path to obtaining the solution. there are problems that challenge a person but not others (laurens, 2010). in learning mathematics students need to be introduced to the problem and how to solve it. problems presented to students need to be studied for their relationship with the reality they are facing, meaning that these problems must be contextual problems that have been experienced by students. mathematical problems that are far from the reach of students will give the impression that mathematics only fills the curriculum without being useful in everyday life. therefore, the mathematical concept introduced must be related to its use in everyday life (basibas, 2020; reyes et al., 2019; root et al., 2020; yee & bostic, 2014). the ability to apply mathematical concepts in everyday reality is known as mathematical literacy. stacey (2011) state that the concept of literacy is closely related to several other concepts discussed in mathematics education, but the most important is modeling (mathematical modeling) which is called the process of mathematics. according to de lange (2003), the mathematical process begins with problems in real life, then the problem solver tries to identify relevant mathematical information, and reorganizes the problem according to the identified mathematical concepts, followed by gradually reducing real situations, the third step takes from real world problems to mathematical problems, and the fourth step is interpreting mathematical solutions in the real world. mathematical literacy is the application of mathematical knowledge, methods and processes in a variety of contexts. mathematical literacy contains more than the use of basic knowledge and procedures that enable individuals to have it. de lange (2003) explains that mathematical literacy is related to (1) numerical literacy which consists of material related to quantity (quantity), (2) quantitative literacy which consists of the form of relationships and changes (change and relationship) and uncertainty (uncertainty) and (3) spatial literacy which consists of form and space (space and shape). numerical literacy is concerned with the ability to use numerical data to evaluate statements relating to problems and situations that require mental processing and estimation of real-world context. quantitative literacy developing numbers includes the use of mathematics as it relates to change, quantitative relationships and uncertainty. spatial literacy emphasizes the ability of individuals in the context of three dimensions that are encountered in their daily lives. this requires an understanding of the properties of objects, the relative positions of objects and a visual perception of the object's dimensions. there is a level of mathematical literacy suggested by bybee (2008). this level is based on reading literacy skills, namely: (1) the lowest level called illiteracy which means the inability to deal with information that is considered relevant, namely ignorance of the basic concepts and methods of mathematics, (2) the second level is nominal literacy, namely literacy that is limited to understanding term or name. in this case the individual understands mathematical terms, questions or topics as part of mathematics, but is minimal in comparing these understandings, (3) the third level is functional literacy, which is the level where a person can use the knowledge gained in the activities required. in this case the standard method of mathematics can be applied to solve simple problems, (4) the fourth level is the conceptual and procedural literacy level. this dimension consists of developing an understanding of interrelated concepts. procedural knowledge leads to a mathematical volume 10, no 2, september 2021, pp. 259-270 261 discovery process, in which individuals understand and use mathematical concepts in the context of mathematical investigations. conceptual and procedural literacy contains some understanding of the structure and central function of mathematical ideas such as optimization and recognition patterns, (5) the highest level is the multidimensional literacy level. this level includes the conceptual understanding of mathematics and the social dimensions of mathematics. individuals may develop some understanding and connections in mathematics, and other knowledge content. its main focus is to apply mathematics functionally in order to solve problems and modeling as well as the transition process from the real world to mathematics and vice versa. based on the understanding of literacy, in the teaching and learning process, students need to be given an understanding of the benefits of studying mathematics, so that they feel that mathematics is not a scary subject but a subject that forms a mindset in analyzing problems faced in everyday life and finding solutions. students need to be faced with context problems that force them to find solutions by utilizing mathematical concepts and procedures that are learned. the use of mathematical concepts and procedures requires simple to complex levels of thought. according to charles and lester (1984) there are 3 aspects that influence solving mathematical problems, namely (1) cognitive aspects, including conceptual knowledge, understanding and strategies for applying this knowledge; (2) affective aspects, which are aspects that affect the tendency of students to solve problems; and (3) metacognition aspects, including the ability to organize one's own thoughts. metacognition aspects are important because they relate to one's awareness of the processes and results of thinking. metacognition is defined variously by experts, but in general the notion of metacognition is related to the awareness, knowledge and control that a person exerts over his own thinking processes and results. according to yong and kiong (2005), “metacognition refers to one's knowledge concerning one's own cognitive processes and products or anything related to them. metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of these processes in relation to the cognitive objects or data on which they bear, usually in the service of some concrete goal". flavell (2004) defines the first aspect of metacognition as one's knowledge of cognitive processes and results or everything related to it, then the second aspect of metacognition is defined as monitoring and self-regulation of one's own cognitive activities. the same understanding was put forward by garofalo and lester (1985) that to describe a person's knowledge and control over his mental processes, including knowledge about himself, his tasks, and the strategies used. for example, a student can find out his ability to solve problems well if he takes the time to check the results of his work, especially if he uses diagrams as one of his strategies. schoenfeld (2016) defines metacognition as follows: “metacognition is thinking about our thinking and it comprises of the following three important aspects: knowledge about our own thought processes, control or self-regulation, and belief and intuition (that is, metacognition is thinking about our own thinking which consists of three important aspects, namely: knowledge of our own thought processes, selfcontrol or self-regulation, and belief and intuition). metacognition is the knowledge, awareness, and control of our cognitive processes. metacognition is very important because our knowledge of cognitive processes can help to organize and choose strategies for solving problems that are being experienced (akben, 2020; desoete & de craene, 2019; lingel, lenhart, & schneider, 2019; yong, gates, & chan, 2019). in relation to literacy, israel et al. (2006) state that metacognitive skills should be taught within the context of authentic literacy engagement, and students should be given sufficient practice in their application that they know, when, why, and how to use them relatively effortlessly. in this study, the metacognition activities that were considered were laamena & laurens, mathematical literacy ability and metacognitive characteristics … 262 (1) how students recognized and identified problems and defined the elements of the situation presented; (2) how do students represent the problem and make connections between the information found; (3) how to plan and decide steps for completion and determine how to achieve these goals; (4) how to check / monitor the results it has obtained; (5) how to evaluate the results and solutions made. the low ability of mathematical literacy is due to the fact that indonesian students are not used to solving problems with the types of pisa and timss questions (ekawati, susanti, & chen, 2020; sulistiyarini, 2021) and metacognition that do not run smoothly. therefore, prospective teachers are the first to have their metacognition improved so that they can help their students in the future. in addition, the characteristics of metacognition based on the level of mathematical literacy have not been carried out, so this study will help educators to encourage the emergence of students' metacognition according to their literacy level. thus, the purpose of this study is to describe the mathematical literacy skills of mathematics pre-service teachers and to analyze the metacognitive characteristics used in mathematical literacy. metacognition characteristic analysis was based on the grouping of students (upper group, middle group and lower group). 2. method the research method used is a mixed method, which uses the results of quantitative research to conduct qualitative research. data from the test results were used as the basis for conducting interviews and document analysis to determine metacognition activities from data sources consisting of 80 students and 6 students were selected to be interviewed and become research subjects. quantitative data in the form of test scores were analyzed using descriptive statistics by calculating the mean value of literacy skills and the percentage of level grouping at each literacy ability level. the research instrument used was a literacy ability test which consisted of 5 questions, namely 1 social mathematics question, 2 work questions, 2 geometric questions as well as interview guidelines and an assessment rubric. interviewed subjects consisted of 6 people representing each group. the analysis of metacognition characteristics that appears is based on the analysis of the test results document and the analysis of the results of the interviews with the data analysis technique used, namely reducing data, presenting data and concluding the results. 3. results and discussion 3.1. results there are 2 types of research data that will be discussed, namely literacy ability test results which are quantitative data and work analysis data and interview transcripts which are qualitative data. the test result data for 3 classes shows that the mean value obtained is class a = 30.8, class b = 27.6 and class c = 22.2. this shows that there are differences in the results of literacy skills tests between subjects in the three groups. based on the results of the test, grouping was carried out in the upper, middle and lower groups in order to obtain the percentage of the upper group (high category) as much as 19%, the middle group (medium category) 60% and the lower group (low category) 21%. furthermore, based on the results of this work and based on the indicator of literacy levels put forward by bybee (2008), then the literacy level is analyzed with the results can be seen in table 1. volume 10, no 2, september 2021, pp. 259-270 263 table 1. data of literacy level (bybee, 2008) no level of literacy number 1 2 3 4 5 1. illiteracy 21.30% 12.50% 13.80% 17.50% 10.00% 2. nominal literacy 37.50% 12.50% 13.80% 23.50% 20.00% 3. functional literacy 12.50% 16.25% 28.60% 28.80% 13.80% 4. conseptual & procedural literacy 20.00% 7.50% 11.30% 11.30% 8.80% 5. multidimensional literacy 0.00% 0.00% 0.00% 0.00% 0.00% based on the analysis of the document results, it is known that not all subjects worked on the questions given. subjects who did not do question number 1 were 10%, question number 2 was 10%, question number 3 was 11.3%, question number 4 was 25% and question number 5 was 48.5%. table 1 shows that the level of literacy of mathematics pre-service teachers on all question numbers does not reach 50%, even if no one is at the multidimensional literacy level. persetance illteracy which only reaches 20% indicates that 80% of mathematics preservice teachers are able to identify information that is considered relevant or already know the basic concepts and mathematical methods that must be used to solve problems. the level of conceptual literacy and procedural literacy only reaches 20% and none of the mathematics pre-service teachers have the ability at the multidimensional literacy level. based on the analysis of worksheets documents (see figure 1), it can be seen that there are indications of the use of metacognition, for example the results of work that are crossed out, deleted then replaced, represent problems visually indicating the use of metacognitive knowledge, especially on the strategy variable. figure 1. example of students’ work based on the work done by the students (see figure 1), six people were selected as respondents to be interviewed. each of a number of two people representing the high, low and medium groups. interviews were conducted to reveal metacognitive characteristics in mathematical literacy. the following is an excerpt of an interview from one of the subjects. r : pay attention to the first problem. after reading this issue what do you think s1 : how to get the answer r : apart from that, is there anything else? s : remember fractions laamena & laurens, mathematical literacy ability and metacognitive characteristics … 264 r : why fractions? s : because what is written (pointing to the results of his work) is one-half, one-third r : what does this matter tell s1 : distribution of money r : did your reading time know what to want in this matter s1 : how much money did dony collect r : the answer? s1 : pointing at the answer with a smile to answer 437, (slowly said "it seems wrong") r : so why was it crossed out? s1 : first, the number must be equal to 1500, but i'm confused, i think it's not like this, then i crossed it out the results of the interview indicated that the metacognitive characteristics that emerged were (1) recalling the mathematical symbols contained in the problem which were metacognitive knowledge specifically referred to as metamemory; (2) know the meaning of the question but do not believe the answer is correct. this indicator is included in the metacognitive experience component, especially with regard to cognitive monitoring and evaluation. consider the following interview excerpt (subject at the conceptual and procedural literacy level) r : why do you draw like this s2 : this only helps me to understand the position of the blocks r : what is the position of the blocks s2 : there is an incoming beam, you need to know its position r : but the solution looks like this (pointing to the block image) s2 : because the block is put into the tub, r : are you sure the answer is correct? s2 : well, because i was asked about volume, i remember the volume of blocks, only because there is 1 block in the water r : do you have other ways besides this s2 : no this interview snippet shows that the subject realizes and then plans strategies that make it easier for him to solve problems. this is included in the metacognition knowledge component, especially individual variables and strategy variables. in relation to the metacognitive experience used, it can be seen that there is monitoring of the cognitive processes that occur by deleting and deleting as well as an evaluation process that raises awareness, for example, lack of confidence in what is done and have no other way than the way that has been made. 3.2. discussion from the results of the analysis of the mathematics teacher candidate literacy level, it can be seen that from the 5 questions given 3 questions related to the quantity content, 2 questions related to the content of space and shape. quantity content is concerned with understanding the relationships and patterns of numbers, including the ability to understand size, number patterns, and everything related to numbers in everyday life, such as counting and measuring certain objects. included in the content of this quantity is the ability to reason quantitatively, present something in numbers, perform systematic calculations. according to prince et al. (2021), the ability students need in quantitative literacy is the ability to: (a) volume 10, no 2, september 2021, pp. 259-270 265 identify and express relationships in an effective symbolic form, (b) use computational tools to process information, and (c) interpret results this calculation. based on the results of the analysis of the level of literacy skills, it can be seen that the average level of literacy skills is still low (because it's less than 30%). this can be seen from the fact that there are no subjects who occupy a special multidimensional level. there are less than 50% of subjects at each literacy level and are more dominant in nominal literacy. this shows that their ability is limited to recognizing symbols in the problem, not understanding relevant information and not understanding basic mathematical concepts. this indicates that there is an incomplete understanding of the problem being solved which is known as instrumental understanding. instrumental understanding is a type of understanding related to the use of methods or rules without knowing (realizing) the reasons for the use. gough (2004) state that instrumental understanding as rules without reasons. percentage of literacy nominal that is higher than other literacy shows that undergraduate students have a higher ability to solve problems using calculations or numbers. laamena, nusantara, irawan, and muksar (2018) mention that students are able to solve problems involving numbers (inductively) but if they are faced with problems that require deductive reasoning then they will have difficulty. so according to laamena and nusantara (2019), if the problem is presented in a statement, students tend to use examples of numbers to test the truth of the statement (numerical backing). they understand the questions but are unable to compare what is understood with the relevant mathematical content. for example, the difficulty in applying the different denominated fraction operation concept. in the functional literacy a group they try to use formulas to solve problems but then experience difficulties in the process of solving them, for example the concept of comparison to be used, but in the process they have difficulty equating the two forms of fractions so that they subtract the variables. this shows that knowledge about a procedure does not always guarantee that someone understands the concepts that underlie the material, it really depends on the metacognitive knowledge one has (akben, 2020; desoete & de craene, 2019; lingel et al., 2019; yong et al., 2019). uncontrolled metacognitive knowledge can lead to errors, as argued veenman, van hout-wolters, and afflerbach (2006) that metacognitive knowledge about how we learn can be wrong or right and this knowledge about ourselves (self-knowledge) is likely to change. this change will occur when there is cognitive monitoring activity that raises awareness (desoete & de craene, 2019; kaune, 2006; smith & mancy, 2018; stillman & mevarech, 2010; veenman & van cleef, 2019). in the conceptual and procedural literacy groups, they try to link some of the concepts needed, for example problem number 1 (an equation involving fractions), the problem is symbolically represented but then cannot evaluate the results obtained with what is needed in the problem. the interesting thing was that some of these groups tried to represent the problem visually so that it was easier to find answers and match them with what was being asked. one example is how to conduct investigations through pictures to find answers to problems, but the problem faced is a lack of knowledge of other concepts needed, for example the concept of physics about the mass of objects. the interesting thing about one of the subjects is trying to interpret the problem by making a pattern of the relationship between the number of workers and the specified time. in the multidimensional literacy a group there is an understanding of mathematical concepts, especially spatial structures so that the spatial abilities of the subjects in this group apply mathematical concepts by symbolically representing problems in sketch form. here they carry out a mathematical process by trying to create a mathematical model (chao, liu, & yeh, 2018; muhammad, kumaidi, & mukminan, 2020; yong et al., 2019; zhong & xia, 2020). laamena & laurens, mathematical literacy ability and metacognitive characteristics … 266 based on the results of the analysis of the interviewed subjects and related to their work results, it can be concluded that almost all metacognition components appear in mathematical literacy, especially those related to metacognition knowledge. the component of metacognitive knowledge that emerges leads to awareness of the concepts they possess, for example the awareness that they do not understand the concept of fractional operations, they do not understand how to relate known geometric concepts to other concepts needed in relation to problem solving. in relation to the level of mathematical literacy, in general the use of metacognition in the illiteracy group and the nominal literacy group is strongly influenced by conceptual knowledge and procedural knowledge. the characteristics of metacognition are realizing that the solution strategy is not right but not improving it, planning the strategy that is made but not realizing the accuracy of the strategy, not showing monitoring activities and not sure what is being thought and tends to not master the concept. for the functional literacy group, the use of metacognition leads to awareness of their knowledge but still has difficulty connecting some related mathematical concepts (ali et al., 2020; özenç & dikici, 2016; zhussupova & kazbekova, 2016). in the use of metacognition activities, the characteristics that are raised are being aware of what is thinking, planning various strategies consciously to increase the accuracy of his thinking, but not always using these strategies, tending to monitor his thinking processes, showing a tendency to master the mathematical concepts underlying the problem. for the conceptual and procedural and multidimensional literacy group, the characteristics of metacognition that arise during problem solving are the use of various strategies to demonstrate or improve the accuracy of their thinking (making sketches, drawing), analyzing problems before solving them, understanding and mastering the mathematical concepts underlying the given problem, even though in some steps, it was difficult to check, but due to a lack of conceptual knowledge, the answers given were not correct (amin & mariani, 2017; bakar & ismail, 2020; kramarski & zoldan, 2008; salam et al., 2020; schneider & artelt, 2010; su et al., 2016; zepeda et al., 2019). 4. conclusion this study concludes that the average mathematical literacy ability of pre-service teachers is in the low category and if it is related to bybee's level (2008), the average is still at the level of nominal illiteracy and literacy, meaning that mastery of mathematical concepts is still low and only limited to introduction. symbols and mastery of mathematical procedures are still lacking. in relation to the use of metacognition, it can be said that those in the low group have difficulties in developing and utilizing their metacognition because of the lack of conceptual knowledge and procedural knowledge of mathematics. subjects in the high literacy group can take advantage of their metacognition in solving several problems by realizing their knowledge regarding mathematical concepts and strategies used in solving problems including monitoring and evaluating their cognitive processes. references akben, n. 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(2016). metacognitive strategies as points in teaching reading comprehension. procedia-social and behavioral sciences, 228, 593-600. https://doi.org/10.1016/j.sbspro.2016.07.091 https://doi.org/10.1016/j.sbspro.2010.12.020 https://doi.org/10.1007/s11858-018-1006-5 https://doi.org/10.1007/s11409-006-6893-0 https://doi.org/10.1016/j.jmathb.2014.08.002 https://doi.org/10.4018/ijgbl.2019010101 https://doi.org/10.1037/edu0000300 https://doi.org/10.1007/s10763-018-09939-y https://doi.org/10.1007/s10763-018-09939-y https://doi.org/10.1016/j.sbspro.2016.07.091 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p1-16 1 collaborative learning through lesson study in pmri training for primary school pre-service teacher: the simulation of polygon matter anna fauziah1,2, ratu ilma indra putri2*, zulkardi2 1stkip pgri lubuklinggau, indonesia 2universitas sriwijaya, indonesia article info abstract article history: received oct 31, 2020 revised july 25, 2021 accepted july 27, 2021 collaborative learning through lesson study has become one of the promising methods for improving the quality of education and improving teachers' quality, likewise with the pmri approach. the combination of the two in the training for primary school pre-service teachers, specifically in the second simulation session, was observed and reported. this article aims to describe the collaboration process in the second session of the simulations about polygon learning at pmri training for primary school pre-service teachers. a design research method of the development type was used in this study, only at the preliminary and development or prototyping phase. the research subjects are students of primary school pre-service teachers of sriwijaya university that consisted of eight students for the small group and 32 students for the field test. data was collected through documentation, observation, and field notes. the result showed that there were good collaboration occurs between researcher-lecturer, lecturer-student, and between students at the plan-do-see-redesign stage of the lesson study. keywords: collaborative learning, lesson study, pmri, polygon this is an open access article under the cc by-sa license. corresponding author: ratu ilma indra putri, departement of mathematics education, universitas sriwijaya jl. masjid al gazali, bukit lama, ilir bar. i, palembang, south sumatra 30128, indonesia email: ratuilma@unsri.ac.id how to cite: fauziah, a., putri, r. i. i., & zulkardi, z. (2022). collaborative learning through lesson study in pmri training for primary school pre-service teacher: the simulation of polygon matter. infinity, 11(1), 1-16. 1. introduction various studies have been developed on collaborative learning and positively impacted at all levels of education. lesson study contributed positively to the teacher's professional development program (chong & kong, 2012; lawrence & chong, 2010). the development of lesson study for teachers has also changed the poses of learning, from teacher-centered to student-centered (kusumah & nurhasanah, 2017). lesson study has also improved the performance and confidence of pre-service teachers (kanellopoulou & darra, https://doi.org/10.22460/infinity.v11i1.p1-16 https://creativecommons.org/licenses/by-sa/4.0/ fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 2 2019). besides, lesson study has also helped improve students' mathematical ability; one is reasoning ability (octriana et al., 2019). this impact because collaborative learning provides an opportunity for each learner to learn. collaborative learning plays an important role in creating quality learning. however, not all learners, especially in higher education, collaborate effectively (fransen et al., 2011). most students prioritize completing assignments rather than teamwork. for this reason, efforts are needed to design learning that makes students interact and collaborate effectively. so, lecturers play an essential role in designing and implementing in the class (de hei et al., 2014). lesson study is a learning system that uses collaborative learning at each stage. in lesson study, teachers collaborate continuously to produce better education through planning, application, and implementation in class, observing, and reporting the results (putri & zulkardi, 2019). lesson study has four stages, namely (1) the planning stage, aims to produce a learning device design; (2) the do stage, aims to carry out the learning that has been designed at the planning stage; (3) the see stage which aims to find the strengths and weaknesses of the learning implementation, and (4) the redesign stage which aims to make improvements to the learning design if there is anything that needs to be improved based on the results at the before step. thus, the lesson study cycle begins with the teacher collaborating in selecting topics, planning and preparing lessons, then one teacher conducts learning and the other teachers observe the class, continue with the teachers discussing the results of their observations and reflecting, and finally re-planning the lesson. the four stages of the lesson study run as a cycle and require collaboration between each party involved. this research is part of developing pendidikan matematika realistik indonesia (pmri) learning environments through lesson studies developed by researchers (fauziah et al., 2020). so, pmri was used as a learning approach in this study. pmri itself is the indonesian version of rme, which is a learning innovation that has reformed education in indonesia (hadi, 2015) and has been widely used in teacher candidates (zulkardi, 2002) and teacher professional development programs (ekawati & kohar, 2017; putri et al., 2015). pmri using the real context as a starting point for learning (zulkardi & putri, 2019). pmri is developed on three basic principles, namely (a) guided reinvention and progressive mathematization; (b) didactical phenomenology; and (c) self-developed models (gravemeijer, 1994). pmri has five characteristics, namely (1) using the real world context as the starting point of learning, (2) using the model as a bridge between the abstract and the real world, (3) using the students' own results or strategies, (4) the interaction as essential elements in learning mathematics and (5) the connection of each learning strand (zulkardi & putri, 2019). as for the principles, van den heuvel-panhuizen and drijvers (2014) explain that each principle reflects every characteristic of pmri, which includes activity principle, reality principle, level principle, intertwinment principle, interactivity principle, and guidance principle. however, this report only focuses on discussing the process of collaboration through lesson study (plan-do-see-redesign) in the second simulation session, namely on polygon learning. a polygon is a closed plane figure bounded by straight line segments as sides. polygon is a part of circumference and area of plane materials in grade 4 of primary school in the k-13 curriculum. students will be asked to analyze the properties of polygons and identify regular polygon and irregular polygons. several studies show that there are still difficulties faced by primary school students when learning about the polygon concept (bernabeu et al., 2018; chiphambo & feza, 2020; fisher et al., 2013). primary school students need scaffolding that facilitates them to learn about polygon. this technique can increase student involvement, focus students' attention and exploration, and encourage students' "sense-making" (alfieri et al., 2011; fisher et al., 2013; honomichl & chen, 2012). volume 11, no 1, february 2022, pp. 1-16 3 because geometry, especially polygons, is all around us, pmri and collaborative methods are expected to help improve student understanding. however, this research was applied to primary school pre-service teachers, not to primary school students, to develop their ability to pmri and collaborative methods, which are expected to be used when they become teachers in the future. thus, this research report aims to discover the collaborative process in developing the pmri learning environment through lesson study in primary school preservice teachers for the second simulation session: the polygon matter. the collaboration process that will be reported is the collaboration process between lecturers and researchers, between students, and between lecturers, researchers, and students. 2. method this study used design research with the type of development study at the preliminary and prototyping stages (plomp, 2013). the research subjects are students of primary school pre-service teachers of sriwijaya university. the research was preceded by curriculum analysis and the design of polygon learning materials conducted by researchers and team teaching who are lecturers of primary school pre-service teacher education. the learning materials produced were validated through self-evaluation, expert review, and oneto-one. they were tested in a small group with eight students with mixed abilities consisting of two men and four women. during the small group, students were divided into two groups consisting of four people. finally, a field test was conducted on 32 students comprised of four men and 28 women. students at this stage are divided into eight groups consisting of four students with different abilities. students who are subject to small groups differ in the field test. the collaborative process seen in this study was during preliminary, small group, and field tests. the data was collected through documentation, observation, and field notes. documentation was done in the form of photos, videos during the research process. documentation and field notes were the findings during the one-to-one, small group, and field test stages. observations were made using observation sheets which included remarks about how students worked individually and collaborated in learning, how students said the sentence, "please teach me" when they had difficulties, actively expressed opinions, showed enthusiasm, and concluded learning. the collected data was then analyzed qualitatively to describe each stage of the collaboration process. observation sheets were analyzed descriptively based on student activities at the small group stage and field tests by processing the scores obtained on the observation sheet and determining the percentage for each component. 3. results and discussion 3.1. results the collaborative process through lesson study in pmri training has begun since the preliminary stage. a team consisting of a research team and two lecturers who are team teaching, design the device. the research team used this first collaboration result in pmri training through lesson study for primary school pre-service teachers in simulation sessions. the learning devices are pmri learning materials with polygon topics consisting of learning implementation plans, student worksheet i, student worksheet ii, and teacher's instructions that were predicting student activities and answers. student worksheet i contain a sharing task, and student worksheet ii has the jumping task. this stage is the first stage of the lesson fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 4 study. the context generated through this collaboration is the "pentagon building" for sharing tasks and "the dome of shakhrah" for jumping tasks. after the device has been designed and validated, the materials are used in a learning simulation session in class through two stages: small group and field test. during the lesson, one pgsd lecturer was appointed as a model lecturer while the research team and other lecturers acted as observers. in the simulation session in class, the learning phase in small groups and field tests, is the "do" stage in lesson study. before the learning process begins, the model lecturer conditions the participant's seat to form the letter “u”. learning continues with the model lecturer explaining the learning process that will take place. if there are students who have difficulties working on student worksheets, they can ask other students for help by first saying, please teach me, and that students have to explain. furthermore, the model lecturer conveyed apperception related to polygon matter. then the model lecturer gives student worksheet i, which contains sharing tasks for students to do individually. the model lecturer and observers go around observing every activity carried out by students. when a model lecturer encounters a student who is having difficulty, the lecturer directs the student to ask his friend. in student worksheet i, students are asked to find and redraw the polygon in the pentagon building that is photographed from above, as shown in figure 1. figure 1. the pentagon building is used as sharing task then name the polygon and give reasons related to the naming of the polygon. this problem is classified as a sharing task. students who have low abilities are predicted can do it. based on the researcher's and observers' observations, most students answered this problem well, both at the small group and field test. following is the students' answer in the small group phase in figure 2. volume 11, no 1, february 2022, pp. 1-16 5 translate version: figure 2. students’ answer on student worksheet i in figure 2, students have been able to find several polygons found in the pentagon building. in the beginning, some students only found two polygons in the picture of the pentagon building, namely quadrilateral and pentagon. the observers saw a collaboration process between students who have not been able to with students who can answer well. figure 3 shows the collaboration process that occurs between students. figure 3. the collaboration process between students in figure 3, a student asked her friend associated with the problems that exist in the worksheet. the following dialogue between that two students: student 1 : masayu, please teach me. how do you find other shapes besides quadrilateral and pentagon? because i only saw two polygons, pentagon, and quadrilateral. student 2 : uhm, try to imagine a polygon’s shape that occurs if you combine several polygons. it will be easier if you redraw the pentagon building and observe it again correctly. student 1 : oo yeah, all right, thanks for the instructions. after a while, the underprivileged students managed to find another plane. pentagon hexagon hexagon rectangle quadrilateral heptagon fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 6 in the next question, the researcher asked students to find two polygons that have the same name but have differences and asked students to find these differences. the following is the students' answers, as shown in figure 4. translate version: figure 4. students’ answer on student worksheet i in figure 4, students can find two shapes with the same name, namely a rectangle and a kite, both of which are part of a quadrilateral. the next question is that students are asked to find two shapes that are the same but have differences, following the student's answers in figure 5 and figure 6. the shape has a difference in the length of the side and the size of the angle. a rectangle has two opposite sides of the same length. a kite has two adjacent sides of equal length. in a rectangle, all angles are the same. in a kite, it has different angles. volume 11, no 1, february 2022, pp. 1-16 7 translate version: figure 5. students’ answer on student worksheet i translate version: figure 6. students’ answer on student worksheet i figure 1. rectangle the shape has four sides where the parallel sides are equal. figure 2. square the shape has four sides where all the sides are equal. so, if there are two shapes that are the same with slightly different shapes. the naming can be seen from the side and the angle. fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 8 students have been able to find two shapes that both the same name but are different, namely a square and a rectangle which is part of a quadrilateral and an isosceles triangle and an equilateral triangle which is a triangle. next, students are asked to provide conclusions from some of the problems above. following are the answers of students who conclude the problem, as shown in figure 7. translate version: figure 7. students’ answer on student worksheet i learning continues with the model lecturer giving student worksheet ii which contains a jumping task. in student worksheet ii, the dome of shakhrah used as a context to define a polygon, in this case, the octagon. it is a golden-domed building located in the middle of the al aqsa mosque complex in palestine, as shown in figure 8. figure 8. the dome of shakhrah is used as jumping task regular polygon: a polygon having equal side lengths and angles irregular polygon: a polygon having unequal side lengths and angles volume 11, no 1, february 2022, pp. 1-16 9 students are asked to find the polygon contained in the picture and draw as many other polygons as they can from it. when finding the polygon, only a few students are confused because a dome covers part of the polygon side. students have been able to find that polygons found in the dome of the rock are octagon and rectangle as shown in figure 9. translate version: figure 9. students’ answer on student worksheet ii however, when asked to draw a lot of other things found in a lot that was found, there were quite a lot of difficulties. most students find a different number of polygons, and only a few students find heptagon, as shown in figure 10 and figure 11. figure 10. students’ answer on student worksheet ii octagon and rectangle fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 10 figure 11. students’ answer on student worksheet ii in figure 10, students only found a hexagonal and in figure 11, students were able to find a heptagonal. figure 10 and figure 11 are the results of student answers at the field test stage. students get polygons in the polygon by multiplying lines from one vertex to another vertex. the observers observed that this result was also obtained through a good collaboration process between students. communication and discussion in groups make students able to draw a heptagon in the octagon. at the end of the lesson, the model lecturer asks the student who was initially struggling to explain the answers. so, the student gained confidence because he was able to answer the problem well. based on observations, the collaborative learning component has appeared well during this much learning simulation. the following are the results of the calculation of the observation sheet during learning as shown in table 1. table 1. the result of observation during learning activity percentage students work on each worksheet individually 100 % students collaborate in learning 85 % students say the sentence "please teach me" when asking for help from friends 90 % students actively express opinions 80 % students show enthusiasm in learning 85 % students conclude learning 85 % average 87.5 % table 1 shows that 100% have worked on worksheets individually, 85% of students have collaborated with other students during learning, 90% of students have said sentences please teach me when they ask their friends for help, 80% of students actively express opinions during discussions together with lecturers, 85% of students showed enthusiasm in learning and 85% of students were able to conclude learning as a result of collaboration. thus, the average student has carried out a very good collaboration process between them, which is equal to 87.5%. volume 11, no 1, february 2022, pp. 1-16 11 after the learning process ends, the research team, teaching team, and students reflect the learning that has occurred. this reflection session in the lesson study cycle is the see stage. students participate in the reflection process as part of the student learning process in pmri training through lesson study. the model lecturer begins the process of reflection by giving impressions, experiences, and obstacles encountered when conducting learning. the model lecturer said that this was the first time she had done collaborative learning in her class and felt something different. lecturer as model teacher, facilitate, motivate and direct students to collaborate in learning. the lecturer motivates students who have difficulty saying sentences, “please teach me” to their friends who have been able to solve problems. this makes students unselfish by teaching each other to their friends, even though worksheets are given individually. the research team, as an observer, explains the activities of the students we observe. researchers also provide input on education that takes place in the right language to offend the model lecturers. students who act as subjects also give their impressions and opinions on the learning that has taken place. thus, at this stage, the collaborative process has also occurred between the research team, the teaching team, and students to produce learning well. the next step is a redesign, and based on the results of reflection, there are improvements to the lesson plans, student worksheets, and teacher’s instruction. 3.2. discussion the first collaboration occurred between researchers and lecturers in the primary school teacher education study program who were the research team teaching. together with lecturers, researchers designed pmri learning tools through lesson studies with the topic of polygons, which were then validated before being used in learning. researchers and lecturers work together in determining the context that will be used in the student worksheets. choosing the right context is very important in pmri learning as a starting point (gravemeijer & doorman, 1999; widjaja et al., 2010). in addition to make it easier for students to understand the material (fauziah et al., 2017, 2019; risdiyanti et al., 2019), the context also plays a role in stimulating students to think, communicate, and collaborate (asari, 2017). this first collaboration of lesson study that requires lecturers and researchers to make instructional learning and choose strategies in teaching, have made lecturers more confident in teaching. this is consistent with the research results that have been carried out relating to lecturers' collaborative learning and self-confidence (chong & kong, 2012). the second collaboration is the collaboration that occurs between students and lecturers, as well as between students. at the time of learning using this collaborative method, the lecturer arranges student seats to be shaped like the letter “u”. it is intended that the model lecturer can accurately monitor each activity carried out by students, and students can see each other (mustadi, 2014). the lecturer directs students to ask their friends if they have difficulty working on student worksheets. according to collaborative learning, this is a method that encourages and invites students to work together in doing the learning task (de hei et al., 2014). this has been seen in the results of research, some students have asked their friends when they encounter difficulties and are finally able to answer questions after hearing explanations from their friends. these results indicate that good collaboration has occurred between the two students. the ability to hear from others becomes an essential element in building dialogue and collaboration (saito et al., 2008). the sentence, please teach me, is also very good to continue to be cultivated among students. students are not used to this sentence when asking their friends. this result suggests that the lesson study has taught students of the philosophy of education by indonesian culture (mustadi, 2014). especially when students work on student worksheet ii which is a jumping task, the process fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 12 of collaborating with each other between students is more common. it is estimated that many students find it challenging to answer problems because jumping task consists of a more difficult task than sharing task (asari, 2017; hobri et al., 2020). thus, this second collaboration have a positive impact on learning motivaton and student learning outcomes (de hei et al., 2014). students have been able to solve problems on the worksheet through dialogue and collaboration between them. the third collaboration occurs between students, lecturers and researchers when reflecting after the end of the lesson. the reflection process aims to evaluate, find weaknesses and strengths of the learning process that has taken place. this stage is a crucial component in lesson study (wessels, 2018). the model teacher initiated the discussion by conveying the impressions experienced during the lesson and then continued with the delivery from the observers. criticisms and suggestions are given wisely without cornering the teacher for future improvements. conversely, those who are criticized must be able to receive input from the obeservers for further enhancement of learning. based on input from the discussion at this stage, learning can be redesign so that further learning can be better. thus, good results on all three collaborations show that collaborative learning through lesson study is recommended to be carried out in classes in tertiary institutions. 4. conclusion collaborative learning has a positive impact on the learning process of primary school pre-service teacher students. at the time of planning, the collaborative approach resulted in a pmri learning device in lesson plan, students' worksheets and teacher’s instruction implemented in the simulation session. the learning process at the time of simulation, both at the small group stage and the field test, also shows a good collaboration between students to complete the given assignment. the average observation score obtained during learning is 87.5%, which indicates that there has been good collaboration during the learning process. finally, it also shows effective collaboration between researchers, model lecturers, and students to provide helpful input for polygon learning 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(2019). new school mathematics curricula, pisa and pmri in indonesia. in c. p. vistro-yu & t. l. toh (eds.), school mathematics curricula: asian perspectives and glimpses of reform (pp. 39-49). springer singapore. https://doi.org/10.1007/978-981-13-6312-2_3 https://doi.org/10.1007/978-3-319-72170-5_41 http://hdl.handle.net/10536/dro/du:30048397 https://repository.unsri.ac.id/871 https://doi.org/10.1007/978-981-13-6312-2_3 fauziah, putri, & zulkardi, collaborative learning through lesson study in pmri training … 16 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p275-286 275 developing an interactive learning model using visual basic applications with ethnomathematical contents to improve primary school students’ mathematical reasoning euis eti rohaeti1*, nelly fitriani1, fadillah akbar2 1insititut keguruan dan ilmu pendidikan siliwangi, indonesia 2smp putra juang, indonesia article info abstract article history: received aug 4, 2020 revised aug 30, 2020 accepted sep 29, 2020 this study aims to examine the development of an interactive learning model using visual basic application for microsoft excel with ethnomathematical content on fractions, to improve primary school students’ mathematical reasoning abilities. the research method used is development through the stages of conducting preliminary studies and literature, designing interactive learning models, conducting fgds, producing initial designs of interactive learning models, conducting limited trials in one primary school, making revisions, conducting extensive trials in four primary schools, producing trial in another primary school. the last obtaining a final model and conducting socialisation. the last, it provides a test of mathematical reasoning ability. the research subjects were teachers and students in six primary schools. the research instruments were interviews, validation sheets, documentation, learning observation sheets, questionnaires and mathematical reasoning ability test. the assessment criteria for the developed learning model include syntax, social system, principle of reaction, supports system and instructional impact. the results showed that 1) development of vba-assisted and ethnomatematically-loaded interactive learning models go through two major stages, namely product development and validation 2) the interactive learning model was declared very valid; 3) the responses of teachers and students were generally positive; 4) the achievement of students’ mathematical reasoning abilities after gave learning with an interactive model using vba for microsoft excel with ethnomathematical contents better than using ordinary learning. keywords: interactive learning, visual basic application, ethnomathematical, primary school, mathematical reasoning copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: euis eti rohaeti, departement of mathematics education, institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, west java 40526, indonesia. email: e2rht@ikipsiliwangi.ac.id how to cite: rohaeti, e. e., fitriani, n., & akbar, f. (2020). developing an interactive learning model using visual basic applications with ethnomathematical contents to improve primary school students’ mathematical reasoning. infinity, 9(2), 275-286. https://doi.org/10.22460/infinity.v9i2.p275-286 mailto:e2rht@ikipsiliwangi.ac.id rohaeti, fitriani, & akbar, developing an interactive learning model using visual basic … 276 1. introduction in primary school from first to the sixth grade, students begin to learn mathematics. in this level of education, one of the problematic materials for students is fractions, because, at fractions, students are required to practice their logical reasoning skills and draw conclusions from some given examples. the reasoning is both a student’s inductive and deductive logical thinking process in solving problems (bernard, 2014, 2015; nugraha et al., 2015). mathematical reasoning has a significant role in students’ thinking process because if students’ reasoning abilities are not developed, mathematics learning will only become material that follows a series of procedures and imitates examples without knowing their meaning (nugraha et al., 2015). the trends international mathematics and science study (timss) survey in 2011 (mullis et al., 2015), indonesia ranked 36 out of 40 participating countries, with the lowest reasoning score of 17 points. based on previous research conducted by nurkhaeriyyah, rohaeti, & yuliani (2018) show that the average level of mathematical reasoning ability in middle school is still low. besides, the results of timss and pisa (program for international students assessment) over the last few years show that the results of student mathematics learning in indonesia, especially regarding mathematical reasoning, are still low and are below malaysia and singapore (chotimah et al., 2018). some of these findings indicate that the reasoning ability of students in indonesia is low. it is needed to initiate the learning process that can train students’ reasoning skills, especially at fractions in primary schools, one of which is the interactive learning model assisted by vba for excel. the interactive learning model emphasises student questions as to its characteristic, in the interactive learning model questions will often appear, and the questions may vary. meanwhile, visual basic application for microsoft excel is a programming language feature that can process data automatically by utilising mathematical functions using visual basic code assistance (chotimah et al., 2018; fitriani et al., 2018; rohaeti et al., 2019). with vba for microsoft excel, we can create interactive learning media that can train students’ reasoning skills in a comfortable, interesting, fun and useful way. the media produced is straightforward to reproduce, develop, understand by students, and be used by both teachers and students at school. besides that the media produced can be made thematic as desired, one of which is by including ethnomathematics elements in it, so that learning will be more meaningful and easily understood by students because it has cultural elements that are very common for students (barton, 1996; d’ambrósio, 2006). 2. method the research conducted is developmental research which aims to develop interactive learning on fractions material using vba for microsoft excel. the resulting product is an interactive learning model on primary school fractions with the help of vba for microsoft excel and contains ethnomathematics. the research stages carried out were adopted from the borg and gall’s development model (borg & gall, 1989), as in figure 1. volume 9, no 2, september 2020, pp. 275-286 277 figure 1. research flow figure 1 shows the stages of compiling an interactive learning model using vba for microsoft excel with ethnomathematics content. in the initial stage, the researcher conducted a preliminary study in the form of a literature study, a school survey and drafting. in the literature study, the researcher conducted a study of the theories relating to the interactive learning model that will be developed. the literature study also examines the characteristics of upper-class students at the primary school, especially in their mathematical reasoning abilities. also, relevant previous research results are also reviewed. the school survey was conducted to observe the learning of fraction which has been carried out by primary school teachers, and interviews also conducted to these teachers so that it is known what learning deficiencies must be improved. based on the result of the literature study and survey, the authors make a draft of an interactive learning model that will be developed. the design of the model was then revised in a forum group discussion (fgd) which was attended by mathematics learning experts, supervisors, media experts and several experienced senior primary school teachers. limited trials were conducted at one of the public primary schools in cianjur after the researchers previously compiled an interactive learning model. based on the findings on a limited trial, the researchers made improvements to the design of the learning model. henceforth, more extensive trials were carried out at four primary schools in jakarta, west java and east java. the results of the trial are then re-validated for product testing, and a final interactive learning model is produced. in the product test at another primary school in cianjur, a mathematical reasoning ability test was carried out on students using interactive learning model products compared to those using ordinary learning. finally, the product was socialised to primary school teachers in west java. the research instruments were interviews, validation sheets, documentation, learning observation sheets, tests and questionnaires. the data analysis technique uses descriptive statistics to analyse the scores on the characteristics of the developed interactive learning model, which also takes into account the input from the validator to improve the learning rohaeti, fitriani, & akbar, developing an interactive learning model using visual basic … 278 model. the assessment criteria for the developed learning model include syntax, social system, the principle of reaction, support system and instructional impact (joyce & weil, 1986) in the form of scores from 1 to 4. the results were then converted into a criteria table to be converted into quantitative data. the learning model validation criteria can be seen in the following table 1. table 1. criteria for the validity of the learning model score validity category interpretation 20-24 very valid very good to use 16-19 valid may be used with minor revisions 11-15 quite valid may be used with major revisions 6-10 invalid should not be used (adopted from akbar, 2013) according to akbar (2013), the learning model is declared valid if the three elements of the validation have been declared valid. the three elements of validation are validation by experts consisting of three learning expert lecturers, user validation by three elementary school teachers who are experienced teaching, and audience validation by students by giving scores on student response questionnaires. in addition, this learning model is said to be useful for improving mathematical thinking if the product test results show that the mathematical reasoning of students using the developed learning model is better than those using ordinary learning. 3. results and discussion 3.1. results 3.1.1. development of an interactive learning model development of interactive learning model with vba for microsoft excel with ethnomathematical contents begins with a preliminary study stage and literature study. from the results of the preliminary study and literature study, an interactive learning model draft was then designed. focus group discussion (fgd) was conducted to get input on the draft model made. the implementation involved four lecturers, five primary teachers and three postgraduate students of mathematics education at ikip siliwangi in which there were validation activities for the developed learning model. the input from the fgd for the improvement of the draft design of the interactive learning model is that the delivery of material is more systematic, designed to be more coherent and efficient so that it becomes more practical to use. after the fgd was carried out, necessary revisions were made to the draft of the interactive learning model design so that it was ready to be used in a limited trial at one of the public primary schools in cianjur. the following images are examples of devices and media used in a limited trial (see figure 2). volume 9, no 2, september 2020, pp. 275-286 279 figure 2. example of media used in a limited trial from the results of the limited trial, more input was obtained so that there could be more cultural elements. for this reason, improvements were made by including the layer legit media and wayang golek into the learning media used. based on the revision of the results on the limited trial, several improvements were made for the extensive trial. these improvements include the visual appearance of vba media, where also the display is designed to be more realistic and more ethnomathematics. also, changes in appearance were made because they adjusted to the students’ level of thinking, which was still very real. result of a revision made and given in the comprehensive trial. there are changes to the device and media (see figure 3). figure 3. example of media in comprehensive trial and product trial after the comprehensive trial was carried out, the validation test was conducted, based on the validation results, no revisions had to be made so that it could be continued to the product test. product testing is carried out by comparing learning and comparing the results of students’ mathematical reasoning abilities between those using interactive learning models and those using ordinary learning. rohaeti, fitriani, & akbar, developing an interactive learning model using visual basic … 280 3.1.2. the validation of interactive learning model using vba for microsoft excel with ethnomatematical contents validation of this interactive learning model consists of logical validation by two learning expert lecturers and one it expert lecturer, user validation by three experienced teachers, and audience validation in the form of student responses obtained from the results of distributing questionnaires. logical validation and user validation in the form of an assessment of the components of the learning model which includes 1) syntax (learning model steps), 2) social systems (teacher and student communication patterns), 3) reaction principles (teacher and student responses), 4) support (learning media with vba and ethnomatematic use), and 5) instructional impact (achievement of desired student abilities). meanwhile, audience validation contains indicators of student responses to teacher material and activities from the beginning to the end of the lesson. the results of logical (expert) validation, user and audience validation of the vba for microsoft excel assisted interactive learning model with ethnomatematical contents are simplified in table 2. table 2. validation result validation elements average score percentage (%) criteria logical expert 21.33 88.89 very valid user 20.66 86.11 very valid audience 20.64 86.00 very valid average 20.88 87.00 very valid table 2 show that the three validators scores are in very valid criteria, according to akbar (2013), it can be concluded that the learning model is in very valid criteria. so the interactive learning model can be used in product testing without having to revise it again. 3.1.3. teachers and students’ responses to learning interactive models in limited trials and more extensive trials, researchers conducted interviews with six teachers in primary schools which were the research sites and distributed questionnaires to students in 6 elementary schools to find out the responses of teachers and students to the developed interactive learning model. the results of interviews with the teachers that reflect the teacher’s responses are described in table 3. table 3. teachers’responses no activity teachers’ code responses 1. limited trials g1 interactive learning is quite exciting and has a relationship with a concept that is well known to students, namely culture. 2. extensive trials g2 this alternative learning should also be made of physical media on the same material, but you do not have to use a laptop because many students are not used to using laptops in learning. volume 9, no 2, september 2020, pp. 275-286 281 no activity teachers’ code responses g3 learning with an interactive model that was developed makes learning more interactive than usual for children to be more excited g4 through this learning, students generally become more motivated to learn fractions and with the media in this learning students are enthusiastic about trying the existing media themselves. g5 learning with this ict-based interactive learning model is very good, exciting and more comfortable to convey material to students, students are more enthusiastic about learning, students also understand the material being taught faster. 3. product testing g6 this interactive learning model is a breakthrough learning model that combines interactive mathematics learning, culture (ethnomathematics) and technology. media is beneficial in teaching students the concept of fractions and practising their reasoning skills. table 3 show that the almost all teachers gave positive responses. the learning model is quite exciting and can help teachers to motivate their students to learn mathematics. based on students’ responses from the results of distributing questionnaires, it was found that the average student responded with an average of 90.25% of the total score. these results indicate that the student’s response is quite positive to this interactive learning model. 3.1.4. students’ mathematical reasoning abilities in product testing students are given a reasoning ability test when testing products and the results are compared with students who use ordinary learning. following are the results of inferential statistical data processing. results of the normality test between the pretest and posttest on students’ mathematical reasoning abilities (see table 4). table 4. result of the normality test kolmogorov-smirnova statistic df sig. pretes 0.197 36 0.001 postes 0.193 36 0.002 rohaeti, fitriani, & akbar, developing an interactive learning model using visual basic … 282 table 4 show that the pretest and posttest have a significance < 0.05, so it can be concluded that the data are not normally distributed. thus the further data processing is to use the non-parametric test for two independent samples special mann whitney u with monte carlo (see table 5). table 5. the result of the non-parametric test mann-whitney u 0.000 monte carlo sig. (1-tailed) sig. 0.000 95% confidence interval lower bound 0.000 upper bound 0.000 based on the inferential statistical test results in table 5, a significance value of <0.05 means that ho is rejected. it means that the achievement of students’ mathematical reasoning abilities after being given learning with interactive models using vba for microsoft excell with ethnomathematics content is better than mathematical reasoning ability which using ordinary learning. 3.2. discussion the overall stages of developing an interactive learning model with vba for microsoft excel with ethnomathematic content in principle consist of two important steps, namely 1) developing a product consisting of preliminary study stages, fgd, limited trials, wider trials, product testing and socialization; 2) validating done by experts, users and student responses. these steps are in line with the opinion of borg & gall (1989). they say that educational development research (r&d) is a process used to develop and validate educational products. the results of development research are not only for developing existing products but also for finding knowledge or answers to practical problems. based on the research procedures carried out, it can be concluded that the stages carried out in research fulfill the steps of a development research. the three scores given by the validator, the results are in very valid criteria (based on the criteria given by akbar (2013), so it can be concluded that the learning model is in very valid criteria. based on the results of these averages, the interactive learning model can be used in product testing without having to do the revision stage again. the effectiveness of learning with this interactive model is in line with several researchers (mutiarni, 2016; rohaeti et al., 2019; maharani, 2015; sukarmin et al., 2018). based on the results obtained, it can be denied that this learning model is quite interesting and can help teachers in motivating their students to learn mathematics. the results obtained indicate that the student response is quite positive to this learning. the results of this study can increase students' interest in learning mathematics, thus learning mathematics becomes more enjoyable in the eyes of students, such as research conducted several researchers (irawan & suryo, 2017; priyambodo et al., 2012). the achievement of students’ mathematical reasoning abilities after being given learning with interactive models using vba for microsoft excell with ethnomathematics content is better than mathematical reasoning ability which using ordinary learning. it happens because learning using an interactive model using vba for microsoft excell with ethnomathematics content leads students to be able to draw messages from a series of concepts that are constructed. hence, students become better as conveyed by kadarisma volume 9, no 2, september 2020, pp. 275-286 283 (2019), first fill in contextual ethnomathematics content so that it makes students more interested, motivated and more comfortable to imagine the media which makes the results of students’ mathematical reasoning better (costu et al., 2009; gainsburg, 2008). the results of this study are also in line with research that has been conducted by several previous researchers (amir et al., 2018; aprisal & abadi, 2018; mulyana & sumarmo, 2015; nunes et al., 2007; saleh et al., 2018). 4. conclusion based on the analysis of the research results, it can be concluded that the development of vba-assisted and ethnomatematically-loaded interactive learning models go through two major stages, namely product development and validation. based on the results of the validation of the three validator elements, the learning model developed is included in the very valid criteria. furthermore, teachers and students’ responses were generally positive. beside that the students’ mathematical reasoning abilities using interactive learning models are better than those using ordinary learning. acknowledgements thank you to the ministry of research, technology and higher education who fund the research. board of rectors of siliwangi teacher training and education institute (ikip siliwangi) who have provided support in research activities. leaders and teachers at one of sate primary school in cimahi who have facilitated the implementation of the research. references akbar, s. 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(2018). improving the reasoning ability of elementary school student through the indonesian realistic mathematics education. journal on mathematics education, 9(1), 41–53. https://doi.org/10.22342/jme.9.1.5049.41-54 maharani, y. (2015). efektivitas multimedia pembelajaran interaktif berbasis kurikulum 2013. indonesian journal of curriculum and educational technology studies, 3(1), 31-40. sukarmin, s., poedjiastoeti, s., novita, d., & lutfi, a. (2018). effectivity of interactive multimedia and student activity sheets with writing-to-learn (wtl) strategy in science learning for hearing impairment students. national seminar on chemistry (snk 2018), 211–217. https://doi.org/10.2991/snk-18.2018.47 https://doi.org/10.22342/jme.10.1.5391.59-68 https://www.learntechlib.org/p/209296/ https://www.learntechlib.org/p/209296/ https://www.learntechlib.org/p/209296/ https://doi.org/10.2991/snk-18.2018.47 rohaeti, fitriani, & akbar, developing an interactive learning model using visual basic … 286 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p159-172 159 reflective thinking characteristics: a study in the proficient mathematics prospective teachers cholis sa’dijah1*, muhammad noor kholid1,2 , erry hidayanto1, hendro permadi1 1universitas negeri malang, indonesia 2universitas muhammadiyah surakarta, indonesia article info abstract article history: received jul 2, 2020 revised aug 31, 2020 accepted sep 1, 2020 reflective thinking begins with repeated confusion and evaluation to solve a problem. there are four aspects to reflective thinking, namely techniques, monitoring, insight, and conceptualization. however, the problem-solvers’ reflective thinking characteristics in mathematical problems have not been discovered. the study describes the reflective thinking characteristics of proficient mathematics prospective teachers based on four aspects. the qualitative research was conducted at universitas muhammadiyah surakarta with a total of 64 reflective thinkers. data collected by test, observation sheets, and interview methods. the tests were administered twice. the instruments developed has been through the validation process and declared valid. data analyzed through the stages of reduction, presentation, and verification. we successfully conclude that proficient mathematics prospective teachers have complete and consistent reflective thinking characteristics. further research can be focused on the characteristics of reflective thinking based on another aspect. keywords: characteristics, reflective thinking, analytical geometry copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: cholis sa’dijah, study program of doctoral of mathematics education, universitas negeri malang jl. semarang no. 5, sumbersari, malang city, east java 65145, indonesia email: cholis.sadijah.fmipa@um.ac.id how to cite: sa’dijah, c., kholid, m. n., hidayanto, e., & permadi, h. (2020). reflective thinking characteristics: a study in the proficient mathematics prospective teachers. infinity, 9(2), 159-172. 1. introduction reflective thinking has been explored by several experts (dewey, 1933; habermas, 2015; schön, 1992). they stated that reflective thinking is a mental activity that employes knowledge and experience in solving a problem. furthermore, reflective thinking begins with the confusion of the problem solver and attempts to solve the problem (rodgers, 2002). suharna (2018) has explored individual differences in overcoming confusion during reflective thinking to solve mathematical problems. prospective mathematics teachers with productive reflective thinking categories overcome confusion by using various ways of solving problems. prospective mathematics teachers with the category of reflective thinking connectively overcome confusion by connecting all mathematical concepts, principles, and processes related to mathematical problems or solutions. prospective mathematics teachers https://doi.org/10.22460/infinity.v9i2.p159-172 sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 160 with the category of reflective thinking overcome confusion by matching solutions with related concepts. zehavi & mann (2005) stated that there are four aspects of reflective thinking, namely techniques, monitoring, insight, and conceptualization. aspects of techniques are individual activities in selecting effective and efficient strategies to solve problems. monitoring is the activity of re-monitoring the steps and solutions of mathematical problems encountered. insight is a condition where an individual uses his ingenuity and emotions in solving problems. this aspect involves how much motivation and persistence of individuals to keep trying to solve problems when experiencing confusion. conceptualization is an individual activity involving his ability to connect several concepts and meanings that have been understood to make the right decision. the authors have reviewed several studies related to reflective thinking. first, researches that focus on developing instruments to measure reflective thinking are reflective thinking scale, questionnaire, rubric for evaluating reflective thinking/report, and a set of mathematical problem (agustan et al., 2017; basol & gencel, 2013; sezer, 2008). second, research described reflection abstraction in solving mathematical problems (djasuli et al., 2017). the research concluded that students’ strategies of problem-solving are not directly proportional to their level of reflective abstraction. next, researches that discovered the role of reflective thinking into problem solvers’ performance (aytekin et al., 2018; hong & choi, 2011; mojžišová & pócsová, 2019; nigrini & karstens, 2019; thompson & thompson, 2018; tutticci et al., 2018). they stated that problem solvers that employe reflective thinking make the fewest mistakes, make the right decisions, and have impressive accomplishments. besides, research by hidajat et al. (2019) concluded that confusion occurs due to the misunderstanding and failure factor in generating new ideas and strategies. based on the review, there is no research focus on the development of reflective thinking indicators and characteristics in mathematical problem solving, especially on analytical geometry content. in the last 2019, researchers had conducted preliminary research to develop reflective thinking indicators based on four aspects. it was a qualitative research employe test, observation, and interview method to collect the data. indicators that have been concluded in each aspect are presented in table 1. table 1. description of aspects and indicators of reflective thinking aspects indicators code techniques 1. understanding given informations t1 2. understanding the questions t2 3. filterring necessary informations t3 4. selecting an effective and efficient solution t4 5. understanding hot to get information t5 monitoring 1. monitoring the steps of solution m1 2. monitoring wether the answers are correct or not m2 3. devising strategies for problem solving m3 4. making a consideration before making decision m4 insight 1. feeling enthusiastic for solving problems i1 2. being ready to correct wrong answers i2 volume 9, no 2, september 2020, pp. 159-172 161 aspects indicators code 3. feeling responsible for the solutions written i3 4. writing down the answers clearly i4 5. understanding how to prevent difficulties i5 conceptualization 1. thinking about strategies for solving problems c1 2. thinking about an alternative way for solving problem c2 3. relating the questions with relevant problems c3 4. relating concepts for problem solving c4 5. understanding the reason for every solution c5 the research’s objective is discovering the characteristics of reflective thinking of prospective mathematics teachers with high mathematical abilities in solving analytic geometry problems in aspects of techniques, monitoring, insight, and conceptualization based on indicators that have been developed. the characteristics of reflective thinking are seen from the tendency of changing patterns of indicators shown by the subject when solving the first and second problems. 2. method 2.1. type the qualitative research describes how the reflective thinking characteristics of proficient mathematics prospective teachers in solving analytical geometry problems. the data described all the facts of data without manipulation so this study employee a descriptive design (sagala et al., 2019). 2.2. participants the subjects in this study were 64 mathematics prospective teachers in mathematics education study program at universitas muhammadiyah surakarta indonesia. they have taken analytical geometry courses, employee reflective thinking in problem-solving, have good communication skills when solving problems with think-aloud techniques, and have mathematical abilities in the high category (proficient). determination of the tiered categories of mathematical abilities of prospective mathematics teachers in proficient, sufficient, and novice categories based on standard deviations and the mean. the boundaries reported in table 2. table 2. the boundaries of mathematical ability (ma) category boundaries number proficient ma ≥ 63,75 17 sufficient 38,33 ≤ ma < 63,75 24 novice ma < 38,33 23 the purposive sampling (putranta & jumadi, 2019) is employed because researchers only focus on reflective thinker subjects. in this paper, the result of two participants (namely sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 162 subject s-1 and s-2) are explained because data already represent the whole data in proficient categories. 2.3. instruments the instruments employed in this study are tests administered for twice, the observation sheet, and interview guidelines. the indicators of test and observation refer to reflective thinking indicators reported in table 1. all instruments have been through the validation process and declared valid. the number of validators is three persons. the experts in mathematics, mathematics education, and educational qualitative research. the suggestions from the validator are editorial improvements to the problem. they asked for an adjustment in mathematical vocabulary so that there were not multi-interpretations. 2.4. data collection method the data explored based on subjects’ answer sheets of problem-solving results, video recording when the subjects solve the problem, interviews, and observation sheets. test instruments are used to determine the characteristics of reflective thinking in solving problems. the problem is presented in figure 1 and figure 2. participants employ a thinkaloud method in solving analytical geometry problem. it is a method for expressing aloud the processes and symptoms of thinking that arise in cognitive (charters, 2003). the thinkaloud method is very suitable to see the thought process of research subjects. figure 1. analytical geometry problem for test round 1 figure 2. analytical geometry problem for test round 2 volume 9, no 2, september 2020, pp. 159-172 163 2.5. data analysis data analysis through the stage of data reduction, data presentation, and concluding. the complete research procedure presented in figure 3. math prospective teachers (64 persons) finish tes round 1 sufficient (24)proficient (17) novice (23) start do they conduct reflective thinking? not subject tes round 2 do they conduct reflective thinking? answer sheet audio visual recording are the data complete? in-depth interview data analysis indicator pattern change characteristic no yes yes no no yes verificationsis presentation reduction figure 3. research procedure 3. results and discussion 3.1. results 3.1.1. subject s-1: data exposure and analysis looking at the s-1 answer sheets, think-aloud transcripts, observation sheet, interview transcripts, and analysis, the s-1 reflective thinking change patterns in terms of test round 1 can be described as follows. in solving test round 1, s-1 gave rise to indicators t1, t2, t3, and t4. this is indicated by s-1 reading the problem repeatedly, rewriting given information, writing questions, and drawing back information to facilitate understanding. indicator t5 marked with s-1 determines the length of the segment ab first, then the length of the segment bc is equal to ab. indicators c2, m2, and i3 appear when s-1 thinking about other solutions in determining the value of h. s-1 experienced confusion so he monitored whether the answer correct or not. besides, this step taken by s-1 to provide confidence in the answer. s-1 performs indicator m1. this can be seen when s-1 monitors the completion steps from question 1 that has been written. in solving question 2, s-1 starts with confusion to investigate whether point c is passed by line bs. however, s-1 seemed enthusiastic about the confusion experienced. this indicates that s-1 performs indicators of i1 and i2. s-1 realizes that it takes several steps before determining the equation of line bs. for this, s-1 draws a plan and consideration to sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 164 answer question 2. his plan is determining the equation of line bs and substituting point c into the equation of line bs. an alternative plan is investigating whether line bs and line bc are coincided. in determining the equation of line bs, he knows how to avoid difficulties. it by simplifying the equations of line bs and bc. his answer in investigating whether line bs crosses c is written very clearly and systematically. this indicates that s-1 performs indicators m3, i4, i5, and m4. in solving question 3 and question 4, s-1 performs indicators c1, c3, c4, and c5. this is marked by s-1 thinking about the way that will be employed before answering questions, relating the results of the questions question 1 and question 2 with the next two questions to be answered, and relating some mathematical concepts. he experienced confusion for several times, but he was able to demonstrate mastery of the material so that it knows every reason for the answers written. looking at the s-1 answer sheets, think-aloud transcripts, observation sheet, interview transcripts, and analysis, the s-1 reflective thinking change patterns in terms of test 2 can be described as follows. in solving test round 2, s-1 raises the indicators t1, t2, t3, and t4. this is indicated by s-1 reading the problem repeatedly, rewriting given information, writing questions, and drawing back given information to facilitate understanding. indicator t5 marked with s-1 determines the gradient of cb, gradients of ad, and equation line ad first before obtaining the coordinates of point d. indicators c2, m2, and i3 appear when s-1 uses two methods in determining gradient cb. he experienced confusion so he employed more than one way to monitor whether the answer correct or not. moreover, this step was taken to provide confidence in the answer. s-1 performs indicator m1. this can be seen when s-1 monitors the completion steps from question 1 that has been written. in solving question 2, s-1 starts with an error that confuses. however, he seemed enthusiastic and willing to correct the mistakes. this indicates that he performs indicators of i1 and i2. s-1 realizes that it takes several steps before determining the coordinates h. for that, s-1 draws up a plan and takes into consideration to answer question 2. his plan to solve question 2 is determining gradient ab, determining gradient ce, determining the equation of line ce, intersection line ce and line ad should be h. in determining h, he knows how to avoid difficulties. the method employed was substituting equations line ce to the equation line ad. the solution in determining the coordinates of the h is written very clearly and systematically. this indicates that s-1 performs indicators m3, i4, i5, and m4. in solving question 3 and question 4, s-1 performs indicators c1, c3, c4, and c5. this is marked by s-1 thinking about the way that will be employed before answering questions, relating the results of question 1 and question 2 with the next two questions to be answered. he also related some mathematical concepts for solving the problem. s-1 experienced confusion several times but he was able to demonstrate mastery of the material so that it knows every reason for the answers written. in solving analytical geometry test both round 1 and 2, s-1 conducted 19 indicators of reflective thinking i.e 5 indicators on the technique aspects, 4 indicators on the monitoring aspects, 5 indicators on the insight aspects, and 5 indicators on the conceptualization aspects. in question 1, he performs indicators t1, t2, t3, t4, t5, c2, m2, i3, m1, in question 2 he performs indicators i1, i2, m3, i4, i5, m4, and in question 3 and question 4, he performs indicators c1, c3, c4, and c5. the changing pattern of indicator in solving analytical geometry test both round 1 and 2 presented in figure 4. the red, blue, green, and yellow squares successively illustrate aspects of techniques, monitoring, insight, and conceptualization. the circle in each box illustrates the indicators of reflective thinking in each aspect. the direction of the arrow indicates the order of change of each indicator. the volume 9, no 2, september 2020, pp. 159-172 165 changing pattern starts from the orange circle on the red square and ends on the orange circle on the yellow square. t1 t2 t3 t4 t5 i1 i2 i3 i4 i5 m1 m2 m3 m4 c1 c2 c3 c4 c5 techniques monitoring monitoring insight figure 4. the reflective thinking change pattern of s-1 figure 4 describes the sequence of reflective thinking change patterns of s-1 in solving both tests. the sequence starts from the orange circle in the red square and ends in the orange circle in the yellow box. the sequence of indicators conducted by s-1 is t1, t2, t3, t4, t5, c2, m2, i3, m1, i1, i2, m3, i4, i5, m4, c1, c3, c4, and c5 (symbol desciption for figure 4 can be seen in table 3). 3.1.2. subject s-2: data exposure and analysis looking at the s-2 answer sheets, think-aloud transcripts, observation sheet, interview transcripts, and analysis, the s-1 reflective thinking change patterns in terms of test round 1 can be described as follows. in solving test round 1, s-2 raises the indicators t1, t2, t3, and t4. this is indicated by s-1 reading the problem repeatedly, rewriting given information, writing questions, determining the length of the segment ab first, then the length of the segment bc equal to the ab. indicator t5 marked by drawing back given information to ascertain the coordinate of point c. indicators m3, m4, c1, m1, and m2 appear when she draws up various settlement plans to solve question 2. she makes various considerations to overcome confusion. the confusion she experienced is making errors in calculating, so she re-monitored the completion steps and written solutions. s-2 performs indicators of i2, i3, and i1. this can be seen when s-2 is willing to correct mistakes with full responsibility and enthusiasm. in solving question 3, she thought about how to avoid the difficulty in determining m. the solution proposed clearly and systematically. this indicates she performs indicators i5 and i4. s-2 realizes that there is more than one solution to solve question 4. however, she prefers to employe one method and does not confirm the answer by another method because of confidence about the answer. during solving question 4, s-2 related questions and concepts she has so that he knows every reason for the solution. this indicates that s-2 performs indicators c2, c3, c4, and c5. looking at the s-2 answer sheets, think-aloud transcripts, observation sheet, interview transcripts, and analysis, the s-1 reflective thinking change patterns in term of test round 2 can be described as follows. in solving test round 2, she performs the indicators of t1, t2, t3, t4, and t5. this is marked by reading the problem repeatedly, rewriting given sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 166 information, writing questions, and drawing back information on the problem to facilitate understanding. indicators m3, m4, c1, m1, and m2 appear when she prepares various settlement plans to solve question 2. she experienced confusion when writing symbols so that she monitors the completion steps and solution. this step conducted to provide confidence in the solution. s-2 performs indicators of i2, i3, and i1. this can be seen when s-2 is willing to correct mistakes with full responsibility and enthusiasm. in solving question 3, s-2 thought of how to prevent the difficulty. in investigating whether d and h crossed by line ag, she substituted d and h into equation ag. the solution written by her seems clear and systematic. this indicates s-2 performs indicators i5 and i4. in solving question 4, s-2 performs indicators c2, c3, c4, and c5. this is indicated by s-2 thinking about various ways of solving to answer question 4. she also related question 1 and question 2 in addressing the next two questions. she experienced confusion several times, but she was able to show mastery of the material. moreover, she knows every reason for the solution. in solving analytical geometry test both round 1 and 2, s-2 conducted 19 indicators of reflective thinking i.e 5 indicators on the technical aspects, 4 indicators on the monitoring aspects, 5 indicators on the insight aspects, and 5 indicators on the conceptualization aspects. in questions 1, she performs indicators t1, t2, t3, t4, and t5. in question 2, s-2 performs indicators m3, m4, c1, m1, m2, i2, i3, and i1. in question 3, she performs indicators i5 and i4. in question 4 s-2 performs indicators c2, c3, c4, and c5. the changing pattern of indicator in solving analytical geometry test both round 1 and 2 presented in figure 5. the red, blue, green, and yellow squares successively illustrate aspects of techniques, monitoring, insight, and conceptualization. the circle in each box illustrates the indicators of reflective thinking in each aspect. the direction of the arrow indicates the order of change of each indicator. the changing pattern starts from the orange circle on the red square and ends on the orange circle on the yellow square. t1 t2 t3 t4 t5 i1 i2 i3 i4 i5 m1 m2 m3 m4 c1 c2 c3 c4 c5 techniques monitoring monitoring insight figure 5. the reflective thinking change pattern of s-2 figure 5 describes the sequence of reflective thinking change patterns of s-2 in solving both tests. the sequence starts from the orange circle in the red square and ends in the orange circle in the yellow box. the sequence of indicators conducted by s-2 is t1, t2, t3, t4, t5, m3, m4, c1, m1, m2, i2, i3, i1, i5, i4, c2, c3, c4, and c5. (symbol desciption for figure 5 can be seen in table 3). volume 9, no 2, september 2020, pp. 159-172 167 table 3. symbol desciption for figure 4 and figure 5 symbol description techniques aspect monitoring aspect insight aspect conceptualization aspect the beginning and end of reflective thinking indicator indicator performed indicator not performed the direction of change pattern 3.2. discussion the data exposure and analysis show that both s-1 and s-2 perform all the reflective thinking indicators. in both tests, s-1 shows the change patterns are t1, t2, t3, t4, t5, c2, m2, i3, m1, i1, i2, m3, i4, i5, m4, c1, c3, c4, and c5. whereas, s-2 shows patterns of t1, t2, t3, t4, t5, m3, m4, c1, m1, m2, i2, i3, i1, i5, i4, c2, c3, c4, and c5. although both proficient mathematics prospective teachers show different change patterns, there are interesting findings that can be revealed. they show a complete and consistent change in both tests round 1 and 2. complete characteristic means the proficient prospective mathematics teachers perform all indicators of reflective thinking on both tests. the consistent characteristic means that the proficient prospective mathematics teachers show the same change pattern between test round 1 and 2. it is relevant to rodgers (2002) stated reflection is a systematic, rigorous, disciplined way of thinking, with its roots in scientific inquiry. reflective thinking is a way of thinking that is systematic, thorough, and disciplined based on the reasons for scientific discovery. sarid (2012) argued that a logical structure is consistent with a systematic thought process. the logical structure is consistent with the process of systematic thinking. the characteristic of complete and consistent reflective thinking described as structured and systematic reflective thinking. complete and consistent reflective thinking performed by proficient prospective mathematics teachers employee a metacognitive process. in employing reflective thinking processes, aspects that appear represent aspects of the metacognitive process. the aspects of metacognitive thinking are awareness, evaluation, and regulation (magiera & zawojewski, 2011). awareness is a situation where an individual is aware of the information he is thinking about (baltaci et al., 2016; purnomo et al., 2017). this is synonymous with confusion and techniques aspect as a characteristic of the emergence of reflective thinking, where the confusion is a symptom of perplexity that appears as the beginning of the reflective thinking process accompanied by individual awareness to overcome the confusion that arises (suharna et al., 2020). evaluation aspect is a condition where individuals conduct monitoring, plan formulation, and consideration for making decisions in problem-solving (callan et al., 2016; maharani et al., 2019). the evaluation aspect of metacognitive thinking is identical to the sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 168 monitoring aspect of reflective thinking namely re-monitoring of information that has been thought to provide confidence in individuals as problem solvers (zehavi & mann, 2005). regulation is a condition where the individual sets the goal of the solution to the problem at hand where this activity arises after an evaluation (kuzle, 2013; purnomo & bekti, 2017). the regulatory aspect of the metacognitive process is identical to the aspect of insight and conceptualization in the reflective thinking process. insight can be an individual effort as a problem solver to overcome emerging confusion or awareness (zehavi & mann, 2005). moreover, conceptualization described as an effort in relating among mathematics concepts (sa’dijah et al., 2020) by employing abstraction (djasuli et al., 2017) or visualization (zayyadi et al., 2020). research focuses on thinking characteristics for problem-solving conducted by purnomo et al. (2017). the research focused on students’ metacognition characteristics in solving a calculus problem. the qualitative research discovered three characteristics among others: complete with the order, complete with no the order, and incomplete. 4. conclusion we successfully concluded that proficient prospective mathematics teachers have complete and consistent reflective thinking characteristics. complete characteristic means the prospective mathematics teacher performs all indicators of reflective thinking on both tests. the consistent characteristic means that the proficient prospective mathematics teachers show the same change pattern between test round 1 and test round 2. the characteristics of reflective thinking that are complete and consistent performed by proficient prospective mathematics teachers on both tests showing the sequence of understanding given pieces of information, understanding the questions, filtering necessary pieces of information, selecting an effective and efficient solution, understanding how to get information, thinking about an alternative way for solving a problem, monitoring whether the answers are correct or not, feeling responsible for the solutions written, monitoring the steps of the solution, feeling enthusiastic for solving problems, being ready to correct wrong answers, devising strategies for problem-solving, writing down the answers clearly, understanding how to prevent difficulties, making a consideration before making a decision, thinking about strategies for solving problems, relating the questions with relevant problems, relating concepts for problem-solving, and understanding the reason for every solution. as for the other sequences shown by proficient prospective mathematics teachers on the tests round 1 and 2, namely understanding given pieces of information, understanding the questions, filtering necessary pieces of information, selecting an effective and efficient solution, understanding how to get information, devising strategies for problem-solving, making a consideration before making a decision, thinking about strategies for solving problems, monitoring the steps of the solution, monitoring weather the answers are correct or not, being ready to correct wrong answers, feeling responsible for the solutions written, feeling enthusiastic for solving problems, understanding how to prevent difficulties, writing down the answers clearly, thinking about an alternative way for solving a problem, relating the questions with relevant problems, relating concepts for problem-solving, understanding the reason for every solution. the change pattern characteristics observed based on aspects of techniques, monitoring, insight, and conceptualization. further research can be focused on the characteristics of reflective thinking based on another aspect. indicators of the study appear in the reflective thinking process in solving analytical geometry problems. there is an opportunity for further research to explore reflective thinking indicators on other contents. volume 9, no 2, september 2020, pp. 159-172 169 acknowledgments the authors are very grateful to director of directorate of research and community service (drpm brin) the republic of indonesia on research funding 2020 on a contract number 10.3.6/un32.14.1/lt/2020. references agustan, s., juniati, d., & siswono, t. y. e. 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(2018). the critically reflective practitioner. macmillan international higher education. https://doi.org/10.22460/infinity.v8i2.p109-116 https://doi.org/10.22460/infinity.v8i2.p109-116 https://doi.org/10.22460/infinity.v8i2.p109-116 https://doi.org/10.1109/carpathiancc.2019.8765961 https://doi.org/10.1109/carpathiancc.2019.8765961 https://doi.org/10.1109/carpathiancc.2019.8765961 https://doi.org/10.1108/maj-09-2017-1640 https://doi.org/10.1108/maj-09-2017-1640 https://doi.org/10.1108/maj-09-2017-1640 https://www.iejme.com/article/patterns-change-of-awareness-process-evaluation-and-regulation-on-mathematics-student https://www.iejme.com/article/patterns-change-of-awareness-process-evaluation-and-regulation-on-mathematics-student https://www.iejme.com/article/patterns-change-of-awareness-process-evaluation-and-regulation-on-mathematics-student https://doi.org/10.5539/ies.v10n5p13 https://doi.org/10.5539/ies.v10n5p13 https://doi.org/10.5539/ies.v10n5p13 https://doi.org/10.17478/jegys.552091 https://d1wqtxts1xzle7.cloudfront.net/33963457/rodgers__c._%282002%29._defining_reflection_another_look_at_john_dewey_and_reflective_thinking._teachers_college_record__104%284%29__842-866..pdf?1402945771=&response-content-disposition=inline%3b+filename%3ddefining_reflection_another_look_at_john.pdf&expires=1601102038&signature=cwl6lnfcc5adr~1mspjtt-pmcocfy~9ucvi3fm3z8cebis47gjxjuy7rbt3viz~bb-s9i4x2h0ipp1hhm27lxrnu-wd5n-rtvw2nb5v~w5gjv7mdylajxdb2qsirzlv6vhxtgoqqabubf6jmftkuupbhjvaj68wbdjyhcsfy3fq-utv1wsxrmxn1cybyoqrusg6omqu59uc-tgfzfyzqj8wtlthykvzzyzr6jlrzfxyhm8klxycauappmhukj5-3vdd7odqhqgwynfneff0ixuotrwbaulyw0fofnc0fb4gcynf57ky09pcsq5btco-g5zy3gd4zk8lb-ogtbvj75w__&key-pair-id=apkajlohf5ggslrbv4za https://d1wqtxts1xzle7.cloudfront.net/33963457/rodgers__c._%282002%29._defining_reflection_another_look_at_john_dewey_and_reflective_thinking._teachers_college_record__104%284%29__842-866..pdf?1402945771=&response-content-disposition=inline%3b+filename%3ddefining_reflection_another_look_at_john.pdf&expires=1601102038&signature=cwl6lnfcc5adr~1mspjtt-pmcocfy~9ucvi3fm3z8cebis47gjxjuy7rbt3viz~bb-s9i4x2h0ipp1hhm27lxrnu-wd5n-rtvw2nb5v~w5gjv7mdylajxdb2qsirzlv6vhxtgoqqabubf6jmftkuupbhjvaj68wbdjyhcsfy3fq-utv1wsxrmxn1cybyoqrusg6omqu59uc-tgfzfyzqj8wtlthykvzzyzr6jlrzfxyhm8klxycauappmhukj5-3vdd7odqhqgwynfneff0ixuotrwbaulyw0fofnc0fb4gcynf57ky09pcsq5btco-g5zy3gd4zk8lb-ogtbvj75w__&key-pair-id=apkajlohf5ggslrbv4za https://doi.org/10.1063/5.0000644 https://doi.org/10.17478/jegys.565454 https://doi.org/10.1111/j.1469-5812.2011.00757.x https://doi.org/10.1111/j.1469-5812.2011.00757.x https://doi.org/10.4324/9781315237473 https://doi.org/10.4324/9781315237473 https://go.gale.com/ps/anonymous?id=gale%7ca177721139&sid=googlescholar&v=2.1&it=r&linkaccess=abs&issn=00131172&p=aone&sw=w https://go.gale.com/ps/anonymous?id=gale%7ca177721139&sid=googlescholar&v=2.1&it=r&linkaccess=abs&issn=00131172&p=aone&sw=w https://books.google.co.id/books?hl=id&lr=&id=9msbdwaaqbaj&oi=fnd&pg=pr5&dq=suharna,+h.+(2018).+teori+berpikir+reflektif+dalam+menyelesaikan+masalah+matematika+(1st+ed.).+deepublish.&ots=dejnlte-mx&sig=eovebhtjwbrpgouttwkifhvmg8g&redir_esc=y#v=onepage&q&f=false https://books.google.co.id/books?hl=id&lr=&id=9msbdwaaqbaj&oi=fnd&pg=pr5&dq=suharna,+h.+(2018).+teori+berpikir+reflektif+dalam+menyelesaikan+masalah+matematika+(1st+ed.).+deepublish.&ots=dejnlte-mx&sig=eovebhtjwbrpgouttwkifhvmg8g&redir_esc=y#v=onepage&q&f=false http://sersc.org/journals/index.php/ijast/article/view/15752 http://sersc.org/journals/index.php/ijast/article/view/15752 http://sersc.org/journals/index.php/ijast/article/view/15752 https://books.google.co.id/books?hl=id&lr=&id=rwzndwaaqbaj&oi=fnd&pg=pr1&dq=thompson,+s.,+%26+thompson,+n.+(2018).+the+critically+reflective+practitioner.+macmillan+international+higher+education.&ots=ihbqzu2x-3&sig=56phonrhontmmuwkekdozvlc0eu&redir_esc=y#v=onepage&q=thompson%2c%20s.%2c%20%26%20thompson%2c%20n.%20(2018).%20the%20critically%20reflective%20practitioner.%20macmillan%20international%20higher%20education.&f=false https://books.google.co.id/books?hl=id&lr=&id=rwzndwaaqbaj&oi=fnd&pg=pr1&dq=thompson,+s.,+%26+thompson,+n.+(2018).+the+critically+reflective+practitioner.+macmillan+international+higher+education.&ots=ihbqzu2x-3&sig=56phonrhontmmuwkekdozvlc0eu&redir_esc=y#v=onepage&q=thompson%2c%20s.%2c%20%26%20thompson%2c%20n.%20(2018).%20the%20critically%20reflective%20practitioner.%20macmillan%20international%20higher%20education.&f=false volume 9, no 2, september 2020, pp. 159-172 171 tutticci, n., ryan, m., coyer, f., & lewis, p. (2018). collaborative facilitation of debrief after high-fidelity simulation and its implications for reflective thinking: student experiences. studies in higher education, 43(9), 1–14. https://doi.org/10.1080/03075079.2017.1281238 zayyadi, m., nusantara, t., hidayanto, e., sulandra, i. m., & sa’dijah, c. (2020). content and pedagogical knowledge of prospective teachers in mathematics learning: commognitive framework. journal for the education of gifted young scientists, 8(march), 515–532. https://doi.org/10.17478/jegys.642131 zehavi, n., & mann, g. (2005). instrumented techniques and reflective thinking in analytic geometry. the montana mathematics enthusiast, 2(22), 1551–3440. https://doi.org/10.1080/03075079.2017.1281238 https://doi.org/10.17478/jegys.642131 https://scholarworks.umt.edu/tme/vol2/iss2/2/ https://scholarworks.umt.edu/tme/vol2/iss2/2/ sa’dijah, kholid, hidayanto, & permadi, reflective thinking characteristics: a study … 172 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p17-30 17 the students' mathematical problem-solving abilities, self-regulated learning, and vba microsoft word in new normal: a development of teaching materials citra megiana pertiwi, euis eti rohaeti*, wahyu hidayat institut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received sep 30, 2020 revised oct 15, 2020 accepted nov 12, 2020 the new normal period makes ict-based education the spearhead of the implementation of learning. this becomes an obstacle and a challenge for students, especially in the mathematical problem-solving ability (mpsa), which is very important for students because it is a prior mathematical ability and is included in hots. self-regulated learning in mathematics (srl) also has a significant role in adapting to learning in the new normal and influences students' mathematical learning outcomes. however, the facts on the ground show that these two abilities are still low. to solve this problem, the researchers developed vba microsoft word-based teaching materials on the relevant polyhedron materials for use so that learning would be more optimal. the method used in this research is an experimental method with a pretestposttest control group design. the population is all the students in cimahi city, while the sample is two classes randomly selected. this study indicates that vba microsoft word-based teaching materials are appropriate to be applied in the new normal period, as indicated by the results of the achievement and improvement of mpsa and srl of students is better than ordinary learning. there is an association between mpsa and srl and positive response even though they still have difficulty making mathematical models on mpsa questions. students are more enthusiastic in learning and don't get bored quickly; in line with the challenges of the new normal, the industrial revolution 4.0, and the learning curriculum, learning is more structured, interactive, effective, and efficient. keywords: mathematical problem-solving, self-regulated learning, virtual basic aplication, microsoft word, teaching material copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: euis eti rohaeti, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, west java 40526, indonesia. email: e2rht@ikipsiliwangi.ac.id how to cite: pertiwi, c. m., rohaeti, e. e., & hidayat, w. (2021). the students' mathematical problem-solving abilities, self-regulated learning, and vba microsoft word in new normal: a development of teaching materials. infinity, 10(1), 17-30. 1. introduction the new normal period was created from adaptation to the covid-19 pandemic, making social distancing and online activities a new lifestyle, including in the field of education. in the new normal era, ict-based education has become the spearhead of the https://doi.org/10.22460/infinity.v10i1.p17-30 pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 18 implementation of learning both directly and at a distance. this becomes an obstacle and a challenge for students in participating in learning, especially aspects that are important to student life such as problem-solving abilities. mathematical problem-solving ability (mpsa) according to harisman, noto and hidayat (2020) is the ability of students to understand problems, plan strategies, and implement problem-solving plans. hendriana, rohaeti and hidayat (2017) state that solving math problems is a basic mathematical ability that must be embedded in high school students. mpsa is very important for students to be able to solve problems in mathematics, learning, and student life at large. this is also because problem-solving abilities are classified as high-order thinking skills. however, the importance of mpsa is not in line with the facts in the field which still show that the mathematical problem solving abilities of students are still low (harisman et al., 2020; hutajulu, wijaya, & hidayat, 2019; kusmaryono, suyitno, dwijanto, & dwidayati, 2018; latifah & widjajanti, 2017; maharani, kholid, pradana, & nusantara, 2019; maulidia, johar, & andariah, 2019; putra, thahiram, ganiati, & nuryana, 2018). in addition to problem-solving skills, another important ability for students is the ability to learn independently. self-regulated learning is a learning process where students can take the initiative, with or without the help of teachers/friends/other people, to diagnose needs, formulate goals, identify sources, select and implement strategies, and evaluate learning outcomes that suits him (adam et al., 2017; gabriel, buckley, & barthakur, 2020; guo, lau, & wei, 2019; guo & wei, 2019; losenno et al., 2020; samo, 2016). selfregulated learning is an attitude with the characteristics of having the initiative, diagnosis, setting goals, monitoring, organizing and controlling, searching for and utilizing relevant sources, selecting and implementing strategies, and evaluating processes. the student learning outcomes, including having a self-concept and viewing adversity as a challenge (cárdenas-robledo & peña-ayala, 2019; peters-burton, cleary, & kitsantas, 2018). self-regulated learning is the ability of students to carry out the learning process without depending on others. ict-based learning in the new normal period makes learning independence have a big role in determining student understanding of learning. this is in line with the statement of kamal (2015) which explains that the self-regulated of student learning in mathematics is one aspect that contributes to the success and achievement of students in learning mathematics. self-regulated learning is one of the factors that affect students' mathematics learning outcomes that originate within students. self-regulated learning is predicted to play a role in the achievement of students' mathematical learning outcomes (adam et al., 2017; cárdenas-robledo & peña-ayala, 2019; gabriel et al., 2020; guo et al., 2019; guo & wei, 2019; losenno et al., 2020; peters-burton et al., 2018; samo, 2016). however, the self-regulated of students' mathematics learning is not easy to be maximally improved because of the many factors that affect students' desire to learn (krisnawati, rohaeti, & maya, 2018; rahmawati, rohaeti, & yuliani, 2018). to overcome this problem, one solution that can be taken is to return to the basics of learning itself, namely regarding the teaching materials used. based on the observations made, the teaching materials used during learning during the new normal period still do not support adequate ict-based learning. so we need ict-based teaching materials that can improve mpsa and student srl. based on these problems and needs, researchers created and developed visual basic application (vba) for microsoft word teaching materials which are expected to be a solution. this teaching material is very relevant for use in online learning which is very dependent on the use of ict so that online learning will be carried out more optimally (wahyudi, ambarwati, & indarini, 2019). this teaching material is a learning tool that contains learning materials, student worksheets, and evaluation tools using vba for microsoft word to improve students' mpsa, especially in polyhedron materials. this teaching material can give an interactive, effective, efficient, modifiable impression, volume 10, no 1, february 2021, pp. 17-30 19 can be adjusted to the desired theme, and provide audio and visual stimulation. besides, it is hoped that it can improve mpsa, especially the polyhedron material and increase students' srl during learning during the new normal period. the objectives and problem formulations in this study are: (1) how is the implementation of vba microsoft word teaching materials for junior high school students during the new normal period? (2) can teaching materials based on vba microsoft word improve the mathematical problem-solving abilities of junior high school students ? (3) can teaching materials based on vba microsoft word improve self-regulated learning for junior high school students? 2. method the method used in this study is the experimental method to shapes of pretest-postest control group design. the experimental class received learning with teaching materials based on visual basic for microsoft word applications, while the control class used ordinary learning (ol), in which both classes learned online. the population in this study were all students of state junior high school in cimahi city, while the sample was two classes randomly selected. the instrument used in this study was a test item in the form of a description of the mpsa as many as 6 questions and a scale of srl as many as 28 statements, as well as the giving of 38 student response questionnaires and interview guidelines for a preliminary study of potential problem analysis. examples of instruments used as references in measuring mathematical problemsolving abilities, self-regulated learning and closed questionnaires about vba teaching materials based on microsoft word are presented in figure 1, figure 2, and figure 3, respectively. figure 1. mpsa instrument figure 2. srl scale pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 20 figure 3. response questionnaire of vba microsoft word 3. results and discussion 3.1. results the low mpsa and srl of students during learning during the new normal period is a problem that must be resolved immediately because there is no clarity on how long the covid-19 pandemic will end. therefore, ict-based teaching materials are needed that can be used in face-to-face, online, and offline learning conditions. given that solving daily problems requires mathematical problem-solving skills and soft skills for students to learn independence because during the new normal period students are required to learn more independently. in addition, teaching materials are needed in accordance with the thinking stage of junior high school students, most of which are still at the concrete stage while the nature of mathematics is abstract. from the interviews with junior high school teachers who experienced learning during the normal period, it is just known as follows: (1) learning in the new normal period gives teachers and students the choice of face-to-face, offline, and online learning. however, to maintain safety, teachers choose online learning. however, learning is not optimal because there is no adequate digital-based teaching material; (2) online learning makes students tired quickly, so that the implementation of learning is not optimal. it takes teaching materials that can make students able to self-regulated learning according to the time that students are interested in; (3) during the new normal learning, students are confused about mastering mpsa. therefore, systematic teaching materials are needed following the meaning of mathematics itself; and (4) following the teacher's experience, mathematics material, which is considered difficult is the geometry materials. this is the teacher's concern in delivering the material because even face-to-face learning is the most difficult. so that it becomes a challenge in itself when learning online. 3.1.1. instructional material design based on visual basic application for microsoft word based on teaching materials vba for microsoft word have some stuffing content. figure 4 is an initial display of vba microsoft word -based teaching materials where there is a title, group name, member name, time. in addition, there is a button to turn on the audio, how many back sounds can be adjusted. volume 10, no 1, february 2021, pp. 17-30 21 figure 4. display a cover of teaching material figure 5 is a sample of teaching material content based on vba microsoft word where there are conceptual findings on the volume and surface area of the cube. in addition, teaching materials contains material from recognizing prisms and pyramids, making characteristics and definitions of prisms and pyramids, distinguishing the sizes of cubes and blocks, proving the volume of cubes and blocks, comparing the volume of prisms and pyramids, finding the formula for the surface area of prisms and pyramids, until as in figure 6, it is an evaluation tool that contains mpsa questions with different level of pain indexes. figure 5. concept discovery content samples pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 22 figure 6. evaluation tool 3.1.2. results of mathematical problem-solving ability, self-regulated learning, as well as the questionnaire responses students table 1. description of mpsa, srl, and questionnaire response variables stat vba material learning (n = 22) ordinary learning (n = 23) pretest postest n-gain pretest posttest ngain mpsa x 3.45 63.05 0.69 4.30 50.91 0.54 % 3.84 70.05 69 4.78 56.57 54 s 2.50 12.33 0.14 3.35 14.72 0.17 srl x 82.09 72.13 % 73.30 64.40 s 13.12 5.55 questionnaire response x 107.59 99.48 % 70.78 65.45 s 12.14 12.19 information: ideal score of mpsa = 90 ideal score of srl = 112 ideal score of questionnaire responses = 152 table 1 show the mpsa pretest for the experimental class was 3.45 (3.84%) lower than the control class value of 4.30 (4.78%) where the two classes were at very low interpretation. meanwhile, the posttest score of the experimental class was 78.18 (69.81%) which was in the high interpretation, which was higher than the control class which was 50.91 (56.57%) which was in sufficient interpretation. at the students’ self-regulated volume 10, no 1, february 2021, pp. 17-30 23 learning given at the end of the meeting for the experimental class, it was 82.09 (73.30 %) which was in good interpretation, which was higher than the control class of 72.13 (64.40%) in good interpretation too. in addition, the positive response of the experimental class students reached 107.59 (70.78%) which was in a good interpretation, which was higher than the control class of 99.48 (65.46%) who were in sufficient interpretation. on average and the percentage of learning outcomes in the experimental class is better than the control class. table 2. testing hypothesis of mean difference of mpsa and srl, and questionnaire response variables teaching method normality homogenity n sig (2-tailed) sig (1-tailed) interpretation mpsa pretes vba 0.007 22 0.356 pre-mpsavba = pre-mpsaol ol 0.017 23 mpsa postes vba 0.200* 0.539 22 0.005 0.002 pos-mpsavba > pos-mpsaol ol 0.200 * 23 n-gain mpsa vba 0.200* 0.555 22 0.004 0.002 n-gain mpsavba > n-gain mpsaol ol 0.200 * 23 srl vba 0.200* 0.000 22 0.002 0.018 srlvba > srlol ol 0.064 23 questionnaire response vba 0.200* 0.373 22 0.031 0.015 response vba > response ol ol 0.101 23 to answer the research hypothesis, a statistical test was carried out to see its significance. the results of the mpsa pretest show that there is no difference in the initial ability of solving mathematical problems between the experimental class and the control class (table 2). after receiving the learning with teaching materials that have been determine the statistical test data shows that the post-test mpsa students experiment is significantly better than the control class. the n-gain data shows that the increase in the experimental class is significantly better than the control class students. on the srl data shows that the experimental class students' srl is significantly better than the control class students. and the positive response data shows that the positive response of the experimental class students is significantly better than the control class. this shows the successful use of vba for microsoft word-based teaching materials in the implementation of learning in the new normal. table 3. testing hypothesis of association betweet mpsa, srl, and questionnaire response variable sig (2-tailed). interpretation q mpsa and srl 0.000 there is an association 0.87 (high) mpsa and response 0.053 no association srl and response 0.075 no association based on table 3, there is an association between mpsa and srl students whose learning uses vba for microsoft word based teaching materials with a high degree of pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 24 association. however, there is no association between the positive responses of students and the results of student mpsa and srl. 3.2. discussion vba microsoft word-based teaching materials has a good impact in learning, shown by the very low initial ability of students to solve mathematical problems. after receiving learning, achievement and improved student mpsa is at a high level and better by significantly compared with students who receive other learning. it because nature of teaching materials based vba microsoft word can be adjusted to the desired content and themes where the content of teaching materials can be customized by the indicators of the mpsa itself. the teaching material guides and requires students to recognize the difference between prisms and pyramids, then there are learning media that can be used to find concepts in the form of proving volume, surface area, nets by conducting experiments. besides, sharpening students 'numeracy skills which is an obstacle causing students to incorrectly solve questions because students' number operations are still lacking, therefore in teaching materials there are counting operation tools. this is in line with research conducted by bernard, sumarna, rolina, & akbar (2019) which shows that learning based on vba microsoft word enables students to easily understand learning. in addition, other research also reveals that ict-based learning can improve mathematical problem-solving abilities better than using ordinary learning (darari, 2017; puadi & habibie, 2018). furthermore, to analyzing the overall student mpsa test results, the students' errors in solving mpsa questions (see figure 7). figure 7. the mistakes of students in solving the mpsa questions figure 7 show the students mostly made mistakes in the calculation process, made mathematical models, and checked their answers again. this is because students do not have good mathematical problem-solving skills. mathematical problem-solving abilities, namely the ability of students to understand problems, develop solving strategies, carry out strategies in solving problems, and finally students can re-examine the results of their work (harisman et al., 2020; hutajulu et al., 2019; kusmaryono et al., 2018; maharani et al., 2019; maulidia et al., 2019). in addition to students' cognitive abilities that need to be improved, students' affective abilities also need to be optimized, especially students' self-regulated learning abilities because they are closely related to learning at home during the new normal period. using teaching materials based on vba microsoft word, the achievement of student srl is at a good level which is significantly better than students who receive regular learning. the volume 10, no 1, february 2021, pp. 17-30 25 results of giving the self-regulated learning scale are analyzed per indicator shown in table 4. table 4. achievement of students’ self-regulated learning indicator percentage (%) experiment control take the initiative to learn with or without the help of others 75.81% 66.15% be able to overcome obstacles or problems 72.24% 62.42% have self-confidence 70.29% 67.55% doing something without the help of others 74.84% 61.49% total 73.30% 64.40% table 4 show the attainment of independence on indicators of student learning initiative to learn with or without the help of others in the class experiment reached 75.81% and the control class is 66.15%. the indicators of being able to overcome obstacles or problems in the experimental class reached 72.24% and the control class was 62.42%. the indicators have self-confidence in the experimental class reaching 70.29% and the control class at 67.55%. the indicators of doing something without the help of others in the experimental class reached 74.84% and the control class was 61.49%. overall the srl of students in the experimental class reached 73.30% and the control class was 64.40%. this shows that online learning using vba microsoft word-based teaching materials makes students have better student srl than students who receive regular learning in interpretation. good. this is in line with research conducted by hendikawati, zahid, & arifudin (2019) and arifin & herman (2018) which reveal that learning using instructional media and/or ictbased teaching materials has self-regulated learning better students. regarding student responses to learning, at the end of the meeting, students were given a response questionnaire according to each class's teaching. the student response analysis is presented in table 5. table 5. recapitulation of learning questionnaire results using vba statement topic percentage experiment control have an interest and enthusiasm in learning mathematics with the teaching materials provided 73.51 67.60 able to overcome difficulties and dare to try new things 70.35 66.60 able to understand the concept of learning through the teaching materials provided 68.56 60.14 able to interact online and have the courage to express opinions 65.68 62.39 total 70.78 65.45 table 5 show the attitudes and responses of students during learning were more positive even though learning was carried out online, whereas junior high school students were very active in their behavior. students are more enthusiastic about learning at each meeting, and students can study independently at home according to the demands of learning in the new normal period. some students even have already understood the material and completed each material content in vba for microsoft word -based teaching materials. it pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 26 can be concluded that learning using vba microsoft word-based teaching materials can foster concept understanding, self-regulated learning, overcoming student difficulties in optimizing learning during the new normal period, besides that ict-based learning makes students close to technology so that students will be ready to face the industrial revolution 4.0. this is in line with research conducted by bernard et al. (2019) and padli & rusdi (2020) which show that students respond positively to ict-based learning, especially vba microsoft word. based on the results of observations on the implementation of learning using teaching materials based on vba microsoft word in terms of cognitive aspects, it can be seen that mastery of the concept of geometry of student shows perfect response and effect. this shows that the students can more quickly distinguish their characteristics and elements in the classification between prism and pyramid. learning starts with formal mathematics, which then finds examples in everyday life that students, so they can distinguish and know the names of prisms and pyramids. furthermore, the students' mathematical problem-solving ability is better. this can be seen in the evaluation stage during the learning process. students can be providing solutions to the questions given. this is because vba microsoft word has been prepared to have complete, structured, applicative content and follow the demands of mpsa itself. the results showed that the mathematical problem solving abilities and self-regulated learning of students who studied using vba microsoft word teaching materials were better than those who studied in the ordinary learning. this is show the vba microsoft word teaching materials have several advantages, namely: (1) microsoft word is very close and familiar to teachers and students so that its use will not be difficult; (2) learning in the new normal period becomes more effective because it can be used in various ways of learning face-to-face, offline and online and gives an interactive impression; (3) can be manipulated according to the desired theme and content; (4) can be made on other mathematical subjects; (5) provide audio and visual stimuli and stimuli; (6) the content of teaching materials is in accordance with the daily lives of students so that learning will be more meaningful; (7) the costs required are relatively small or even unnecessary; (8) everyday necessities that are accustomed to using microsoft word make it easy to access by users; (9) students have become closer to ict; (10) learning during the new normal does not easily become bored for students; (11) ict-based teaching materials are not easily damaged, easy to use, can be disseminated through various data sending media; (12) shorter learning time; (13) increasing the creativity of teachers in arranging teaching materials that can be adjusted under various conditions; (14) easy to store, save storage memory; (15) the implementation of learning is more effective, efficient, systematic, and structured; and (16) the implementation of learning is centered on student contributions so that students are always ready and enthusiastic in learning. 4. conclusion the conclusions of this study are: (1) the results of using vba microsoft wordbased teaching materials show that it is appropriate to apply in a new normal period, one of which is mathematics for junior high school students, especially in the most difficult material, namely wake polyhedron materials; (2) the achievement and improvement of students' mathematical problem-solving abilities who get learning with vba microsoft word is better than those who get ordinary learning. the achievement and improvement of students’ mathematical problem-solving abilities at a high level; (3) the students' selfregulated learning who get learning with vba microsoft word is better than those who get ordinary learning. the students’ self-regulated learning at a good level; (4) there is an volume 10, no 1, february 2021, pp. 17-30 27 association between mpsa and srl, and no association betweet mpsa and srl with student response levels; (5) using vba microsoft word-based teaching materials, students give a positive response amounting to 70.78%; (6) most of the student's difficulties in making a mathematical model in a matter of mpsa; and (7) students more enthusiastic in learning and are not easily bored, in line with the challenges of the new normal period, revolutionary industrial 4.0, and the learning curriculum, learning is more structured, interactive, effective, and efficient. acknowledgments the author would like to thank the director of the directorate of research and community service (drpm brin) of the republic of indonesia and ikip siliwangi for research funding in 2020 with contract number 080/sp2h/amd/lt/drpm/2020, 144/sp2h/amd/lt-mono/ll4/2020, and 03/lppm ikip-slw/kp-pt/v/2020. references adam, n. l., alzahri, f. b., soh, s. c., bakar, n. a., & kamal, n. a. m. 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(2019). development of web game learning materials for primary school students. infinity journal, 8(2), 199-208. https://doi.org/10.22460/infinity.v8i2.p199-208 https://doi.org/10.30738/indomath.v1i1.2091 http://doi.org/10.25273/jipm.v6i2.2007 https://doi.org/10.22460/jpmi.v1i4.p607-616 https://doi.org/10.22460/infinity.v5i2.p67-74 https://doi.org/10.22460/infinity.v8i2.p199-208 pertiwi, rohaeti, & hidayat, the students' mathematical problem-solving abilities … 30 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p109-120 109 pre service teachers’ perception on the implementation of project based learning in mathematic class marah doly nasution1, ahmad*2, zulkifley mohamed3 1universitas muhammadiyah sumatera utara, indonesia 2universitas muhammadiyah purwokerto, indonesia 3sultan idris education university, malaysia article info abstract article history: received nov 6, 2020 revised jan 16, 2021 accepted jan 18, 2021 project-based learning is one of the 21st-century methods that can increase the students' ability to have useful competence in their knowledge, especially in learning mathematic. hence, the teachers' perception of the implementation of pbl is essential to know. the purpose of this study was to determine student perceptions about the application of project-based learning models. this research is a survey research with a quantitative descriptive approach. the samples used in this study amounted to 63 students of the sixth semester in the department of mathematical education, muhammadiyah university north sumatra. the data collection technique used a questionnaire. the data analysis technique consists of quantitative analysis techniques with a statistical approach. the results of this study indicate that: (1) the students' perceptions of the interaction aspects of students and teachers are in the very good category with a mean score of 85.32%. (2) students' perceptions of motivation / increasing student interest in learning are in the very high category, with an average score of 85.53%. (3) students' perceptions on the competency aspect of understanding subject matter are in the very good category with an average score of 85.48%. (4) students' perceptions of the competency aspects of critical, effective, and efficient thinking are in the category of strongly agree with a mean score of 82.62%. (5) students' perceptions of good time management competence are in the good category with an average score of 79.10%. (6) students' perceptions of good student learning outcomes are in the very good category, with a mean score of 82.67%. (7) students' perceptions of the learning model's conformity aspect with subject characteristics are in the very high category with a mean score of 84.05%. keywords: mathematics education, pre service teacher, project based learning copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: ahmad, department of mathematics education universitas muhammadiyah purwokerto, jl. kh. ahmad dahlan, kembaran, banyumas, central java 53182, indonesia email: ahmad@ump.ac.id how to cite: nasution, m. d., ahmad, a., & mohamed, z. (2021). pre service teachers’ perception on the implementation of project based learning in mathematic class. infinity, 10(1), 109-120. 1. introduction teachers are expected to be able to choose and use learning strategies in accordance with the material to be delivered (english & kitsantas, 2013). every learning strategy has https://doi.org/10.22460/infinity.v10i1.p109-120 nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 110 strengths and weaknesses seen from various angles, but in essence any learning strategies, method or model used must have clear objectives to be achieved (kurzel & rath, 2007). because students have very heterogeneous interests, ideally a teacher should use a multimethod, that is, varying the use of the learning model used in the classroom. this is intended to avoid boredom experienced by students (tamim & grant, 2013). in this process, students use all their basic abilities and knowledges to obtain information and the learning outcomes they get. students try to find out and solve solutions to any existing problems (amamou & cheniti-belcadhi, 2018). so that students get experience and knowledge that are truly meaningful, not only learning outcomes in terms of values. project-based learning (pbl) is a learning model that is conceptualized on a process, timed, problem-focused, meaningful learning unit by integrating concepts from a number of components of knowledge, discipline and collaborative learning activities. so that in its implementation the teachers act as a facilitator whose task is to help provide experience for students in designing problem solving related to the subject matter (efstratia, 2014). students are expected to be able to interact with teachers and study groups to find solutions of the problems (van rooij, 2009). pbl contains project-based complex tasks based on the questions and problems that are very challenging, and requires students to design, solve problems, make decisions, carry out investigative activities, well as provide the opportunity for students to work independently. the goal is that students have independence in completing the tasks they face (barron et al., 1998). in the implementation of learning, pbl teachers pose problems in the form of sheets student activities then students carry out activities to complete problems and teachers oversee student performance (amamou & cheniti-belcadhi, 2018). when finished, one of the students representing the group presented the results of the discussion in front of the group then asked each other questions. the teacher leads the class discussion. this shows that with the pbl model students can develop conceptual understanding, procedural skills, ways of thinking of launching a related problem so that they can understand the problem setting and find out the next steps to take through discussion (sart, 2014). as a result of the discussion, students will become more skilled at using ideas and techniques that produce experiences of the problems at hand (anazifa & djukri, 2017). this research tried to explore more deeply the teachers’ perception about using pbl. the deep analysis by exploring the previous research and the current literature review enrich the novelty of this research. the reasons in conducting this research know the teacher’s perception on pbl model will give a new paradigm on how to use pbl in teaching and learning process, especially in learning mathematic, thus the possession of this research is very important for futher research the same topic. based on the considerations of the thoughts and problems, the researcher took the title "pre service teachers’ perception on the implementation of project based learning in mathematic class”. 2. method the type of research used is a survey with a quantitative descriptive approach, where the phenomena to be studied are events that have passed or are ongoing (design-based research collective, 2003). the procedure of the research follows the scientific model of descriptive analysis in which it is conducted using some stages, namely: deciding the best model of research, selecting the suitable respondents, creating and validating the instrument and doing the scientific analysis methods to guarantee the output of this research. this type selected for this study intends to reveal how the process of applying and perceptions of students about the model of project-based learning (project-based learning) in mathematics volume 10, no 1, february 2021, pp. 109-120 111 samples used in this study amounted to 63 respondents student 6th semester mathematics education, university of muhammadiyah sumatra utara. in collecting information about student perceptions, researchers used a questionnaire in the form of a closed questionnaire with alternative answers given to the indicators, namely strongly agree, agree, disagree), and strongly disagree. there are 7 indicators used to measure student perceptions with a total of 30 questions (see table 1). table 1. research instrument grid of project-based learning in mathematics indicator item number total number 1. interaction aspects students and teachers 1, 2, 3, 4, 5 5 2. ability to motivate / increase interest student learning 6, 7, 8, 9 4 3. understanding the subject matter 10, 11, 12, 13, 14 5 4. emerging think critically, effectively and efficiently 15, 16, 17,18, 19 5 5. effective time management 20, 21, 22 3 6. student learning outcomes 23, 24, 25 3 7. the suitability of the application of the learning model with subject characteristics 26, 27, 28, 29, 30 5 number of questions / statements 30 descriptive statistical analysis, including mean price, standard deviation, median value, mode, range, highest score, lowest score and frequency distribution were used for each research variable / indicator. 3. results and discussion in this research, there are 7 indicators that can measure the success of implementing the project-based learning model. the following is an analysis of the results of research on student perceptions about the application of project-based learning in subjects (see table 2). table 2. results of analysis of students' perceptions indicator percentage category student perceptions in terms of student and teachers interaction aspects 85.32% very good student perceptions in terms of motivation / increase student learning interest 83.53% very good perception of subject matter 85.48% very high student perceptions are reviewed from critical, effective and efficient thinking competencies 82.62% very good nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 112 indicator percentage category perception of students judging from good time management competence 79.10% good perception of students judging from results good student learning 82.67% very good student perceptions about the suitability of the application of learning models with subject characteristics 84.05% very high table 2 shows that: (1) students' perceptions of the aspects of student and teachers’ interaction are in the very good category with an average score of 85.32%; (2) students' perceptions on the aspect of motivation / increasing student interest in learning are in the very high category with an average score of 85.53%; (3) students' perceptions on the competency aspect of understanding subject matter are in the very good category with an average score of 85.48%; (4) students' perceptions of the competency aspects of critical, effective and efficient thinking are in the category of strongly agree with a mean score of 82.62%; (5) students' perceptions of the aspects of good time management competence are in the good category with an average score of 79.10%; (6) students' perceptions on aspects of good student learning outcomes are in the very good category with a mean score of 82.67%; (7) students' perceptions on the conformity aspect of the learning model with subject characteristics are in the very high category with a mean score of 84.05%. 3.1. student’ perception in terms of student and teachers’ interaction the results of data analysis on student perceptions about project-based learning models on the indicators of student and teachers’ interaction aspects can be seen in the table 3. table 3. student perception of student and teachers’ interaction ideal score interval frequency % category > 16.25 35 55.56 very good 12.5 to 16.25 28 44.44 good 8.75 to 12.5 0 0 bad < 8.75 0 0 very bad table 3 shows that there are 35 students (55.56%) who are in the very good category, 28 students (44.44%) are in the good category and there are no students who are in the sufficient and very bad category. from the analysis of each question item of the indicators of student and teachers’ interaction aspects which consist of 5 questions. the lowest mean score was 3.32, which is in question item number 1 with the question: "i feel more active in working on drawing assignments with a project-based learning model". about 5 the questions raised, only item number 1 has the lowest mean, but that does not mean intercourse students and faculty are located in the poor category. this is evidenced by the interaction between students and teachers on question items 2-5 still going very good. so that it can support the deficiencies in item number 1. volume 10, no 1, february 2021, pp. 109-120 113 data obtained from questionnaires given to 63 respondents on indicators of the interaction of students and professors showed that the tendency of the average score (mean) of 17.06 (85.32%) lies in the class interval > 16.25 in the excellent category. from the findings that have been explained, that students strongly agree with the application of the project-based learning model on the aspects of student and teachers’ interaction (mills & treagust, 2003). this is in accordance with the objectives of implementing the project-based learning model, namely to create a more active condition of student and teachers’ interaction (condliffe et al., 2017), so that in dealing with problems in carrying out assignments and subject matter can run smoothly. the teachers as a facilitator in this learning model can play a role, namely by knowing the progress of the learning activities that students carry out from the interaction process. it is stated that the learning method is a method used by teachers to establish relationships / interactions with students. 3.2. students’ perception of the term motivation and learning interest the results of the data analysis, perceptions of students about project-based learning model on indicators of motivation / growing interest in learning of students (see table 4). table 4. students’ perception in the term of motivation and learning interest ideal score interval frequency % category > 13 32 50.79 very high 10 to 13 31 49.21 high 7 to 10 0 0 moderate < 7 0 0 low table 4 shows that there are 32 students (50.79%) in the very high category, 31 students (49.21%) in the high category and no students who are in the medium and low categories. from the analysis of each question item of this indicator it can be seen that the lowest mean score was 3.21, which is in question item 9 with the question: "i don't feel tired in doing every assignment given by the teachers ". with these findings, this statement implies that students still feel tired in doing their very bad on work. this is influenced by the length of the meeting duration for each one-time meeting. and the number of each very bad on competency that must be achieved. however, of the 4 question items, there are weaknesses in question number 9 that can be overcome with questionable items 6-8, that students remain motivated, increase interest in learning and are enthusiastic in implementing project-based learning models in mathematics subjects. data obtained from questionnaires given to 63 respondents on indicators of motivation interest to learn the students showed that the tendency of the average score (mean) of 13.37 (83.53%) lies in the class interval > 13 in the category are very high. from the findings that have been explained, that students strongly agree with the application of the project-based learning model in mathematics in the aspect of motivation / increasing student interest in learning. this indicator itself is a development of the analysis of student characteristics in the planning process of the learning model. so that in its application, it can be known or can be determined by the learning model used which can motivate / increase student interest in learning. and with this project-based learning model, it has been proven that students can be motivated / increase their interest in learning. nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 114 3.3. students’ perception of the term of understanding subject matter the results of data analysis on students' perceptions about project-based learning models on the competency indicators of understanding the subject matter (see table 5). table 5. students’ perception understanding subject matter ideal score interval frequency % category > 16.25 37 58.73 very good 12.5 to 16.25 24 38.10 good 8.75 to 12.5 2 3.17 bad < 8.75 0 0 very bad table 5 show that, there are 37 students (58.73%) who are in the very good category, 24 students (38.10%) are in the good category and there are no students who are in the sufficient and poor category. from the analysis of each question item of the competency indicator in understanding the subject matter which consists of 5 questions. obtained the lowest average score (mean) of 3.32, which is in question item number 13 with the question: "from the application of the project-based learning model, i have come to understand the meaning of each line in the mathematical picture". this question item has a role as a form of deepening students in understanding the material from mathematics subjects. overall, the existing data on the indicators of material understanding, students can be said to understand and understand the content of mathematical subject matter by applying the mathematics learning model. this is supported by data analysis on this indicator, there are no students who disagree and disagree with the application of the project-based learning model in mathematics. data obtained from questionnaires given to 63 respondents to the indicators of competence to understand the subject matter shows that the propensity score average (mean) of 17.10 (83.48%) lies in the class interval > 16.25 excellent category. from the findings that have been explained, that students strongly agree with the application of the project-based learning model in mathematics in the aspects of understanding the subject matter. this indicator is intended to measure the extent to which students understand understanding of the material considering the purpose of implementing a project-based learning model is to provide a memorable learning experience for students. as the definition stated that project based learning is a method that foster abstract, intellectual tasks to explore complex issues (alacapinar, 2008). pbl is a learning approach that pays attention to understanding the subject matter. so that students are required to explore, assess, interpret and synthesize learning information in meaningful ways (mcdonald, 2008). 3.4. students’ perception of the term of critical, effective and efficient thinking competencies the results of data analysis on students' perceptions about project-based learning models on the indicators of competency in critical thinking, effective and efficient (see table 6). volume 10, no 1, february 2021, pp. 109-120 115 table 6. student perception in terms of critical, effective and efficient thinking competencies ideal score interval frequency % category > 16.25 31 49.21 very good 12.5 to 16.25 32 50.79 good 8.75 to 12.5 0 0 bad < 8.75 0 0 very bad table 6 shows that there are 31 students (49.21%) who are in the very good category, 32 students (50.79%) are in the good category and there are no students who are in the sufficient and poor category. from the analysis of each question item of the competency indicator to think critically, effectively and efficiently which consists of 5 questions. obtained the lowest average score (mean) of 3.11, which is in the item question number 19 with the question: "the project-based learning model made me find new ideas to work on math picture assignments ". this statement means that students have not been able to find renewable ideas by doing math assignments. even though the project-based learning model in its application requires students to develop student ideas to find new knowledge in dealing with the work they face. however, the other four statements have shown good results, namely students are able to think creatively, can find solutions to any existing problems, can apply their drawing assignments into existing assignment exercises and can explore the potential that exists in students. data obtained from questionnaires given to 63 respondents to the indicators of competence critical thinking, effective and efficient show that the propensity score average (mean) of 16.52 (82.62%) lies in the class interval > 16.25 in the excellent category. from the findings that have been explained, that students strongly agree with the application of the project-based learning model in mathematics subjects in the competency aspects of critical, effective and efficient thinking. so that in its application, the projectbased learning model has achieved self-development potential. this shows that the problems given by educators as a source of learning can train students to think and develop their potential and personality (harisman et al., 2020; hidayat & sariningsih, 2020; putra et al., 2020; widodo et al., 2020). 3.5. students’ perception of the term of time management the results of data analysis on student perceptions about project-based learning models on indicators of time management competence are good (see table 7). table 7. student perceptions in terms of time management ideal score interval frequency % category > 9.75 35 55.56 very good 7.5 to 9.75 19 30.16 good 5.25 to 7.5 9 14.29 bad < 5.25 0 0 very bad nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 116 table 7 shows that there are 35 students (55.56%) who are in the very good category, 19 students (30.16%) are in the good category, 9 students (14.29%) are in the sufficient category and there are no students who are in the poor category. from the analysis of each question item on the indicator of time management competence good which consists of 3 questions. the lowest mean score was 2.98, which is in the item question number 21 with the question: "i fill my spare time at home by doing math assignments". this statement has the lowest average score among the statements in the questionnaire regarding the application of the project-based learning model. this can happen because of the duration of the class meeting the students feel that they have had bad, because in one face-to-face duration the duration lasts 10 very bad for hours. this means that from morning class hours to completion, students are faced with one subject. this is what makes students sometimes feel bored and bored. therefore, the teachers applies a project-based learning model to overcome this problem. given this learning model, students are given the freedom to think and learn while adhering to the learning objectives. data obtained from questionnaires given to 63 respondents at the time of management competency indicators demonstrated that the tendency of the average score (mean) of 9.49 (79.10%) lies in the class interval 7.5 – 9.75 in both categories. from the findings that have been explained, it shows that students agree with the application of the project-based learning model in mathematics for the competency aspects of time management good. the data show that students only agree and do not really agree, this is because the duration of studying at school is too long and students tend to feel bored. from this indicator, it can be measured that students work on mathematics only during class meetings. so, in this aspect needs more attention when considering that the project-based learning model has the goal of making students more independent in doing tasks including managing study time students with good (scarbrough, bresnen, et al., 2004). 3.6. students’ perception of the term learning outcomes the results of data analysis on student perceptions about project-based learning models on indicators of student learning outcomes (see table 8). table 8. students’ perception in the term of learning outcomes ideal score interval frequency % category > 9.75 39 61.90 very good 7.5 to 9.75 20 31.75 good 5.25 to 7.5 4 6.35 bad < 5.25 0 0 very bad table 8 show that there are 39 students (61.90%) who are in the very good category, 20 students (31.75%) are in the good category, 4 students (6.35%) are in the sufficient category and there are no students who are in the poor category. from the analysis of each question item of the indicators of student learning outcomes which consist of 3 questions. obtained the lowest average score (mean) of 3.22, which is in question item 23 with the question: "i am satisfied with my score results". from this statement, there were 4 students who disagreed with the results of their learning scores. however, from the results of the evaluation conducted by the teachers, the overall student learning competency achievement has met the complete limit, namely with a minimum score of 75. volume 10, no 1, february 2021, pp. 109-120 117 data obtained from questionnaires given to 63 respondents on indicators of learning outcomes of students who demonstrated that the tendency of the average score (mean) of 9.92 (82.67%) lies in the class interval > 9.75 in the excellent category. from the findings that have been explained, that students strongly agree with the application of project-based learning models in mathematics as seen from the aspect of good student learning outcomes. this is also corroborated by the results of the teacher competency evaluation. that all students reach the minimum completeness limit of 75. so that in its application the project-based learning model has been said to be successful and has achieved the learning objectives. 3.7. students’ perception of the term of suitability of learning model application with subject characteristics the results of data analysis on student perceptions about project-based learning models on indicators of suitability of the application of learning models with subject characteristics (see table 9). table 9. student perceptions in terms of suitability of learning model application with subject characteristics ideal score interval frequency % category > 16.25 33 52.38 very high 12.5 to 16.25 30 47.62% high 8.75 to 12.5 0 0 moderate < 8.75 0 0 low table 9 shows that there are 33 students (52.38%) who are in the very high category, 30 students (47.62%) are in the high category and there are no students who are in the medium and low categories. from the analysis of each question item on the indicator of the suitability of the application of the learning model with the characteristics of the subject which consists of 5 questions. the lowest mean score was 3.29, which is in question item number 27 with the question: "the application of project-based learning models makes mathematics more interesting". and item number 28 with the question: "the picture i am working on making my math assignment more real". from these two statements, it can be seen that students think that using a project-based learning model has not been able to make mathematics more interesting and make it real. however, that statement is on a small scale. so that it does not really affect the application of project-based learning models in mathematics subjects. data obtained from questionnaires given to 63 respondents in conformity indicator learning model application with the characteristics of the subjects showed that the tendency of the average score (mean) amounted to 16.81 (84.05%) lies in the class interval > 16.25 in very high category. from the findings of the data that have been explained, that students strongly agree with the application of the project-based learning model in mathematics on the aspects of the suitability of the characteristics of the learning model with the subjects (gary, 2015). so that the project-based learning model on the subject has achieved success. the learning model is successful if it includes 3 stages, namely, planning, implementation and evaluation. the third phase of the project-based learning model in the eyes of subjects of mathematics has gone good and get a positive response from students (scarbrough, swan, et al., 2004). nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 118 4. conclusion the results of this study indicate that: (1) the students' perceptions of the interaction aspects of students and lecturers are in the very good category with a mean score of 85.32%. (2) students' perceptions on the aspect of motivation / increasing student interest in learning are in the very high category with an average score of 85.53%. (3) students' perceptions on the competency aspect of understanding subject matter are in the very good category with an average score of 85.48%. (4) students' perceptions of the competency aspects of critical, effective and efficient thinking are in the category of strongly agree with a mean score of 82.62%. (5) students' perceptions of the aspects of good time management competence are in the good category with an average score of 79.10%. (6) students' perceptions on aspects of good student learning outcomes are in the very good category with a mean score of 82.67%. (7) students' perceptions on the conformity aspect of the learning model with subject characteristics are in the very high category with a mean score of 84.05%. acknowledgments the authors would like to express my special thanks to university of muhammadiyah north sumatra, university of muhammadiyah purwokerto and university of sultan idris for supporting the facilities and finances to accomplish this paper. references alacapınar, f. (2008). effectiveness of project-based learning. eurasian journal of educational research (ejer), 33, 17-34. amamou, s., & cheniti-belcadhi, l. (2018). tutoring in project-based learning. procedia computer science, 126, 176-185. https://doi.org/10.1016/j.procs.2018.07.221 anazifa, r. d., & djukri, d. 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(2014). experiential education through project based learning. procedia-social and behavioral sciences, 152, 1256-1260. https://doi.org/10.1016/j.sbspro.2014.09.362 english, m. c., & kitsantas, a. (2013). supporting student self-regulated learning in problem-and project-based learning. interdisciplinary journal of problem-based learning, 7(2), 128-150. https://doi.org/10.7771/1541-5015.1339 https://doi.org/10.1016/j.procs.2018.07.221 https://doi.org/10.15294/jpii.v6i2.11100 https://doi.org/10.1080/10508406.1998.9672056 https://doi.org/10.3102/0013189x032001005 https://doi.org/10.1016/j.sbspro.2014.09.362 https://doi.org/10.7771/1541-5015.1339 volume 10, no 1, february 2021, pp. 109-120 119 gary, k. (2015). project-based learning. computer, 48(9), 98-100. https://doi.org/10.1109/mc.2015.268 harisman, y., noto, m. s., & hidayat, w. 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(2020). profile of mathematical representation ability of junior high school students in indonesia. journal of physics: conference series, 1657(1), 012003. https://doi.org/10.1088/1742-6596/1657/1/012003 sart, g. (2014). the effects of the development of metacognition on project-based learning. procedia-social and behavioral sciences, 152, 131-136. https://doi.org/10.1016/j.sbspro.2014.09.169 scarbrough, h., bresnen, m., edelman, l. f., laurent, s., newell, s., & swan, j. (2004). the processes of project-based learning: an exploratory study. management learning, 35(4), 491-506. https://doi.org/10.1177/1350507604048275 scarbrough, h., swan, j., laurent, s., bresnen, m., edelman, l., & newell, s. (2004). project-based learning and the role of learning boundaries. organization studies, 25(9), 1579-1600. https://doi.org/10.1177/0170840604048001 tamim, s. r., & grant, m. m. (2013). definitions and uses: case study of teachers implementing project-based learning. interdisciplinary journal of problem-based learning, 7(2), 72-101. https://doi.org/10.7771/1541-5015.1323 van rooij, s. w. (2009). scaffolding project-based learning with the project management body of knowledge (pmbok®). computers & education, 52(1), 210-219. https://doi.org/10.1016/j.compedu.2008.07.012 widodo, s. a., irfan, m., trisniawati, t., hidayat, w., perbowo, k. s., noto, m. s., & prahmana, r. c. i. (2020). process of algebra problem-solving in formal student. journal of physics: conference series, 1657(1), 012092. https://doi.org/10.1088/1742-6596/1657/1/012092 https://doi.org/10.1109/mc.2015.268 https://doi.org/10.22460/infinity.v9i1.p59-68 https://doi.org/10.28945/967 https://doi.org/10.18848/1447-9494/cgp/v14i10/45493 https://doi.org/10.18848/1447-9494/cgp/v14i10/45493 https://doi.org/10.1088/1742-6596/1657/1/012003 https://doi.org/10.1016/j.sbspro.2014.09.169 https://doi.org/10.1177/1350507604048275 https://doi.org/10.1177/0170840604048001 https://doi.org/10.7771/1541-5015.1323 https://doi.org/10.1016/j.compedu.2008.07.012 https://doi.org/10.1088/1742-6596/1657/1/012092 nasution, ahmad, & mohamed, pre service teachers’ perception on the implementation … 120 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p133-144 133 the development of online learning game on linear program courses zainnur wijayanto, dafid slamet setiana, betty kusumaningrum* universitas sarjanawiyata tamansiswa, yogyakarta, indonesia article info abstract article history: received dec 19, 2020 revised jul 14, 2021 accepted jul 16, 2021 this study aims to produce online learning games on linear program subjects for 3rd-semester students that are valid, practical, efective and can increase student learning motivation. students are less interested in linear program subject because it requires high analysis skill and feel boring. this research and development model consisting of the stages of define (literature study and field survey), design (early product design activities), develop (expert validation, product revisions, and development testing), and disseminate (final product revision) with descriptive analysis techniques qualitatively and quantitatively. qualitative descriptive analysis techniques are used to describe the stages of development that describe the results of observations of the implementation and effectiveness of online learning games that have been developed in the field. quantitative descriptive analysis techniques are used at the development stages. this study has achieved the expected goal: producing online learning games for linear program subjects that are valid (very good category), practical (very good category), and effective (85.71% of students had achieve a minimum score of c). keywords: learning games, linear programs, online this is an open access article under the cc by-sa license. corresponding author: betty kusumaningrum, department of mathematics education, universitas sarjanawiyata tamansiswa, yogyakarta jl. batikan, tahunan, umbulharjo, daerah istimewa yogyakarta 55167, indonesia email: betty.kusumaningrum@ustjogja.ac.id how to cite: wijayanto, z., setiana, d. s., & kusumaningrum, b. (2022). the development of online learning game on linear program courses. infinity, 11(1), 133-144. 1. introduction the development of science and technology encourages the learning process to be more applicable and attractive as an effort to improve the quality of education (hughes, 2004). appropriate teaching methods will help the students understanding process, so students can apply the knowledge gained in everyday life (ganyaupfu, 2013; kalyani & rajasekaran, 2018) . one of the ways to encourage the achievement of effective learning, is to use learning aids or what are commonly called media (wahid et al., 2020; wardani et al., 2018). one of the media that can be used in learning is a game (wahyudi et al., 2019). games can be used as learning media (becker, 2007). one of the effective learning methods https://doi.org/10.22460/infinity.v11i1.p133-144 https://creativecommons.org/licenses/by-sa/4.0/ wijayanto, setiana, & kusumaningrum, the development of online learning game … 134 to achieve learning goals is to use games, or the learning process through gaming (aleksić et al., 2016; ding et al., 2017). gaming is one of the main methods in the learning process for adults (hidayat, 2018). therefore the role of the media in the learning process is important because it will make the learning process more varied and less boring (pivec & kearney, 2007; wijaya, 2020). the use of games as learning media can be applied to technologies that are widely used today, such as mobile phones (ekanayake & wishart, 2014). a mobile phone is a communication tool that can be used to make calls or send text messages. mobile phones that are currently developing have supporting features, including internet, game applications, and others. this mobile phone can be used as a device to play mobile games (godwin-jones, 2011). implementation of games on mobile phones and internet networks as learning media at the higher education are not fully used yet (yu & conway, 2012). therefore, it is necessary to design online learning games as an innovation in mathematics learning that can increase motivation to learn mathematics in the subject that are considered difficult by students, one ofthem is a linear program. liner program requires analysis and long processing steps (ariawan, 2015) and also requires high reasoning and communication skills (saparwadi & aini, 2016). it can be seen in the final exam, there were still many students who get c+ (17.85%), c (14.28%), and d (25%) grades. by using online games, it is expected to increase student motivation which can improve student learning outcomes.the results of this study are expected to be used as a reference in developing learning media that is suitable for internet generation learning. in addition, learning games that have been designed can be used by educational institutions as a basis for improving the quality of the process and student learning outcomes. the need for more urgent references is associated with the lack of references that discuss online game development in mathematics learning at the higher education. 2. method 2.1. research design the procedure for developing a linear program online learning game consists of define, design, develop, and disseminate stages.the define stage includes literature study and field surveys to identify problems and analyse needs; the design stage includes early product design activities; the develop stage includes expert validation, product revisions, and development testing, while the dissemination stage includes final product revision. the development procedure are presented in figure 1. figure 1. the development procedure volume 11, no 1, february 2022, pp. 133-144 135 2.2. participants research subjects are grouped based on two activities: product development and product development testing. at the product development stage, research subjects consisted of 2 expertswhile at the product development testing stage, the research subjects were 28 students. 2.3. data collection in the define stage, data was collected by observation, interviews, and literature studies to obtain information about the use of technology and the internet in learning among students; (2) the design stage: data was collected by literature study to produce linear program questions as material for making online learning games; (3) the development stage: data was collected by instrument validation (at the early product development stage), a questionnaire (at the development testing and the experiment testing) and student learning outcomes. 2.4. data analysis the data were analysed by descriptive qualitative and quantitative. qualitative descriptive analysis was used to describe the development implementation of linear program online learning game, and describe the effectiveness of linear program online learning games that have been developed in the field. quantitative data analysis was used at the development stage. the data analysis approach used was: (1) the implementation and results of the online learning game development linear program and the validity of the linear program questions described in the form of a data presentation, then analyzed qualitatively and quantitatively; (2) at the development testing, the test results of the application of the linear program online learning game were analyzed by a quantitative approach; (3) experiment testing using a quantitative analysis approach. the following were the steps that used to determine the quality criteria for the developed game-based learning media products: (1) data in the form of expert scores obtained through the validation sheet are added up; (2) the actual total score was obtained converted into qualitative data on a scale of four shown in table 1. table 1. conversion of quantitative data to qualitative data of scale 5 interval category x > mi + 1.5 sbi verry good mi + 0.5 sbi < x ≤ mi + 1.5 sbi good mi 0.5 sbi < m ≤ mi + 0.5 sbi enough mi 1.5 sbi < m ≤ mi 0.5 sbi less x ≤ mi 1.5 sbi verry less the analysis technique for measuring validity was to provide the instruments that would be validated to the expert deemed appropriate to provide an assessment. the aspects that were assessed in term of material/content, construction, and language. the criterion used to decide that an instrument has an adequate degree of validity is if the mean (x) of the results of the assessment for all minimum aspects is in the "valid" category. if this is not the case, it is necessary to revise it based on the suggestions of the validators or by looking back at the aspects that are lacking, re-validated then re-analyzed. and so on until it meets the minimum average value in the valid category. wijayanto, setiana, & kusumaningrum, the development of online learning game … 136 linear program online learning game are considered practical if the results of student assessments for all minimal aspects are in the "good" category. these results indicate that the online games developed are practical and can be applied at universities, while it can be said effective if it meets the effectiveness indicator: the achievement of student competence classically reaches the minimum category c (passed). the data analysis steps on the effectiveness of the linear program online learning game in terms of achieving student learning competencies were: (1) recapitulating the scores that appear automatically at the end of the game (2) calculating the (number) of students who reach the specified level of learning outcomes: minimum c; and (3) determine the achievement of the objectives or classical learning outcomes (all students in one class): minimum 75% of students get a c grade. 3. results and discussion 3.1. results the result of this research was a product in the form of an online learning game “kahoot” in the linear program subject. before developing a product, a needs analysis must be carried out first. needs analysis was carried out to obtain information about linear program learning of 3rd semester mathematics education students and the use of technology and internet in mathematics learning among the students. the needs analysis obtain in this research process are linear program learning has taken advantage of technology by using lindo software (linear interactive discreat optimizer). this software is used to solve linear program problems easily, quickly, and accurately. however, in linear program learning, internet-based technology has not been utilized, such as the use of online game-based applications that encourage motivation and interest in learning linear program. online game applications using kahoot have never been used in linear program subject. kahoot is a free online game-based learning platform that can be accessed via a web browser or kahoot application using a computer or gadget that is easily accessible to students. after conducting a needs analysis, the next step was design. the first design stage in this research was a literature study that used for making questions about linear program in online learning games. the next stage was make an online learning game using kahoot. the questions were made in 2 forms, 5 multiple choice questions and 5 true and false questions. multiple choice and true-false questions are selected based on their ability to cover a wide range of subjects in an exam. before making the questions, the researcher determined the question lattices first, so that the questions were in accordance with the indicators of cognitive ability to be measured. the grid of questions is presented in table 2. table 2. the grid of questions indicators number form of problem can determine the area of the set of solutions of a known system of two-variable linear inequalities 1, 2 multiple choice can determine a two-variable linear inequality system from the set area of known solutions in the problem 3,4,5 multiple choice can determine the area of the set of solutions of a known system of two-variable linear inequalities 6,7 true-false can determine the minimum value of a function if the area of the set of solutions is known 8 true-false volume 11, no 1, february 2022, pp. 133-144 137 indicators number form of problem can determine the maximum revenue from a sale if the area of the set of solutions is known 9 true-false can determine the intersection point between two lines 10 true-false the questions that have been made then validated. the purpose of validation was to find out whether the instrument could be measuring what is desired. validation was carried out by 2 lecturers. the validation process was validating ten linear program questions in the kahoot. the results of the question validation for each component can be seen in table 3. table 3. results of validation no components v1 v2 a. content 1 problems in accordance with the measured cognitive domain 5 5 2 there is only one correct answer 5 5 total 10 10 average 10 b. construction 1. problem is clearly formulated 5 4 2. the question provides no clues to the answer 5 5 3. the images/graphs/diagrams are presented clearly 4 5 4. the item does not depend on the answer to the previous question 5 5 5. problem does not contain double negative statements 5 5 total 24 24 average 24 c. language 1. the language used is in accordance with indonesian rules 5 5 2. communicative language 5 5 3. no local language is spoken 5 5 total 15 15 average 15 the criteria and limit values for content were determined (see table 4). table 4. the criteria and limit values of content content the criteria and limit values of content the max score of each item: 5 the min score of each item: 1 number of item: 2 ideal average = 6 2 210 = + standard deviation = 3 4 6 210 = − =1.3 very good x > 7.95 good 6.65 < x ≤ 7.95 enough 5.35 < x ≤ 6.65 less 4.05 < x ≤ 5.35 very less x ≤ 4.05 average : 10 (very good) the criteria and limit values for construction were determined (see table 5). wijayanto, setiana, & kusumaningrum, the development of online learning game … 138 table 5. the criteria and limit values of construction construction the criteria and limit values of construction the max score of each item: 5 the min score of each item: 1 number of item: 5 ideal average = 15 2 525 = + standard deviation = 6 525 − = 3.3 very good x > 19.95 good 16.65 < x ≤ 19.95 enough 13.35 < x ≤ 16.65 less 10.05 < x ≤ 13.35 very less x ≤ 10.05 average : 24 (very good) the criteria and limit values for language were determined (see table 6). table 6. the criteria and limit values of language language the criteria and limit values of language the max score of each item: 5 the min score of each item: 1 number of item: 3 ideal average = 9 2 315 = + standard deviation = 6 315 − = 2 very good x > 12 good 10 < x ≤ 12 enough 8 < x ≤ 10 less 6 < x ≤ 8 very less x ≤ 6 average : 15 (very good) the overall results of test validation were very good (see table 7), so the linear program test proper to use in the kahoot. the validator also provides suggestion: questions are formulated more clearly and the pictures/graphs are presented more clearly. the questions that have been validated are then used in the kahoot learning game. table 7. summary of test validation results no component average criteria 1 content 10 very good 2 construction 24 very good 3 language 15 very good development testing the subjects in this stage were 5 students of the mathematics education study program who have been taking linear program lectures. the students were asked to become a player in the online learning game kahoot. the researcher simulated by showing the game using a projector and the five students were asked to take part in the game until finish using their smartphones. then, students filled out a questionnaire and provide a suggestion on the linear program online learning game that has been made. the results of the questionnaire can be seen on table 8. table 8. development testing result components respondents total 1 2 3 4 5 game online 1 increase interest on linear programming 5 5 5 5 5 25 2 increase motivation to learn 5 5 5 5 5 25 volume 11, no 1, february 2022, pp. 133-144 139 components respondents total 1 2 3 4 5 3 improve communication to exchange information among studens 5 5 5 5 5 25 4 the set time duration can increase the speed and accuracy in answering questions 5 5 5 5 5 25 5 easy to use 4 5 4 4 5 22 average 24.4 appearance 1 easy to read 5 5 5 5 5 25 2 images/graphics are presented clearly 5 4 4 5 5 23 3 the order of the questions is adjusted to the difficulty level of the questions 5 5 5 5 5 25 4 the duration of answering the questions is sufficient 5 5 5 5 5 25 5 music can support competition and concentration 5 5 5 5 5 25 average 24.6 the criteria and limit values of development testing were determined (see table 9). table 9. the criteria and limit values of development testing development testing the criteria and limit values of language the max score of each item: 5 the min score of each item: 1 number of item: 5 ideal average = 15 2 525 = + standard deviation = 6 525 − = 3.3 very good x > 19.95 good 16.65 < x ≤ 19.95 enough 13.35 < x ≤ 16.65 less 10.05 < x ≤ 13.35 very less x ≤ 10.05 overall, the result of development testing gots very good criteria (see table 10). however, there was a less than optimal score on the “easy to use” component, because the ability to access the internet varies depending on the cellular internet provider of each student. in addition, the weakness of this linear program online game was the graphics were not clearly visible. table 10. summary of development testing results no components average criteria 1 game online 24.4 very good 2 appearance 24.6 very good experiment testing after the kahoot is implemented, students then filled out 30 statements on a questionnaire to find out their responses to the kahoot online learning game that has been developed. this questionnaire aims to determine the practicality of kahoot online learning games. the questionnaire results are presented in table 11. wijayanto, setiana, & kusumaningrum, the development of online learning game … 140 table 11. student questionnaire results no aspect item practicality grade x category 1 content 1-8 very good x > 67.35 good 65.18 < x ≤ 67.35 enough 63.01 < x ≤ 65.18 less 60.84 < x ≤ 63.01 very less x ≤ 60.84 68.3 very good 2 learning 9-14 very good x > 72.83 good 65.83 < x ≤ 72.83 enough 58.83 < x ≤ 65.83 less 51.83 < x ≤ 58.83 very less x ≤ 51.83 72.85 very good 3 technical 15-21 very good x > 69.18 good 63.63 < x ≤ 69.18 enough 58.08 < x ≤ 63.63 less 52.53 < x ≤ 58.08 very less x ≤ 52.53 69.8 very good 4 overall evaluation 22-30 very good x > 67.27 good 63.54 < x ≤ 67.27 enough 59.8 < x ≤ 63.54 less 56.06 < x ≤ 59.8 very less x ≤ 56.06 67.32 very good all aspects of the assessment were “very good” category, it can be concluded that the online learning game is practical for use in learning. media can be accessed and used by students easily without expensive costs. after the students have finished working on 10 questions on the kahoot, a summary of the student's work can be seen immediately. from the results of the summary analysis, the level of difficulty of each question item can be determined (see table 12). table 12. results of test analysis number the number of students who answered correctly percentage of students who answered correctly test difficulty level 1 18 64.29 easy 2 13 46.43 easy 3 18 64.29 easy 4 15 53.57 moderate 5 14 50 moderate 6 21 75 easy 7 14 50 moderate 8 8 25 difficult 9 10 35.71 difficult 10 22 75 easy the scores obtained by the students were then converted into letter grades according to the university's academic guidelines (see table 13). volume 11, no 1, february 2022, pp. 133-144 141 table 13. conversion of learning outcomes interval class letter grades score category 90 – 100 a 4.00 very excellent 80 – 89 a3.80 excellent 75 – 79 b+ 3.30 very good 68 – 74 b 3.00 good 64 – 67 b2.80 pretty good 60 – 63 c+ 2.30 enough 56 – 59 c 2.00 not enough 40 – 55 d 1.00 less 0 – 39 e 0 very less student scores obtained from the application of the kahoot online learning game are shown in table 14. table 14. student score no name correct score letter grades no name correct score letter grades 1 a 9 90 a 15 o 7 70 b 2 b 9 90 a 16 p 7 70 b 3 c 9 90 a 17 q 7 70 b 4 d 9 90 a 18 r 7 70 b 5 e 8 80 a 19 s 7 70 b 6 f 8 80 a 20 t 7 70 b 7 g 8 80 a 21 u 6 60 c+ 8 h 8 80 a 22 v 6 60 c+ 9 i 8 80 a 23 w 6 60 c+ 10 j 8 80 a 24 x 6 60 c+ 11 k 8 80 a 25 y 5 50 d 12 l 7 70 b 26 z 5 50 d 13 m 7 70 b 27 aa 4 40 d 14 n 7 70 b 28 ab 4 40 d 3.2. discussion the teaching-learning process is an important element in education, it includes the planning, implementation and assessment processes. assessment refers to the process of observing the changes in the lives of the people (çalışkan & kaşıkçı, 2010). during a pandemic, the right way to assess is to do a digital assessment. digital assessments provide instant feedback and can be used to conduct individual or group assessments in a competitive environment (yilmaz & baydas, 2017). so, digital assessment in education is important in terms of feedback and control of learning. one of the digital assessments that can be used in learning, especially in learning mathematics is kahoot!. the kahoot! is a free online game application that can be used in every learning session to increase student participation and can be used as a formative assessment. based on the results of development testing, it was concluded that kahoot! easy to apply in learning because it does not require special training for students and can be easily accessed via a smartphone or pc (plump & larosa, 2017). the set time duration can increase the speed and accuracy in answering questions. the points earned by students are wijayanto, setiana, & kusumaningrum, the development of online learning game … 142 determined by how quickly students respond to questions as well as to answer question correctly. kahoot! can be used to increase students' concentration during the course, especially when their concentration starts to decrease after the first ten minutes. in this study, kahoot! proven to increase student interest in linear program lectures. they are motivated to attend lectures by communicating with others to exchange information about the subject. in terms of appearance, kahoot! can attract students' attention because of kahoot! equipped with features that support the learning process. during the quiz, there is a music feature that can increase the enthusiasm of students in competing. in each quiz, kahoot comes with customizable response times. when the time was up, a voice sounded and the names of the top 5 students were displayed on the board (bawa, 2019; çeti̇n, 2018). quiz featured on kahoot! can vary, it can be in the form of quizzes, discussions or questionnaire, so that students do not get bored easily with the monotonous form of questions. in this study, the researcher used multiple choice questions and true-false questions with a sequence of questions adjusted to the level of difficulty. based on the results of the test analysis, it is known that students who use kahoot! in learning activities achieve good grades in exams. it means that the kahoot! application had the potential to improve and develop the high scores in exams. our outcomes show that kahoot! motivated students to be engaged the knowledge and encourage interaction in the classroom. student stated that kahoot! had a positive impact on their knowledge and skills. features on kahoot! which can increase attention and student involvement really supports students in learning. 4. conclusion this study has achieved the expected goal: producing valid, practical, and effective online learning games for linear program subjects. validity is seen from the results of expert validation. overall from the results of expert validation, it can be said that this product was very good or feasible to use. practicality of this product is seen from the results of development testing and experiment testing. it show that the student's assessment of the product being developed was also very good or practical for use in learning. effectiveness is obtained from student scores at the end of the game. students who achieve a minimum score of c were 85.71%. the linear program online learning game is said to be effective if the overall score obtained by students who reaches a minimum value of c is at least 75%. it can be concluded that the linear program online learning game was effectively used in learning (apsari & rizki, 2018; ariawan et al., 2017). based on the results of the study it can be concluded that the development of kahoot! meet the criteria of validity, practicality, and effectiveness. by the results of the validity test, the development of kahoot! it has very good criteria in terms of content, construction, and language. kahoot! also meet practical requirements in terms of content, technical learning, and overall evaluation. it means that kahoot! easy to read, can present pictures/graphics clearly, enough time for students to answer the questions, the music features of kahoot! can support competition and concentration, and the form of the questions presented is appropriate and adjusted to the level of difficulty of the questions. application of kahoot! also able to support students to achieve good scores on tests. based on the results of the study, it is recommended that teachers can use creative learning media that utilize technology so that students become accustomed to using computers or laptops and smartphone not only to play game but also enrich their insights. the use of educational games in the classroom can also minimize interference, so as to improve the quality of learning. volume 11, no 1, february 2022, pp. 133-144 143 references aleksić, v., ivanović, m., budimac, z., & popescu, e. 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(2019). development of web game learning materials for primary school students. infinity journal, 8(2), 199-208. https://doi.org/10.22460/infinity.v8i2.p199-208 wardani, d. k., martono, t., pratomo, l. c., rusydi, d. s., & kusuma, d. h. (2018). online learning in higher education to encourage critical thinking skills in the 21st century. international journal of educational research review, 4(2), 146-153. https://doi.org/10.24331/ijere.517973 wijaya, t. t. (2020). how chinese students learn mathematics during the coronavirus pandemic. ijeri: international journal of educational research and innovation, 15, 1-16. https://doi.org/10.46661/ijeri.4950 yilmaz, r. m., & baydas, o. (2017). an examination of undergraduates’ metacognitive strategies in pre-class asynchronous activity in a flipped classroom. educational technology research and development, 65(6), 1547-1567. https://doi.org/10.1007/s11423-017-9534-1 yu, f., & conway, a. (2012). mobile/smartphone use in higher education. proceedings of the 2012 southwest decision sciences institute, 831-839. https://doi.org/10.1177/2379298116689783 https://doi.org/10.20414/jtq.v14i1.20 https://doi.org/10.1088/1742-6596/1594/1/012047 https://doi.org/10.1088/1742-6596/1594/1/012047 https://doi.org/10.22460/infinity.v8i2.p199-208 https://doi.org/10.24331/ijere.517973 https://doi.org/10.46661/ijeri.4950 https://doi.org/10.1007/s11423-017-9534-1 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p167-178 167 implementation of realistic mathematics education based on adobe flash professional cs6 to improve mathematical literacy uba umbara *1 , zuli nuraeni 2 1,2 stkip muhammadiyah kuningan article info abstract article history: received nov 13, 2018 revised june 17, 2019 accepted sept 2, 2019 students' mathematical literacy abilities are important to master, especially to formulate mathematical concepts that can be used in everyday life. mathematical literacy has become an important issue lately to be developed in the study of mathematics learning. however, mathematical literacy has not become the main goal in the implementation of the learning carried out. the purpose of this study was to examine the comparison of students 'mathematical literacy skills with adobe flash professional cs6-based rme learning with conventional learning and to examine the comparison of improvement in students' mathematical literacy skills with adobe flash professional cs6-based rme learning with conventional learning. the research method used in this study was quasi-experimental with the design of a non-equivalent control group design. the results of the two research hypotheses were accepted. it is possible that the increase in students' mathematical literacy skills is triggered by the adobe flash professional cs6-based rme learning which in the implementation of learning always focuses on self regulated learning. keywords: adobe flash profesional cs 6, mathematical literacy, rme copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: uba umbara, departement of mathematics education, stkip muhammadiyah kuningan, jl. r.a. moertasiah soepomo no. 28 b, kuningan, jawa barat 45511, indonesia email: uba.bara@upmk.ac.id how to cite: umbara, u., & nuraeni, z. (2019). implementation of realistic mathematics education based on adobe flash professional cs6 to improve mathematical literacy. infinity, 8(2), 167-178. 1. introduction one of the goals that students need to achieve in order to obtain deep and meaningful literacy in mathematics learning is to understand the mathematics they learn through constructing students' abilities regarding various mathematical concepts. constructing these capabilities, can be done if students have the ability to read and write mathematical symbols known as mathematical literacy abilities. mathematical literacy skills help individuals to recognize that mathematics plays a role in every aspect of life. mathematical literacy is an individual’s capacity to formulate, employ and interpret mathematics in a variety of contexts. it includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. it assists individuals to recognise the role that mathematics plays in the world mailto:uba.bara@upmk.ac.id umbara & nuraeni, implementation of realistic mathematics education … 168 and to make the well-founded judgements and decisions needed by constructive, engaged and reflective citizens (oecd, 2017). by looking at this, literacy can be placed as a very important ability, not only in mathematical learning but also mathematical literacy further in life. mathematical literacy helps one to understand the role or usefulness of mathematics in everyday life. in a discussion, mathematical literacy is not limited to the ability to apply quantitative aspects of mathematics but involves mathematical knowledge in a broad sense (de lange, 2003). this action has mandated the teacher to think differently about what learning means and understanding mathematical concepts for students to align more closely with mathematical literacy (nctm, 2000). meanwhile, gallimore and tharp (draper, 2002) stated in the most general sense that the ability to read, write, speak, calculate, reason, and manipulate symbols, concepts of verbal and visual as a form of literacy must be taught in school . in daily life, the equalization of students and teachers' perceptions of the importance of mathematical literacy is so important to do (venkat, 2010). thus, we consider someone to be mathematicsally illiterate, if he is unable to read and write. this mathematical literacy is the impact of demands that encourage people to survive under the culture and civilization they have in an area. mathematical literacy refers to the ability and knowledge of students to take and apply the knowledge and abilities acquired from the classroom into their reallife understanding and experience in situations involving mathematical concepts (sumirattana, makanong, & thipkong, 2017). in particular, mathematical literacy can be interpreted as the ability of an individual to understand and use mathematics in daily life activities. the subject that is specifically driven by the application of mathematics in life is at the core of a discussion of mathematical literacy that must be owned by someone (julie, 2006). in this connection, someone who studies mathematics is required to have and develop the ability to solve mathematical problems, especially those related to daily activities. some activities that can encourage mathematical literacy are (1) mathematical and mathematical concepts of reasoning, (2) recognizing the roles played by mathematicians in the world, (3) making reasonable decisions and decisions, (4) solving problems set in pupils of the world life context (sandström, nilsson, & lilja, 2013). in mathematics, students are expected to build knowledge by reading, analyzing, and writing mathematical texts (for example, numbers, symbols, graphics) to be considered mathematical literacy (siebert & draper, 2012). mathematical literacy has become a serious oecd study through pisa which began in 2000. nevertheless, rico (sáenz, 2009) explained that pisa is often not directly related to the mathematics curriculum taught in schools, but revolves around the functional model of mathematics learning by developing competencies that are directly related to real world. this concept is then seen to be highly related to realistic mathematical concepts that are focused on the application of contextual mathematics (gravemeijer, 1994). furthermore, larochelle & bednarz (colwell & enderson, 2016) integrates literacy into mathematics, and promotes the vision of constructivist learning. this ability can help students formulate mathematical concepts in solving problems in their daily lives in a structured and systematic way. however, even though it has been stated that mathematical literacy skills are very much needed by students, in reality this field of ability is still not fully controlled by students. one effort that can be implemented in improving students 'mathematical literacy skills is learning realistic mathematics education (rme) which is supposed to be able to generate students' enthusiasm in learning so that in the end it is expected to be able to improve their literacy skills. the main idea of rme learning is that students must be given the opportunity to rediscover mathematical concepts. volume 8, no 2, september 2019, pp. 167-178 169 therefore, rme is a potential model that integrates open problem-based learning, collaborative learning, error analysis and problem solving in the real world (hidayat & iksan, 2015). rme can be understood as an approach that emphasizes social processes, because learning is based on the principle of mutual respect for students' ideas in completing mathematical problems (umbara, 2015). furthermore, as a form of development of rme learning, in this study computer-assisted rme learning will be used using adobe flash professional cs6 software. there are many benefits to using flash animation in learning especially in helping students understand mathematics more meaningfully, connecting mathematics with the real world, visualizing, and understanding the importance of mathematics (salim & tiawa, 2015). the use of animated courseware is seen as very useful to see concrete examples of abstract concepts that require a deep understanding of mathematics learning (pesonen, 2003). learning material compiled using adobe flash software is expected to facilitate students in understanding contextual problems presented in learning. previous research was conducted the mathematical thinking process based on the rme approach assisted by vba excel helps students bridge the horizontal mathematization process towards vertical mathematization mathematical formations (fitriani, suryadi, & darhim, 2018). furthermore, salomon states that integrating ict in education has the motivation to support students' ability to explore constructive thinking that enables them to have the ability to transcend cognitive limitations that they might not be able to do before (lim et al., 2003). integrating ict in learning is seen as being able to develop a culture of thinking by involving students with problems that are challenging but meaningful personally through the use of the world conceptually and the learning culture of the students themselves. however, one challenge is to find methods that can increase the effectiveness of learning in learning and teaching through ict (cañas, novak, gonzález, & hammond, 2004). learning with rme based on adobe flash professional cs6 is expected to be able to improve mathematical literacy skills significantly because it is based on the process of providing learning experiences to students and is not limited to the process of transferring knowledge and knowledge with display of contextual problems that are not only imagined by students but real for students. based on the background above, research aims to review whether mathematical literacy of students with rme based on adobe flash professional cs6 is better than students who obtain learning with conventional learning and assessing the increasing of mathematical literacy of students who apply the rme based on adobe flash professional cs6 learning better than students who obtain learning with conventional learning. 2. method the research method chosen was quasi-experimental research, because the subjects were not randomly grouped but the researchers accepted the condition of the subject as a minimum. the design used in this study is the design of non-equivalent control groups. in this experimental design there were two sample classes, the pretest, the different treatment and the posttest. the sample in the first class is an experimental class that uses rme learning based on adobe flash professional cs6. meanwhile the second class as a control class that gets learning using ordinary learning. the existence of this control class is a comparison, to what extent changes occur due to the treatment of the experimental class. the design diagram of this study is as follows (ruseffendi, 2005). umbara & nuraeni, implementation of realistic mathematics education … 170 o x o o o information : o : pretest and posttest in the form of tests of mathematical literacy skills x : the treatment uses the rme learning based on adobe flash professional cs6. --: subjects are not randomly selected the population in this study were all eighth grade students of smp negeri 3 kuningan totaling 284 students. the sampling technique is done by using purposive sampling technique. the reason for choosing the sample was purposive sampling because the two groups were not actually randomized, only based on the existing class. the sample chosen from the class viii-a students was used as the experimental class and class viii-c which was used as the control class with the number of students in both classes amounting to 65 students. the instrument that will be used in the research is a mathematical literacy test. to provide an objective assessment, the criteria for scoring the test questions for mathematical literacy skills researchers adopted from quasar general rubric (maryanti, 2012), as shown in table 1. table 1. scoring guidelines mathematical literacy score student response mathematical knowledge strategy communication 0 does not show literacy concepts and mathematical principles of the problem. using irrelevant information, it fails to identify an approach that can be used to answer questions, copy some problems without any solution being given. ineffective communication, words do not describe the problem, not completely illustrate the problem. 1 very little shows the literacy of mathematical concepts and principles, one fails in mathematical terms and the majority of calculations are wrong. using irrelevant information, failing to identify an important part, the strategy used is incorrect, the facts provided are incomplete, difficult to identify or not systematic. some parts are explained but not complete and do not pay attention to the important parts of the problem, explanations are less and difficult to understand, the concepts given do not represent problems or are not clear (difficult to interpret) 2 understanding some mathematical concepts and principles, there are still many mistakes in calculations. identify the important parts of the problem, but only show a little literacy about the relationship between the two parts, showing the facts of the calculation process but not complete and not systematic. some of the sections explained have led to problems, but some explanations are still ambiguous or unclear, the mathematical concepts presented are not precise or unclear, the arguments provided are incomplete and the explanation does not enter the basic logic of the problem. volume 8, no 2, september 2019, pp. 167-178 171 score student response mathematical knowledge strategy communication 3 most literacy concepts and mathematical principles, the use of terms and mathematical notation are close to true, outline calculations are correct but there are some calculations that are still wrong. using relevant information, identifying a number of parts and showing in general the relationship between these parts, giving clear facts in the calculation and systematic process, the answers are close to true. the responses given are close to complete, with clear explanations and descriptions, mathematical concepts are presented in full, answers are generally communicated effectively so that they are easily understood by others, provide supporting arguments and the arguments given are reasonable but there are some small parts that are omitted and not explained 4 showing correct literacy concepts and mathematical principles, the use of terms and mathematical notation is correct, calculation and use of complete and correct algorithms. using relevant information, identifying all the important parts and showing the general relationship between these parts, describing the systematic approach and strategy, presenting facts clearly in the calculation process, correct and systematic answers. providing complete and clear responses, unambiguous explanations and descriptions, mathematical concepts are presented in full, effectively communicated so that they are easily understood by others. give a strong argument where the argument is reasonable and complete. furthermore, at the stage of processing research data, the procedure is carried out as follows: (1) give a score on student answers according to alternative answers and scoring system used; (2) make a table of the pretest and posttest students scores of the experimental class and the control class; (3) calculate the average test score for each class; (4) calculate the standard deviation to find out the distribution of groups and show the level of variance in the data group; (5) conduct normality tests to determine the normality of the pretest, posttest and n-gain scores as a whole using the saphiro-wilk test; (6) if the data is not normally distributed, then the mann-whitney nonparametric test can be directly carried out; (7) if the data meets normal assumptions, then the variance homogeneity test can be carried out using the lavene statistics test. if the variance of the two classes is not homogeneous, a t test can be done directly; (8) after a normal and homogeneous assumption is fulfilled, then it can then test the two average differences (t-test) using compare mean independent samples test; (9) next to test the difference between two data gain averages, in this case between the data gain of the experimental class and the data gain of the control class. if the data is normally distributed and homogeneous, then to see whether there are differences in the increase in mathematical literacy skills viewed from kam, a two-way anova test can be carried out. if the data is not normally distributed, umbara & nuraeni, implementation of realistic mathematics education … 172 it can be continued with the kruskal-wallis test. meanwhile, the statistical test used to determine the interaction between the factors of the learning model provided with the category factors of increasing students' mathematical literacy skills can be carried out by two-way anova test using the general linear model univariate analysis. 3. results and discussion 3.1. results after processing the pretest and posttest score data on the aspects of mathematical comprehension ability in the experimental and control groups, descriptive statistics were obtained as shown in the following table 2. table 2. descriptive statistics mathematical literacy capability score test experiment class control class n xmin xmax s n xmin xmax s pretest 31 44 72 52.26 4.12 34 32 76 54.97 10.74 postest 31 60 92 75.10 7.48 34 35 92 66.24 12.89 table 2 presents students' mathematical literacy descriptive statistics. the statistical data states that students' abilities are obtained through pretest and posttest consisting of: number of subjects (n), lowest score (xmin), highest score (xmax), average ( ̅) and standard deviation (sd). based on table 5.4 the average value of the experimental class pretest is 52.26 and the control class average value is 54.97. the control class has an average greater than the experimental class. meanwhile, the experimental class posttest score was 75.10 and the control class was 66.24. the experimental class has a higher average value than the control class. the initial analysis carried out in this study was to conduct a pretest score analysis. analysis of the pretest score was done to see the students' initial abilities or find out whether the difference in the average score of the pretest students in the experimental group and the control group was done using the mann-whitney non parametric test, because based on the normality and homogeneity tests that had been done previously, it was known that one the data are distributed abnormally but are homogeneously distributed, so that the similarity test of the average pretest score is done. testing the hypothesis with a one-way test with α = 0.05 with the testing criteria is accept h0, if asymp.sig (1-tailed)> α, besides h0 is rejected. the test results of the difference in post-test average mathematical literacy ability are shown in table 3. table 3. test results differences in average post-score mathematical literacy ability test statistics a post mathematical literacy mann-whitney u 287.000 wilcoxon w 882.000 z -3.175 asymp. sig. (2-tailed) .001 a. grouping variable: research class volume 8, no 2, september 2019, pp. 167-178 173 table 3 show the asymp. sig (2-tailed) value for the posttest data of students' mathematical literacy ability is 0.001. if taken α = 0.05 then asymp.sig (1-tailed) <α so that h0 is rejected. in conclusion, one research hypothesis was accepted, that: students' mathematical literacy skills using rme based on adobe flash professional cs6 were better than students who used conventional learning. to determine the significance of the difference in the average of the two data classes, a two-way analysis of variance (anova) was conducted. this analysis was conducted to see the direct effect of two different treatments given to students 'mathematical literacy abilities, as well as the interaction between the learning approaches used to the students' ability categories. the results of the calculation of variance analysis test using the general linear model (glm) univariate carried out at a significance level of 5% (α = 0.05). the results of the analysis are shown in the following table 4. table 4. analysis of gain variance in mathematical literacy source type iii sum of squares df mean square f sig. corrected model 1.576 a 5 .315 10.876 .000 intercept 3.695 1 3.695 127.480 .000 kam .690 2 .345 11.898 .000 model .479 1 .479 16.516 .000 kam * model .018 2 .009 .309 .735 error 1.710 59 .029 total 11.588 65 corrected total 3.286 64 a. r squared = .480 (adjusted r squared = .436) table 4 show the value of fcount = 16.516 with a significance level (sig.) is 0.000 smaller than α = 0.05. therefore, the null hypothesis is rejected, meaning that the increase in mathematical literacy skills of students who get learning using the rme based on adobe flash professional cs6 is better than students who get learning using conventional approaches. in other words, it can be concluded that the second hypothesis is accepted. 3.2. discussion the results showed that students with rme learning based on adobe flash professional cs6 had higher average mathematical literacy abilities than students who used conventional learning. this result is possible because through this learning students are able to learn independently, the teacher as a facilitator who provides clues and suggestions in group discussions conducted by students when students feel difficulty in understanding and resolving contextual problems so that students get an understanding of the mathematical concepts better. rme seems to be a promising teaching approach that meets indonesia's needs to improve mathematics teaching (lestari & surya, 2017). on the other hand, the rapid development of ict supports the development of multimedia learning, one of which can be done using adobe flash. the effort aims to improve the quality of learning better. in this ever-changing era, schools must maximize the contribution of new technologies that can be implemented in teaching and learning (psycharis & kallia, 2017). computer technology devices as learning tools have great potential if they can be utilized in the learning process (suartama, 2010). umbara & nuraeni, implementation of realistic mathematics education … 174 rme learning based on adobe flash professional cs6 makes students active in mathematical learning and can rediscover mathematical concepts in their own way. rme was a movement to reform mathematical education, so it was not only a mathematical learning method, but also an attempt to carry out social transformation (sembiring, 2010). mathematical learning with adobe flash professional cs6-based rme is learning that takes advantage of contextual problems that are easily understood by students and students are given the widest opportunity to solve problems given independently according to their initial knowledge. this activity means that students are given the opportunity to describe, interpret and look for appropriate strategies. meanwhile, these activities did not occur in mathematical learning using conventional learning. in general, the learning process that occurs in the experimental class is in accordance with the guidelines and criteria and characteristics of rme learning. this is reflected in the active process of students in discussions, asking questions, answering problems in more than one way, explaining and displaying the results of their work in front of the class. student activities during the learning process seem to run smoothly, even though at first the students have not been able to adapt optimally. this is understandable because the learning process carried out is somewhat different from the learning they have been used to. rme which is carried out based on the principle of constructivism provides space for students to provide their abilities in carrying out mathematical activities. the final benefit of this adaptation is an active and fun math class based on constructivist understanding by accommodating student needs and involving communication between students and teachers (draper, 2002). the enthusiasm of students in learning is seen when they begin to understand the contextual problems related to daily activities, they really feel the aspects of the usefulness of mathematics. this corresponds to the mathematical literacy aspects that are trying to be developed. in mathematics, to improve mathematical literacy, change is needed to focus on communicative and language-centric activities to hone students' mathematical literacy skills (colwell & enderson, 2016). in this study, rme learning based on adobe flash professional cs6 was seen to be able to influence the way students learn so that they can improve their learning achievement. the use of rme increases students 'mathematical achievements and encourages students to actively participate in mathematics teaching and learning, but students' attitudes towards mathematics are still the same as conventional learning (zakaria & syamaun, 2017). collaboration between rme as an approach and the use of adobe flash in bringing ict-based learning into its own advantages that complement each other. this is consistent with a similar study which provides the conclusion that animated content developed using macromedia flash students is far better than conventional approaches to high school students' mathematical connection skills (rohendi, 2012), and make mathematics learning more active and fun (chotimah, bernard, & wulandari, 2018). the influence of the learning method in question is primarily on the independence of learning by context-based learning in learning. furthermore, if you look at the results of the research that has been stated, it shows that rme learning based on adobe flash professional cs6 is significantly better in improving students' mathematical understanding skills compared to learning using conventional learning. rme learning based on adobe flash professional cs6 provides an interesting learning experience for students, because the problems presented in learning are relevant to their daily activities, technology support allows students to easily explore the concepts being studied. from this perspective, the use of technology in learning has a central role (mills, 2003). specifically, that new knowledge can be possessed by students through a systematic and ongoing process with the help of animation media used in learning (taylor, pountney, & baskett, 2008). volume 8, no 2, september 2019, pp. 167-178 175 rme learning based on adobe flash professional cs6 designed by displaying animation applications seems to provide significant changes in learning. in other words, this technology-based learning has great potential in learning especially helping in the development of mathematical concepts and improving student achievement (kurz, middleton, & bahadir yanik, 2004). in addition, ict has a role in helping visualize abstract mathematical concepts (oktavianingtyas, salama, fatahillah, monalisa, & setiawan, 2018). this concept is in line with the theory of cognitive development, where children at the formal-operational stage have the capacity to use abstract hypotheses and principles both simultaneously and sequentially (umbara, 2017). on the other hand, the basic reason that can be stated is that in the implementation of learning in the control class that uses conventional learning, the teacher only provides informative learning. so that the learning carried out tends to be passive and the ability of students to develop mathematical concepts cannot be explored to the maximum. duranti & goodwin (mkhwanazi & bansilal, 2014) states that when context problems are raised, it is usually debated that focus events cannot be properly understood, interpreted correctly, or explained in a relevant way, unless someone looks beyond the event itself to another phenomenon (eg cultural settings, speech situations, share the background assumptions) where the event is pinned, or alternatively that the features of the lecture itself use certain background assumptions that are relevant to the organization of the next interaction. overall, based on the implementation of rme learning based on adobe flash professional cs6, students' mathematical literacy abilities appear simultaneously because students are trained to reason and solve problems. two factors suggested as the development of literacy skills centers are mathematical reasoning and problem solving (venkat, graven, lampen, nalube, & chitera, 2009). 4. conclusion based on the results of the study, obtained several research conclusions, including mathematical literacy of students with rme based on adobe flash professional cs6 are better than students who obtain learning with conventional learning and increasing of mathematical literacy of students who apply the rme based on adobe flash professional cs6 learning better than students who obtain learning with conventional learning. acknowledgements we are very grateful to the drpm ministry of research, 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(2016). an analysis of self-regulated learning on mathematics education student fkip undana. infinity, 5 (2), 67-74. samo, an analysis of self-regulated learning on mathematics education … 68 introduction deductive reasoning is one of the characteristics of mathematics. this particular character which later makes mathematics is considered as a difficult subject for those who learn mathematics. in the context of learning mathematics, there are several main factors that influence mathematics achievement including academic achievement, self-efficacy, selfregulated, learning resources and learning styles (murray, 2013). in this regard, schraw & brooks (2015) reveal that the students can show a gradual but steady progress, if they follow four steps plan outlined below: 1. spare sufficient time (for example, put more effort and never give up on learning when there is any obstacle appeared) 2. compile an integrated basic knowledge 3. develop a set of appropriate strategies for learning math, and 4. believe that they can succeed if they are able to pass through the previous three steps. those four steps above can be illustrated as follows: figure 1. four steps concept of self regulated the four steps above describe the concept of self-regulation which is a concept that consists of four major components namely time, strategy, basic knowledge and belief. using the foursteps plan above will help the students become independent since it gives them a clear plan to increase their success in learning mathematics and science, as well as help them to understand the integral relationship between knowledge, strategy and motivation. without the selfregulatory skills, the students have greater risk of dropping out or failing because of their learning problems and lack of ability (graham, 1991). according to zimmerman (1990), in general, the students can be described as having a (be) self-regulated that cover metacognition, motivation, and active behavior in their own learning process. the students personally initiate and direct their own efforts to acquire knowledge and skills rather than relying on teachers, parents, or other colleagues to achieve academic goals based on the perception of self-efficacy. this definition assumes the importance of three elements: student self-learning strategies, self-efficacy perception of performance skills, and commitment to academic purposes. self-efficacy refers to the perception of a person's ability to organize and carry out the necessary actions to achieve desired performance of specific tasks skills (bandura, 1986). murray (2013) explains that self-regulated learning can be defined as the ability of the students to monitor, evaluate and make best plans for their learning. these three capabilities will greatly support the students’ progress because in each learning activity, the students not only learn to accept but also learn with plan and self control that is carried out continuously. volume 5, no. 2, september 2016 pp 67-74 69 fadlelmula, cakiroglu & sungur (2013) state that self-regulated learning is a complex and multidimensional construction that involves a number of cognitive, motivational, and behavioral aspects. self-regulated learning theory shows that in order to have better understanding of how students become active agents of their own learning process, it is important to understand how the interaction between the motivational factors that may be associated with self-regulated and academic quality of the students. self-regulated learning functions as a comprehensive framework for understanding how the students become active agents of their own learning process (fadlelmula, cakiroglu & sungur, 2013). srl development has a significance meaning in terms of increasing the students’ math performance. several studies have shown the fact that the students’ good srl will support their performance which eventualy lead them to the good learning achievement (murray, 2013; pape, bell & yetkin, 2003; malpas et al, 1999). tang (2012) in his research on srl mathematics class in pre university students shows that a high-ability subject on an easy mathematical topic has better srl than the high-ability subject difficult mathematical topics. the high-ability subject is able to manage anxieties, organize themselves, and time better than the low-ability subject. this study seeks to explore and describe srl students from the first to the third level with the existing three ability categories. the study involved three main parts namely planning (praaction), execution (action) and evaluation (postaction) divided into 10 aspects of measurement. the purpose of this study is exploring the difference of every aspect of self regulated in mathematics education students of fkip undana by involving three groups of students which are the first level (the first semester), second level (fifth semester) and third level (ninth semester) to review the individual ability. this study, in particular answers the following questions: is there any difference between self-regulated learning ability of students of high, average and low ability? and whether there is any difference of self-regulated learning of students at first level, second level and third level? method this research is a survey research. the data collection was done by distributing questionnaires on self-regulated learning to those three groups. valid questionnaire data obtained from the students of the first level which are 60 students with 18 represent high-ability students, 27 representing average-ability students and 15 represent low-ability students. the second group which is the second level consists of 64 students with 16 represent high-ability students, 30 represent average-ability students and 18 represent low-ability students. the third group which is the third level consists of 43 students with 6 represent high-ability students, 24 represent average-ability students and 13 represent low-ability students. thus, the total of the sample were 167 students. the instrument used in this study was a questionnaire on selfregulated learning who have met the criteria of validity and reliability. the data were analyzed by using two-way anova. the questionnaire contains 10 self-regulated learning aspects as follows: 1. goals setting a. set goals and targets to be achieved b. make a work plan c. prepare supporting learning aids before the lecture is held 2. motivation a. interest in mathematics b. encouragement that make students enjoy to learn math c. belief in the importance of mathematics samo, an analysis of self-regulated learning on mathematics education … 70 3. analysis of learning difficulties a. recognize the internal difficulties b. strive to overcome the difficulties 4. self efficacy a. confidence in solving problems b. anxiety 5. election strategy a. use your own strategy b. focus on problem solving c. discuss with friends and lecturers 6. metacognition a. awareness of the problem solving process b. awareness of learning 7. management of resources learning from a variety of sources 8. evaluation of performance a. review on learning activities that have been done b. assess the learning progress c. observe the achievement of the learning objectives 9. evaluation of understanding measure of understanding 10. the self-satisfaction satisfaction in the learning process results and discussion results the valid questionnaire data were analyzed using two-way anova with a review of semester levels and individual capabilities. tests carried out on each srl aspect with the students’ average data in each category for selecting each item in that dimension. the results of descriptive statistical analysis using spss for the first srl aspect of goal setting, which consist of nine statements of measurement for the first level student, second level student and third level student as follows : table 1. descriptive statistic iq level mean sd n iq level mean sd n high one two three total 3.1481 2.8272 2.6667 2.9259 .23632 .36993 .19876 .34889 15 18 6 39 low one two three total 3.0370 2.8611 3.0513 2.9811 .37728 .11111 .24265 .28216 18 16 13 47 average one two three total 3.1934 2.9074 2.7361 2.9520 .43098 .43971 .20412 .42051 27 30 24 81 sum one two three total 3.1352 2.8733 2.8217 2.9541 .37504 .36065 .26130 .36816 60 64 43 167 the data above is the measurement of srl of goal setting aspect on the first, second, and third level students with the high-ability, average-ability and low-ability review. data mean volume 5, no. 2, september 2016 pp 67-74 71 indicates, high-ability students at the first level have higher srl of goal setting aspect is than the second and third level, it also happen to average-ability. different things shown in the low-ability students where the first and the third level had srl of goal setting aspect which are relatively similar and higher than the second level students. if we compare the mean among those three abilities, it shows the results are relatively the same, which means there is no difference between srl of goal setting aspect among the three existing abilities. however, if we deal with the level (semester) point of view, it is shown that the first semester students (first level) has better srl than the fifth (second level) and ninth semester (third level). that means there is no srl difference among the firsts, second and third level students in learning goal setting and relevant activity aspects. results of hypothesis testing population are presented as follows: table 2. anova analysis for the first srl aspect of goal setting tests of between-subjects effects source type iii sum of squares df mean square f sig. corrected model 4.488 a 8 .561 4.921 .000 intercept 1172.096 1 1172.096 1.028e4 .000 ability .196 2 .098 .859 .425 level 2.727 2 1.364 11.962 .000 ability * level 1.258 4 .315 2.759 .030 error 18.012 158 .114 total 1479.852 167 corrected total 22.500 166 a. r squared = .199 (adjusted r squared = .159) the first line shows the corrected model of the combined effects (together) between level and ability. value of f = 4,921 and ftable (0.05) (8.158) = 1.997437. because f > ftable then ho is rejected, which means there is a different srl of goal setting aspect score among high, average and low-ability students with the first, second and third level students. for the factor of ability, f = 0859 and ftabel (0.05) (2.158) = 3.053257. because f < ftable then ho is accepted, which means there is no difference in the value of srl of goal setting aspect between students who have high, average and low-ability. for the level factors, f = 11.962 dan ftabel (0.05) (2,158) = 3.053257. because f > ftable then ho is rejected, which means that there is at least one level which is different from the others. in other words, each semester level has a significant role to the value of srl of goal setting aspect. based on the average value of srl of goal setting aspect among low-ability students at the first, second and third level, it can be said that low-ability students at the first levels are able to set learning targets, create a lesson plan and prepare advice supporting learn better than low-ability students in the same level. for the ability and level interaction factor f = 2.759 and ftable (0.05) (4,158) = 2.428885. because f > ftable then ho is rejected, which means there are differences in self-regulated learning caused by the interaction between different students’ abilities and level. hypothesis testing for the tenth aspects of the srl can be presented in the following table: samo, an analysis of self-regulated learning on mathematics education … 72 table 3. summary analysis of hypothesis testing of ten srl aspects aspect of srl first level student second level student third level student ha, ma & kr (α=0,05) fl, sl & tl (α=0,05) ha aa la ha aa la ha aa la mean mean mean mean mean mean mean mean mean gs 3.148 3.193 3.037 2.827 2.907 2.861 2.667 2.736 3.051 0.859* 11.962 mt 2.961 3.025 2.944 2.778 2.953 2.912 2.639 2.698 2.891 3.407 12.348 lda 2.947 2.756 2.733 2.467 2.827 2.463 2.667 2.6 2.785 0.81* 8.956 se 2.389 2.272 2.222 2.5 2.333 2.208 2.111 2.083 2.551 1.428* 0.766* es 2.948 3.029 2.444 2.852 3.026 2.778 2.704 2.889 2.897 18.289 1.484* mc 3 2.852 2.833 2.667 3.083 2.531 2.917 3.063 3 4.012 3.49 mr 2.667 2.787 2.667 2.417 2.617 2.531 2.25 2.344 2.327 3.523 24.78 epf 2.833 2.885 2.633 2.611 2.753 2.506 2.633 2.738 2.6 14.734 8.798 eu 2.778 2.42 2.333 2.556 3.056 2.646 2.111 2.417 2.539 3.424 17.131 ssf 3 2.796 3 3 2.5 2.656 2.667 2.625 3.039 10.48 5.648 1) ha = high ability, aa = average ability, la = low ability 2) gs = goal setting, mt = motivation, lda = learning difficulties ability, se = self efficacy, es = election strategy, mc = metacognition, mr = management resources, epf = evaluation of performace, eu = evaluation of understanding, ssf = self satisfaction furthermore, for the ability factor of the second aspect of srl, f = 3.407 and ftable (0.05) (2.158) = 3.053257. because f > f table then ho is rejected, which means that there is at least one level of ability which is different from the others. based on the average, low-ability students have better srl motivation than the average and high-ability students. in other words, the level of ability has a significant role to the value srl motivational aspects. for the semester level, f = 12.348 and ftable (0.05) (2.158) = 3.053257. because f > f table then ho is rejected, which means that there is at least one level which is different from the others. in other words, level of the semester has a significant role to the value srl aspect motivation. discussion summary of hypothesis testing of ten srl aspects, with the ability review, showed that whether high, average and low-ability students have different motivation, strategies election, metacognition, resource management, performance evaluation, understanding evaluation, and self-satisfaction aspects of srl. in the average comparison for those three ability groups, low-ability students excel at motivation aspect of srl, high-ability students excel at selfsatisfaction aspect of srl, while average-ability students lead in the election strategy, metacognition, resource management, performance evaluation, evaluation of the understanding aspects. seven quantitatively different aspects with five dominant aspects of the average-ability students show possible interpretation that the average ability students were able to arrange themselves and the resources as well as evaluate them in learning better. thi s condition is possible because the average ability students were aware that it needs more effort to reach a high ability and at the same time do not fall into the low-ability level. qualitative interpretation overview of this condition can be explained by qualitative research later. this finding is in contrast to some previous findings (tang, 2012; tang 2013; yip, 2009; yip and chung, 2005) which reveal that the high-ability students have better srl than low one. this finding opposes against the facts of a common research which states that srl has a good volume 5, no. 2, september 2016 pp 67-74 73 contribution in improving math skills (murray, 2013; pape, bell & yetkin, 2003; malpas et al, 1999). at the review of semester level, there are also seven significantly different aspects of srl namely motivation, analysis of learning difficulties, metacognition, resource management, performance evaluation, evaluation of the understanding and self-satisfaction. in average comparison of those three semester levels, the first level (first semester) excels in the motivation, the analysis of learning difficulties, metacognition, resource management, performance evaluation and self-satisfaction aspect of srl. the second level (fifth semester) leads in evaluating of understanding aspect while the third level (ninth semester) does not excel in any aspect of srl. students of the first level which are in the first year at the university seem to have better motivation, setting, and self-evaluation in learning. this may be due to a great motivation to enter the university that makes the students organize themselves well in learning. it eventually appear that they are different from the second and third level students. it is interesting that in every srl aspect analysis which done by using anova, goal setting and self efficacy have no difference in both ability and level reviews. the possible interpretation is that the goal setting aspect is done by almost all of the students in different level and ability. while self-efficacy which is more specific to the domain aspects of perceived anxiety becomes an aspect that felt by all of the students in different level and ability. conclusion the main objective of this research is to analyze the srl difference among the first, second and third level students with the students’ individual capabilities review. the results of the analysis showed that the average ratio of the three semester level. first level students excel at srl than the students of two other levels. in the capabilities review, the average comparison of all three groups showed that the average-ability students were superior in srl. some studies show that srl has a good mathematical performance impact. increasing srl contributes in improving math ability. this particular study reflects that there is an equal ratio between individual abilities and srl (murray, 2013; pape, bell & yetkin, 2003; malpas et al, 1999). the low-ability students, should be supported by srl development which in this study include strategies, metacognition, resource management, performance evaluation, evaluation of the understanding and self-satisfaction that still quite low. this development has to be conducted so that the low-ability students will be able to have better learning achievement. in line with the low-ability students, high-ability students should be supported to improve their srl to be able to improve their learning performance. students at the second and the third level should always be supported to develop their decreasing srl acknowledgments thanks to students of mathematics education fkip undana kupang who have taken the time been the subject of research and honestly fill this instrument. samo, an analysis of self-regulated learning on mathematics education … 74 references bandura, a. (1986). social foundations of thought and action: a social cognitive theory. englewood cliffs, nj: prentice-hall. fadlelmula, k. f., cakiroglu, e., & sungur, s. (2013). developing a structural model on the relationship among motivational beliefs, self-regulated learning strategies, and achievement in mathematics. international journal of science and mathematics education 2013, national science council, taiwan 2013. graham, s. (1991). a review of attribution theory in achievement contexts. educational psychology review, 3 (1), pp. 5-39. murray, j. (2013). the factors that influence mathematics achievement at the berbice campus. international journal of business and social science. 4 (10). pp.150-164. pape, s.j., bell, c.v., & yetkin, i. e. (2003). developing mathematical thinking and selfregulated learning: a teaching experiment in a seventh-grade mathematics classroom. educational studies in mathematics, kluwer academic publishers, pp.179–202. schraw, g., & brooks, d.w. (2005). helping students self-regulate in math and sciences courses: improving the will and the skill. university of nebraska-lincoln, lincoln, ne. tang, e. l. (2012). self-regulated learning between low-, average-, and high-math achievers among pre-university international students in malaysia. european journal of social sciences, 30(2), pp.302-312. tang, e. l. (2013). self regulated learning of pre university students in mathematics classroom. journal of education science & phychology. 3 (2), pp. 40-47. yip, m. c. w. (2009). differences between high and low academic achieving university students in learning and study strategies: a further investigation. educational research and evaluation, 15(6), pp. 561-570. yip, m. c. w., & chung, o. l. l. (2005). relationship of study strategies and academic performance in different learning phases of higher education in hong kong. educational research and evaluation, 11(1), pp. 61-70. zimmerman, b. j. (1990). self-regulated learning and academic. journal of educational psychology, 25 (1), pp. 3-17. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p121-132 121 studying conditions and problems for developing mathematics learning model of undergraduate students in thailand puchong praekhaow*1,2, tweesak chindanurak3, sureerat areeraksakul konglok3, kritsana sokhuma4 1doctor of philosophy in education, sukhothai thammathirat open university, thailand 2king mongkut's university of technology thonburi, thailand 3sukhothai thammathirat open university, thailand 4phranakhon rajabhat university, thailand article info abstract article history: received jan 8, 2021 revised jan 22, 2021 accepted jan 27, 2021 this research intends to study the conditions and problems of learning management in mathematics for undergraduate students. research problem is that students have low achievement and ability problem-solving in mathematics. the research method used is development through the stages of conducting preliminary studies and quantitative survey research, producing initial designs of integrative learning models. the results of this research were used to develop the mathematics learning model. the research was conducted over a one-year period considering two groups. the first sample was collected from the group with 376 students studying mathematics in the academic year 2020. the second sample was collected from the group with 116 professors of public universities in thailand. questionnaires were used as a tool of the research. the data analysis was divided into 2 stages. the first stage was to analyze supporting factors with factor analysis. the second stage was to design the learning management of students and professors with regression analysis. the results have shown that the opinions of students and professors on conditions and problems of learning management can be summarized as follows: (1) the students’ opinions for corrections in the aspects were group learning and teamwork, steps of solving problems, a learning model that is real situations, and the problem-based learning, respectively. (2) the professors’ opinions for corrections in the aspects were student interaction, academic achievement, problem-based learning, and learning management model that is current situations, respectively. (3) the supporting factors for the development of the learning management model that professors and students were consistent in solving problems. it was found that there were three main factors as follows; group learning, problem-based learning, and active learning. the learning management model should be developed by integrating group learning, problem-based learning, and step of mathematical problem-solving to enhance problem-solving ability and mathematics learning achievement. keywords: factor analysis, mathematics learning model, regression analysis copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: puchong praekhaow, department of mathematics, king mongkut's university of technology thonburi, 126 pracha uthit rd., bang mod, thung khru, bangkok 10140, thailand. email: puchong.pra@kmutt.ac.th how to cite: praekhaow, p., chindanurak, t., konglok, s. a., & sokhuma, k. (2021). studying conditions and problems for developing mathematics learning model of undergraduate students in thailand. infinity, 10(1), 121-132. https://doi.org/10.22460/infinity.v10i1.p121-132 praekhaow, chindanurak, konglok, & sokhuma, studying conditions and problems … 122 1. introduction nowadays, all countries are at the beginning of the 21st century. the conditions and problems on learning management in higher education need to be adjusted and review in order to solve the problems that arise. some of these findings indicate that the achievement and problem-solving ability of students in thailand is low. the research report on graduate employment in the academic year 2015 of king mongkut's university of technology thonburi found the five problems of graduates as follows, working together with colleagues, socializing, problem solving, applications in work, ethics, and human relations (temsiripot, 2015). the results of students' mathematical ability test when they first entered king mongkut's university of technology thonburi at the bachelor's degree level for the academic year 2017. it was shown that students had problems with low mathematical achievement. they had an average score of 15.72 points out of 54, and there was 21.11 percent of students whom have to study general mathematics (predasawat, 2017), corresponds to the mathematics exam results in the past three years. the students failed examinations in the mathematics since 2014-2016, 10.12%, 13.05%, and 14.59%, respectively. in one academic year, approximately 2000 students are required to enroll in general mathematics. these problems are caused by learning management. based on the above information, the student's problems were mathematical achievement and ability to solve mathematical problems, which was a national student's problem. this issue is presented as the main topic of research in the thailand educational development plan 2017-2032 (office of the education council, 2017). this problem is a global problem in the world of education, especially for students in schools and college students and graduates with problem-solving abilities (harisman et al., 2020; hutajulu et al., 2019; maharani et al., 2019). robert king wrote in the student handbook for 21st century learning that new problem solving involves solving mathematical problems. problem solving consists of critical thinking, creative thinking, collaboration, and communication (king, 2018). these things can be achieved through working together as a brainstorming group to define a problem, analyze the problem, then plan to find the answer, bring relevant knowledge together to develop problem solving and evaluation modes as shown in figure 1. figure 1. steps of troubleshooting figure 1 shows that in line with the national education association and partnership for the 21st century skills, 3r+4c is defined the core elements of the 21st century learning volume 10, no 1, february 2021, pp. 121-132 123 framework, where mathematics is one of the 3r subjects (phanich, 2012). because of creative thinking, logical thinking, and problem analysis, this can be made in humans to help in solving the problems, forecast, plan, make decisions and apply them in daily life. it is also a tool on improving the quality of life (ministry of education, 2017). the problems of higher education in thailand and abroad were presented to higher education institutions. it is needed to initiate the learning process that can train students’ problem-solving skills, especially at mathematics in university, one of which is the integrative learning model. to find a new learning management model that can solve current conditions, researchers agree that we should develop a learning management model in mathematics. the advantages of developing a learning management model in mathematics are to enhance problem-solving ability and mathematics learning achievement. the conditions and problems of learning management in the mathematics should be studied on the first thing, in accordance with the concepts and principles of the systematic model development that consisting of 1) analysis, 2) design, 3) develop, 4) implement, and 5) evaluate (dick, carey, & carey, 2005). therefore, the first thing that researchers should be interested in is developing a learning management model. study the conditions and problems of mathematics learning management of the undergraduate level. because that study from student and professors' perspectives are challenges in mathematics learning.the research question is what the students and professors want to solve in learning management of mathematics. we can define it a modern and visible learning management. by studying the conditions and problems of mathematics learning management of undergraduate students to propose a guideline for the development of learning management model in mathematics courses for the university. 2. method this research study was based on the methodology of quantitative survey research. the survey was conducted to observe the learning of mathematics which has been carried out by professors, and questionnaires also conducted to these students and professors so that it is known what learning deficiencies must be improved. the resulting data for each problem was analyzed using r-studio software to find the results. those composed of the details were as follow: 2.1. population and sample the research population consists of two parts. first, the population was 6,178 undergraduate students of studying mathematics at the faculty of engineering in thailand. second, population was 164 professors of mathematics department in public universities. the samples of students and professors were selected by stratified sampling that the stratums are bangkok and region of thailand. the sample size was calculated by taro yamane formula, with the sampling error can occur not more than 5% and then divided in equal proportions as follows. first, 379 undergraduate students were selected by sampling from the faculty of engineering in thailand. second, 116 professors were selected by sampling from mathematics department in public universities of thailand. 2.2. research variables and study duration the independent variables in this research are conditions and problems of mathematics learning management in university, and for the dependent variables consisted of the problem level of mathematics learning management in each area. the period of study is the 2020 academic year. praekhaow, chindanurak, konglok, & sokhuma, studying conditions and problems … 124 2.3. research instruments the research instrument had two parts. first, part a was the background information of the respondents who were the students and professors. second, part b was the student and professors’ questionnaires of learning management in mathematics were constructed by using problem situations at present with rubric scoring for assessment. the 15 questions of questionnaire were in the format of likert rating scale 5 levels, where the quality of the questionnaires had been evaluated by relevant experts and try-out in the 30 students. the item-objective congruence (ioc) was used to evaluate the items of the questionnaire based on the score range from -1 to +1, congruent = + 1, questionable = 0 and incongruent = -1.the items that had scored lower than 0.5 were revised. on the other hand, the items that had scores higher than or equal to 0.5 were reserved. it was found that the index of item objective congruence (ioc) of questionnaires, each question was in the range of 0.8 -1.0. the reliability value was calculated by using cronbach‟s alpha to ensure whether there was internal consistency within the items. mallery and george (2000) illustrated the value of coefficient cronbach‟s alpha as the following: ≥ 0.9 = excellent, ≥ 0.8 = good, ≥ 0.7 = acceptable, ≥ 0.6 = questionable, ≥ 0.5 = poor, and ≤ 0.5 =unacceptable. therefore, in order for the research questionnaire to be reliable, its value of coefficient cronbach‟s alpha must be at least 0.7. according to the pre-test, the cronbach‟s alpha was found that the value of cronbach's alpha coefficient was 0.821 for students' questionnaires and 0.812 for professors' questionnaires. it was shown that quality of all tools was more than the required criteria, so the questionnaire was highly reliable. 2.4. data collection the researcher made a request for permission to the mathematics department of the university that was randomly selected as the sample. the questionnaire was distributed to samples in the university. student and professor volunteers answered the questionnaire according to the volunteer's consent document of the human research ethics committee, king mongkut's university of technology thonburi. which has conducted the research project evaluation of the researcher and agreed. to pass the human research ethics assessment by certificate number kmutt-irb-coa-2020-025. 2.5. data analysis calculate the mean of each question of the conditions and problems in the questionnaire.the mean is interpreted by the 5-level likert scale rating the likert criteria. can be seen in the following table 1 (wongratana, 2017). table 1. the evaluation criteria for likert scale score interval (mean) evaluation criteria 1.00 – 1.79 1.80 – 2.59 2.60 – 3.39 3.40 – 4.19 4.20 – 5.00 very low level low level medium level high level very high level the conditions and problems of learning management in questionnaires were used to calculate the factor components by factor analysis. to summarize the problems and needs of students and professors in the development of learning management model. volume 10, no 1, february 2021, pp. 121-132 125 3. results and discussion 3.1. results 3.1.1. professors analysis results the analysis is shown on the table 2 of the 15 questions from professors. it is found that the relevant study of the learning management problems to be solved of professors in mathematics, the results have shown that the vast majority of professors thought the current learning management problems at a high level (mean = 3.51, sd. = 0.39). the problems that should be developed in the following order: helping each other in learning, low academic achievement, the learning management style is not for the current situation, interaction with peers and professors, and problem-based learning. table 2. the interpretation of the professor's effects on the conditions and problems questions mean sd interpretation 1. teaching assessments have clear criteria 4.41 0.79 high 2. questions are used to stimulate interest in teaching 3.94 3.08 high 3. the learning outcome is clearly defined in the learning objective 3.88 1.05 high 4. learning are self-learning from the exercises 3.82 0.63 high 5. your teaching approach is a lecture style 3.76 0.56 high 6. teaching mathematics should be solving problems from real situations 3.65 0.70 high 7. the learning style of the students lacked the search for knowledge 3.53 0.62 high 8. the professor's learning management model is suitable for the current situation 3.47 0.80 high 9. your learning management style opens up opportunities for interaction between learners 3.47 0.71 high 10. your learning management model is problembased learning 3.29 0.84 medium 11. your students do not have a step-by-step solution to math problems 3.24 0.83 medium 12. your learning management style does not offer group learning in classroom 3.18 1.01 medium 13. my students have low academic achievement 3.12 0.85 medium 14. your teaching does not offer solutions by students in the classroom 3.12 1.16 medium 15. your students lack mutual assistance in learning 2.82 0.80 medium total 3.51 0.39 high 3.1.2. students analysis results the analysis shown on the table 3 of the 15 questions from students. it is found that the relevant study of the learning management problems to be solved of students in mathematics, the results have shown that the vast majority of students thought the current praekhaow, chindanurak, konglok, & sokhuma, studying conditions and problems … 126 learning management problems at medium level.(mean = 3.30, sd. = 0.34) the problems that should be developed in the following order: team collaboration, experience in the presentation, steps of solving math problems, the unsuitable learning management style for the current situation, interaction with peers and professors, and problem-based learning. table 3. the interpretation of the student's effects on the conditions and problems questions mean sd interpretation 1. mathematics creates a curiosity and interest in problem solving 3.77 0.86 high 2. the students are trained to have a problem-solving process in mathematics 3.70 0.84 high 3. mathematics focuses on solving mathematical problems 3.66 0.80 high 4. mathematics is a subject that taught me a lot of methodology for solving problems 3.58 0.94 high 5. mathematics is taught mostly by lectures 3.55 0.91 high 6. mathematics is taught to build knowledge by students 3.49 1.04 high 7. mathematics should provide to learn by solving problems 3.43 0.95 high 8. teaching mathematics should be interactive with peers and professors 3.34 0.91 medium 9. mathematics is a subject that i can apply to in my daily life 3.30 1.04 medium 10. mathematics gives students the opportunity to present their work 3.26 1.05 medium 11. problem-based learning is used in mathematics teaching 3.19 0.92 medium 12. i still lack the ability to solve math problems in mathematics 2.83 0.97 medium 13. i still lack to do a well-structured in math subjective test 2.83 0.99 medium 14. mathematics is a subject that trains learners to have experience in presentations 2.75 1.07 medium 15. mathematics is a subject that trains learners who know how to work together in groups 2.75 1.10 medium total 3.30 0.34 medium 3.1.3. factors to develop new learning management the main ideas to create a learning management model of professors were shown by analysis results in table 4. table 4. the results of factor analysis of professors factors question number variance % variance 1. group learning and problem-based learning 6, 8, 7, 3, 15 4.47 29.81 2. stimulating interest and research 5, 4, 1, 9 3.08 20.55 3. solving problems in mathematics 14, 13, 12, 2 1.82 12.14 volume 10, no 1, february 2021, pp. 121-132 127 factors question number variance % variance 4. presentation and interaction with peers and professors 10, 11 1.21 8.04 total 15 70.55 kmo = 0.50, bartlett's test has p-value < 0.01 from table 4, fifteen questions relating to reasons for learning management model were factor analyzed using principal component analysis with varimax(orthogonal) rotation. the analysis yielded four factors explaining a total of 70.55% of the variance for the entre set of variables. factor 1 was group learning and problem-based learning. this first factor explained 29.81% of the variance. the second factor was stimulating interest and research. this second factor explained 20.55% of the variance. the third factor was solving problems in mathematics. this third factor explained that, 12.14% of the variance. and the fourth factor was presentation and interaction with peers and professors. the variance explained by this factor was 8.041%. the kmo (kaisser-meyer-olkin measure of sampling adequacy) was 0.50 and bartlett’s test of sphericity (p-value < 0.01) both indicate that the set of variables are at least adequately related for factor analysis. the main ideas to create a learning management model of students were shown by analysis results in table 5. table 5. the results of factor analysis of students factors question number variance % variance 1. group learning and problem-based learning 9, 11, 12, 5, 14 5.41 36.08 2. solving problems in mathematics 4, 3, 7, 8 2.21 14.75 3. solving problems in real situations 15, 13, 6, 10 1.43 9.54 4. learn from problem solving 2, 1 1.30 8.69 total 15 69.06 kmo = 0.72, bartlett's test has p-value < 0.01 table 5 shows that fifteen questions relating to reasons for learning management model were factor analyzed using principal component analysis with varimax (orthogonal) rotation. the analysis yields four factors explaining a total of 69.06% of the variance for the entre set of variables. factor 1 was group learning and problem-based learning. this first factor explained 36.08% of the variance. the second factor was solving problems in mathematics. this second factor explained 14.75% of the variance. the third factor was solving problems in real situations. this third factor explained 9.54% of the variance. and the fourth factor was learned from problem solving. the variance explained by this factor was 8.69%. the kmo (kaisser-meyer-olkin measure of sampling adequacy) was 0.72 and bartlett’s test of sphericity (p-value < 0.01) both indicate that the set of variables are at least adequately related for factor analysis. the main ideas relationship between professors and students were used to develop mathematics learning management model from factor analysis. these are shown and confirmed by multiple regression analyzes in table 6 and table 7. praekhaow, chindanurak, konglok, & sokhuma, studying conditions and problems … 128 table 6. the anova calculation of professors model sum of squares df mean squares f sig regression 6.729 4 1.682 117.378 < 0.001 residual 1.691 111 0.014 total 8.320 115 adjusted r square = 0.802, durbin watson = 1.31 table 6 shows that a multiple linear regression was calculated to predict y (problem level of learning mathematics) based on independent factors (group learning and problembased learning, stimulating interest and research, solving problems in mathematics, and presentation and interaction with peers and professors). a significant regression equation was found (f(4,111) = 117.378, p < 0.001), with an r2 of 0.802. the p-values of f statistics for the coefficients indicate whether these relationships are statistically significant. the statistical significance indicates that changes in the independent variables correlate with shifts in the dependent variable. the r-squared is 0.802. therefore, the problem level of learning mathematics that the independent variables explain collectively equal 80.20%. table 7. the anova calculation of students model sum of squares df mean squares f sig regression 100.191 4 25.048 14338.79 < 0.001 residual 0.646 370 0.002 total 100.838 374 adjusted r square = 0.894, durbin watson = 1.882 table 7 shows that a multiple linear regression was calculated to predict y (problem level of learning math) based on independent factors (group learning and problem-based learning, solving problems in mathematics, solving problems in real situations, and learn from problem solving). a significant regression equation was found (f(4,370) = 14338.79, p < 0.001), with an r2 of 0.894. the p-values of f statistics for the coefficients indicate whether these relationships are statistically significant. the statistical significance indicates that changes in the independent variables correlate with shifts in the dependent variable. the rsquared is 0.894. therefore, the problem level of learning mathematics that the independent variables explain collectively equal 89.40%. table 6 and table 7 shows that the results of multiple regression analysis support the ideas of students and professors. there is consistency in the learning management problem that should be addressed by the new learning management model. the model should consist of the first elements being group learning and problem-based learning, the steps of problem solving in mathematics. these will be stimulated by questions, search, actions, presentation, and interaction with peers and professors. 3.2. discussion levels of learning management problems for professors and undergraduate students. it appears that overall mathematics students had a medium level of satisfaction in mathematics learning management problems. while the professors were at a high level. that was, students need to adjust to conditions and problems more than professors. this was volume 10, no 1, february 2021, pp. 121-132 129 consistent with the students' academic achievement and math problem solving ability, which was the problems. these problems arise from lecture learning management. it is also consistent with the hypothesis that the conditions and problems of mathematics learning management of students greater than the professors. because those students did not learn by practice themselves following the constructivist learning theory (piaget, 2002) that states that the learners actively construct or make their own knowledge by engaging in activities in their surrounding environment and society; in accordance with the sociocultural theory (eggen & kauch, 2011) that states that the learning of human beings occurs as a result of the exchange and comparison of one’s own knowledge with those of the others in the society; in accordance with the pragmatism theory (eggen & kauch, 2011) that states that students can learn from their own real experience in doing all activities; and in accordance with the cooperative learning theory (johnson & johnson, 2009) and the multiple intelligences learning theory (gardner & moran, 2006; sener & çokçaliskan, 2018) that state that the group process of the groups that have members with different abilities and characteristics results in the cooperative relationship, competitive relationship, and each individual’s independent working. when considering the conditions and problems of mathematics learning management in various fields from the factor analysis found that there were interesting observations as follows: the professors and students would like to revise their learning management in the following areas of (1) group learning and problem-based learning, (2) steps of problemsolving, (3) learning in real situations, and (4) presentation and interaction with peers and professors. thus, researchers seek for contextualized teaching situations, where the student’s environment takes on greater relevance (romero & gómez, 2014), and such educational model should offer students tools to understand and interpret the world around them (hegedus et al., 2017; valencia-arias et al., 2019) these research results are consistent with the concepts and principles of the 21st century. the skills are often referred to as the 3rs+4cs, the 3rs are reading, writing and arithmetic, the 4cs are critical thinking and problem-solving; effective communication; collaboration and team building; and, creativity and innovation. where mathematics is in the 3rs of core components (phanich, 2012). the reason that mathematics makes students creative think logically, work systematically, able to analyze problems, help to solve problems. it helps to predict events, plan, make decisions and apply them in daily life. it was also a tool for studying other sciences, improving the quality of life (ministry of education, 2017). it was consistent on issues that we should developed in the 21st century. those were essential for students and higher education graduates to solve new problems that have not been encountered by myself. which the problem-solving ability in mathematics problem related to solving real-life problems (griffin & care, 2014; larson & miller, 2011; saavedra & opfer, 2012). the components of the learning management model were presented by professors and students based on the analysis results. several learning theories support the concept of a new learning management model: constructivism theory (piaget, 2002). students can learn through social and environmental interactions in different ways. sociocultural theory, human learning is the result of exchanging knowledge and comparing one's thoughts with others. pragmatism theory (eggen & kauch, 2011), students can learn from real experiences and activities. theory of cooperative or collaborative learning (johnson & johnson, 2009), group processes in subgroups with different members provide mutual assistance in learning, there is brainstorming within each group enabling the group members to see the problem solving approach and learn the steps of problem solving. praekhaow, chindanurak, konglok, & sokhuma, studying conditions and problems … 130 the results of research concluded that the new learning management model as shown in figure 2. figure 2. components of a developed learning management model figure 2 shows the components of knowledge that will be integrated into the development of a learning management model in mathematics. the learning management model should be developed by integrating group learning, studying with problem-based learning and mathematical problem-solving, and enhancing problem-solving and mathematics learning achievement. 4. conclusion based on research that has been done and discussion of research results, it can be seen that the instructors perceived the need for improvement for the lack of group learning and the lack of problem-based learning. while the students perceived that the instructors did not organize the learning management model that put emphasis on problem-based learning and group learning. thus, it can be concluded that the instructors and students were in agreement on their needs for the learning management model with an emphasis on group learning, problem-based learning, learning based on thinking for problem-solving with the use of questions to motivate students to search for knowledge, and the classroom presentation for interaction with classmates. developed learning management styles should arise from the integration of these elements. acknowledgments the authors would like to thank the human research ethics committee of king mongkut's university of technology thonburi, which has considered the research project evaluation of the researcher and agreed. to pass the human research ethics 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(2017). techniques for building research tools. bangkok: amon publication. https://doi.org/10.11114/jets.v6i2.2643 https://doi.org/10.1007/s10639-018-9815-2 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p81-92 81 fraction cipher: a way to enhance student ability in addition and subtraction fraction mohd afifi bahurudin setambah1*, anis norma jaafar1, mohammad ikhwan mat saad1, mohd faiz mohd yaakob2 1sultan idris education university, tanjong malim, perak, malaysia 2universiti utara malaysia university, sintok, kedah, malaysia article info abstract article history: received jan 7, 2021 revised jan 15, 2021 accepted jan 16, 2021 learning the concept of fractions can be one of the most difficult skills to master for primary school students. fractions are also seen to affect other mathematical knowledge, such as algebra. researchers have introduced an innovation called fraction cipher to help students learn fractions. fraction cipher is an innovation in the arena of education that involves learning the malay language and mathematics. design research is used as the research method to solve this problem consisting of three phases: preliminary design, teaching experiment, and retrospective analysis. the instruments used are fraction cipher, fraction test, and observation checklist. the results of the pair's comparative study by controlling the type i error using the bonferroni method show that the mean values of the mathematical achievement of the experimental group and the control group are significantly different. the results show that fraction cipher impacts students to understand and master the concept of fractional addition and fraction subtraction operations. this research also explains the "sake-beda" strategy to make it easier for students to solve fractional operation problems. besides, this study also shows the change in students' attitudes from negative to more positive. thus, students understand and are more motivated to learn the concept of fractions. keywords: design research, fraction, fraction addition, fraction subtraction, mathematics education copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: mohd afifi bahurudin setambah, faculty of human development sultan idris education university, 35900 tanjong malim, perak, malaysia email: mohdafifi@fpm.upsi.edu.my how to cite: setambah, m. a. b., jaafar, a. n., saad, m. i. m., & yaakob, m. f. m. (2021). fraction cipher: a way to enhance student ability in addition and subtraction fraction. infinity, 10(1), 81-92. 1. introduction mathematics is an appropriate instrument used to develop individual intellectual proficiency. among the aspects that need to be developed are logical reasoning, spatial visualization, analysis and abstract thinking of students. this can be done when they learn numeracy, reasoning, thinking and problem solving skills through learning mathematics (malaysia education ministry, 2014). there are various mathematical topics that students learn from year 1 to year 6. these topics form the basis of their future mathematical development. among the topics they studied were numbers and operations, measurements and geometry, relationships and algebra, statistics and algebra (malaysia education https://doi.org/10.22460/infinity.v10i1.p81-92 setambah, jaafar, saad, & yaakob, fraction cipher: a way to enhance student ability … 82 ministry, 2014). within those topics, there are smaller subtopics. for example the topics of numbers and operations contain smaller subtopics such as whole numbers, fractions, decimals, percentages and money. therefore, this study only focuses on fractions. fractions have been seen as numbers that have unique properties compared to whole numbers that students have learned before. the uniqueness of its nature has made it difficult to understand (braithwaite et al., 2018). this topic often occurs, there are four things that students often do when answering addition and subtraction fraction operation questions, namely systematic errors, random errors, negligence errors and not knowing how to answer fraction questions (braithwaite et al., 2018; loc et al., 2017; purnomo et al., 2019; salleh et al., 2013; saparwadi et al., 2017; tian & siegler, 2017). learning the concept of fractions can be one of the most difficult skills to master for elementary school students (gaetano, 2014; nurhani et al., 2018). fractions are also seen to affect other mathematical knowledge such as algebra. this in turn will affect mathematic achievement (siegler & lortie-forgues, 2015). if viewed over a long period of time, such knowledge will also affect their mathematical ability in high school (siegler & pyke, 2013). this is necessary and can be overcome through the teaching and learning process. one of the aspects that can improve students' understanding is through the use of effective teaching aids (noh et al., 2016; rohaeti et al., 2020). therefore, innovation and transformation must be done through the development and construction of teaching aids. the use of teaching aids is very important so that teachers can explain things more accurately and clearly compared to oral explanations only. this can ensure that the delivery of teaching and learning can be implemented more effectively (noh et al., 2016; rohaeti et al., 2020). the need to develop these teaching aids is very significant as described by jones et al. (2011) and mcneil and jarvin (2007). the use of aids can change the teaching and learning methods of the teacher for the better and give internal motivation to students to learn something (gaetano, 2014). there are several teaching methods of teachers that are often practiced by mathematics teachers in order to cultivate effective teaching and learning practices. among them are lecture methods (oral teaching and presentation of materials), discussions, inquiry methods, problem based methods, cooperative learning, project methods (idris, 2005; mok, 1993; setambah, 2017). however, teachers still maintain teaching practice with the method of reviewing training answers, lectures and individual exercises while conducting math classes. this is because they are more focused on improving academic achievement (koh et al., 2008; mariani & ismail, 2013). teachers are also said to still practice teacher-centered methods as informants and demonstrators, while students as observers and recipients of knowledge (bahuruddin et al., 2016). for the teaching and learning of mathematics, there are seven frequently used math teacher teaching practices. two of the seven practices are practiced by most teachers, namely the emphasis on understanding the concept and use of the polya model in teaching and learning. in addition, teachers also often use appropriate examples when explaining a mathematical topic. the use of easy-to-understand mathematical terms as well as existing materials has become commonplace by mathematics teachers. there are also teachers who use mind maps such as i-think and heuristic models during their teaching and learning sessions. some of them also take into account the factors of students' abilities when planning their teaching and learning sessions. according to idris (2005), there are several factors that hinder the learning of mathematics namely mind set (mind set), less effective drills, memorization before comprehension, less active student involvement, undiagnostic student doubts and unchallenged training. this has become the practice of mathematics teachers and has been identified by him while conducting research. in conclusion, the researcher would like to emphasize that the teaching practices presented are based on several surveys conducted by researchers, especially the teaching and volume 10, no 1, february 2021, pp. 81-92 83 learning practices of mathematics. therefore, teachers are expected to make transformations and reforms in order to practice teaching and learning practices that cultivate skills and increase the added value of human capital. this can be started by building the right teaching and learning materials. innovative materials are able to give a better effect. the effect of the material being built should be tested for its effectiveness. therefore, this study aims to identify the effects of fraction cipher (fc) to improve mathematical achievement for fraction topics. fraction cipher (fc) (figure 1) is an innovation in the arena of education that involves learning the malay language and mathematics. this absorption element is an incentive proposed by the ministry of education malaysia. fc is a combination of fraction and cipher words. the fraction is a topic chosen for improvement. cipher is a key code used by a person for the purpose of conveying information in secret (yeoh et al., 2015). the combination of these two words forms fc. the integration is carried out so that the concept of numerical mathematics familiar with malay. it can improve students' vocabulary related to malay although fc is specially developed for fractions in math. this is very well done because it can combine two subjects in one material. in other words, fc is a teaching aid that takes into account cross-curricular elements as recommended (malaysia education ministry, 2014). fc is a teaching aid in the form of board that contains the hidden words. each board contains a specific theme. students are required to find the hidden words in the fc board. next the word is converted into fraction number form as in the given guide. students are required to solve the fraction question using the concept of "sake beda". the group that successfully completes all the questions will move on to the next station. fraction reinforcement process for repeated fc use by station until students successfully complete all fc themes. this teaching process begins with forming a group of students. the concept of adventure-based teaching (explore race) is used where students will move according to the checkpoint. for example; figure 1. fraction cipher fc needs to be practiced using a fun learning method in order to further enhance its effectiveness through the application of the concept of gamification. gamification is meant to combine teaching and learning fractions in math operations and the strengthening of malay vocabulary through crossword puzzle game version of the cryptographic code. j a l a n setambah, jaafar, saad, & yaakob, fraction cipher: a way to enhance student ability … 84 according to huang and soman (2013), gamification is a craft that produces the fun and addictive elements found in games and uses them for everyday life activities. this will help teachers attract the interest and attention of students towards the teaching and learning sessions delivered. gamification refers to the application of game design elements to nongame activities and has been used for a variety of contexts including education (nah et al., 2014). fc holds that the concept of learning mathematics should involve fun activities, interest in things enjoyed, students be actively involved, and involve daily life. four of the six elements suggested by bahuruddin et al. (2016) and setambah et al. (2019) were applied during the construction of fc. this concept of gamification was chosen because of its good effect on improving student learning (pradhana & latifah, 2013) in particular the improvement of vocabulary and fraction knowledge of students. apart from that, fc also uses a fun learning method which is adventure based learning (abl). through a review of the literature that has been made, this method is rarely carried out for the purpose of learning the malay language and mathematics. this abl method was selected based on proven impact based on studies that can produce a student who is competitive, able to lead, improve communication skills, and help strengthen students' critical thinking skills. this method also coincides with 21st century learning methods (bahuruddin et al., 2016). this innovation also emphasizes the concept of "sake beda". the word sake refers to "sama=equal means permanent" while the word beda refers to "beza=difference mean multiply". this technique is a very important element in solving the problem of fractional basic operations among primary school students. in addition, students will also be able to add english vocabulary through this game indirectly. for example, among the words supplied such as transitive verbs are buy, give and many more. the word supplied refers to the theme of each game set. it is thus able to improve students' knowledge of the malay language. in conclusion, fc is a teaching aid development project that emphasizes some important elements. first, it involves a combination of mathematics and the malay language. second, applying the concept of gamification in order to increase the interest of students. third, use the abl method during teaching and learning. fc provides a more engaging learning environment when it incorporates two curriculum elements. fc also provides unique game learning methods that form a deep interest and motivate students. fc also forms a collaboration between students to think to complete assignments. fc emphasizes the concept of adherence to time while learning as well as emphasizing the element of discipline during learning. finally, emphasize the concept of "sake beda" while performing fraction addition and subtraction operations. therefore, it is hoped that fc will be able to give implications to students in order to improve the addition and subtraction operations for fraction topics. fc is expected to be able to impact the human capital aspects of students, especially thinking skills and leadership skills. however, this study focuses on mathematical achievement for fraction topics only. 2. method design research is used as a research method. the design research consists of three main phases namely experimental preparation, design experimental, and retrospective analysis (aris et al., 2017; gravemeijer & cobb, 2006). there are two important aspects related to design research namely hypothetical learning trajectory (hlt) and local instruction theory (lit). learning activities as a learning path taken by students in learning activities they must have hlt and lit. hlt consists of three components (hendriana et al., 2019). volume 10, no 1, february 2021, pp. 81-92 85 in the first phase, the researcher conducted a content analysis for fraction topics. in addition, the researcher also set the objectives and purpose of teaching fractions. then, determine the material to be used, concepts to be applied, teaching methods to be used and discuss the development of interventions to solve problems. besides that, researchers also review the results of student training, teacher teaching sessions, and assessments instrument used by teachers. researchers have also interviewed teachers to help with problems that arise. after identifying all of these, a fc was developed for use in the second phase. in the second phase, the researchers tested the fraction cipher with the help of a teacher and 8 students. a run test was performed as required by the researcher. the running test involves the process of improving the fc, the teaching period to be carried out, the method chosen during the use of fc. the results of the first round test were used for the purpose of improvement in the second round experiment. researchers have found that teaching takes a long time because students are slow to find words on fc. second, the concept of the fraction cannot be properly applied. third, the process of teaching and learning journeys seems unsystematic during activities. this has been overcome by setting the time of each fraction cipher and more marks are given if students can give the most and fastest answers. researchers also introduced the concept of "sake-beda" to facilitate students to implement the process of addition and subtraction of fractions. sake means “sama-kekal refer to equal value of the denominator value”, so that the value of the denominator is permanent. beda mean “beza=darab different denominator value”, so student needs to multiply each other or make the denominator to equal value. next, the researchers have arranged the student movement system according to a predetermined fc theme. linear and rotation systems are used during student movement. for example, fc a will change to fc b, fc b will change to fc c, fc c will change to fc d. eventually fc d will change to fc a. this system can help launch teaching and learning sessions using fc. after improvements were made, the second experiment was carried out smoothly. in the third phase, all data implemented during the experimental process were collected and analyzed. analysis using descriptive statistics and inferences using statistical package for the social sciences (spss) software. thus, the findings can answer the research questions and fulfill the purpose of the study that has been formed. the intervention (figure 2) was carried out on 30 students in which 2 groups were formed namely 15 experimental groups and 15 control groups. the experimental group was given a fraction cipher while the treatment group was given a conventional learning method. students are selected based on using cluster techniques based on the following procedures: (a) write the name of the class on a piece of paper; (b) the paper is put in a box; (c) one paper is randomly drawn; (d) the class name listed is used as a sample; (e) students who are in the class are used as a study sample. there are the differences between the fraction cipher learning method and the conventional method are as follows: (a) teaching and learning across the curriculum for mathematics and malay language for the experimental group, while the control group only learning math; (b) learning uses fc teaching aids for experimental groups, while conventional groups do not involve fc; (c) using adventure-based learning methods (bahuruddin et al., 2016) for experimental group, control group using lecture, drill and teacher-centered methods. setambah, jaafar, saad, & yaakob, fraction cipher: a way to enhance student ability … 86 figure 2. implementation of fraction cipher intervention pre-tests were given before the intervention and post-tests were given after the intervention. observation sessions were also conducted before and after the intervention. students for the experimental group were divided into four groups. each group is given a different fc theme. students are given 15 minutes to find the hidden words on fc board. when they have finished finding the word, students need to decipher the word based on color. each color found in fc represents a fractional value. for example, the color yellow represents value ½. red represents value 0. students are then asked to sum up or subtract fractions depending on the decipher they do. the concept of "sake beda" is used during the process of addition and fraction breakdown. students are given marks based on the number of hidden words found and the implementation of correct and accurate addition and subtraction operations. this intervention process was carried out repeatedly using fc for 4 weeks. this means that students are given interventions during that period. 3. results and discussion the learning that is implemented is to overcome the problem of students who have problems in addition and subtraction of fractions. the study began with a review and observation of the students. when investigated the main cause of their errors is lack of understanding of the process involved, difficulty in subtracting fractions, difficulty converting fractions to the same denominator, errors in calculations, difficulty converting volume 10, no 1, february 2021, pp. 81-92 87 improper fractions to mixed numbers and using incorrect processes. students' errors in the process of adding fractions are shown through figure 3. this occurs when misconceptions occur from one concept to another. students found to have brought the concept of operations of addition of whole numbers to the addition topic fraction. this can be seen through the answers to questions 1, 2, 3, 4, 6, 7 and 8 that students provide. they were found to add numerators and numerators, as well as denominators and denominators. this clearly shows that a misconception has happened to them. the same thing happens when researchers review student exercises in subtraction operation. this review is similar to the study that has been conducted by salleh et al. (2013). the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. figure 3. student error on the process of addition of fractions when the researchers reflect back, the researchers also found that the methods that the researchers implemented during the teaching and learning sessions did not have a positive impact on them. researchers simply use examples and write step-by-step operations of addition and subtraction fractions without the help of teaching aids. this can be seen in the effect when researchers often find that students always complain of misunderstanding, re-ask the process of addition and subtraction of fractions this can be recorded through the expression of their words such as "i do not understand the teacher", "then how is the teacher", "ouch, why cannot be like that teacher". this proves that they often rely on the teacher’s answers and no longer understand the concepts that have been taught by their teacher. findings show that students have an improvement in their learning for the topic of addition and subtraction of fractions. these findings can be seen through figure 4. significant improvement can be seen through students 4, students 5, students 7, students 8 and students 15. this shows that fc is able to have a good impact on students. the findings setambah, jaafar, saad, & yaakob, fraction cipher: a way to enhance student ability … 88 of this study are in line with the study aris et al. (2017) that the intervention process enhances student interest is able to have a positive impact on student mathematical achievement. figure 4. differences of pre-test and post-test of experimental group comparisons between the experimental group and the control group were also performed through t-test. the findings show that there is a significant difference [f(1,28)=4.61, p<0.05] with large effect size based on value partial eta square = 0.141 (cohen et al., 2007). the mean of the experimental group was higher than that of the control group. the results of the comparative analysis of the pair by controlling the type i error using the bonferroni method show that the mean values of the mathematical achievement of the experimental group and the control group are significantly different (mean difference = 4.2, p<0.05). the results of this study further strengthen the findings of the study of noh et al. (2016) which shows that innovative teaching aids can have an impact on student achievement. observation findings also indicate a change in good behavior. this study coincides with the study of jones et al. (2011) and mcneil and jarvin (2007) when fc had a positive impact on student attitudes. before the intervention is performed. students were found to exhibit negative behaviors such as (1) talking unrelated to the topic, (2) thinking elsewhere (chestnuts), (3) walking from one place to another, (4) doing other work, (5) interrupting other students physically, (6) try to attract attention, (7) sharpen a pencil and (8) leave the classroom. after the intervention is implemented, negative behavior can be reduced from 40% to 10%. this can be seen more clearly through the observation table made by the researcher. table 1 shows the observations that were made before the intervention was performed. for example, student 1, negative behavior is reduced to 10%. student 2 reduced by 20%. volume 10, no 1, february 2021, pp. 81-92 89 table 1. observation of student behavior student observation percentage 1 2 3 4 5 6 7 8 9 10 1 3 4 3 8 40% 2 2 2 2 30% 3 2 1 4 4 40% 4 1 1 4 4 40% 5 7 4 4 30% 6 6 10% 7 5 10% 8 1 5 5 30% 9 1 1 1 1 8 1 60% 10 1 10% 11 3 7 2 30% 12 1 2 2 30% 13 3 3 5 30% 14 1 10% 15 2 3 20% researchers also conducted interviews on teachers who had been exposed to fc (figure 5). figure 5. discussion with mathematics teacher the results of interviews with teachers found that fc has attracted their interest. they think that fc is a good innovation. they also stated that fc is able to give them a difference in terms of teaching and learning. fc changed the teacher-centered approach to the student setambah, jaafar, saad, & yaakob, fraction cipher: a way to enhance student ability … 90 centered approach. they feel excited and want to use fc in their teaching and learning. teaching aids give a good impact on teaching methods. thus, fc gives a good perception on teachers and students. fc is seen to have a good impact on students (noh et al., 2016). 4. conclusion learning operations of addition and subtraction of fractions using the fc have a positive impact on students in terms of understanding, interest and motivation. this study also proves that the construction of mathematical concepts is not seen as something that needs to be moved passively, but rather needs to be built by students actively through concrete experience. in general, this study successfully solves the problem of student learning from the aspect of fractions, especially in addition and subtraction operations. through this fc-based learning, it is hoped that the six mistakes that students often make while completing the addition and subtraction operations of fractions can also be reduced. such errors are such as lack of understanding of the process involved, difficulty in converting to the same denominator before performing addition operations, errors in calculation, and difficulty in converting improper fractions to mixed numbers. fc is expected to have positive implications on the mission and aspirations of malaysian education. fc is seen to have other potentials such as providing more enjoyable teaching and learning methods for teachers, providing useful experiences to students, enhancing student cooperation during group activities, improving student concentration style to find words, changing student character to be punctual and disciplined according to game rules and improve thinking skills through reasoning. future studies can be conducted to determine these various aspects. references aris, r. m., putri, r. i. i., & susanti, e. 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(2015). aplikasi matematik. bangi: pelangi professional publishing sdn. bhd. https://doi.org/10.33474/jpm.v3i2.715 https://doi.org/10.1037/edu0000025 https://doi.org/10.1037/a0031200 https://doi.org/10.1177/0022219416662032 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p15-30 15 ontological misconception in mathematics teaching in elementary schools imam kusmaryono* 1 , mochamad abdul basir 2 , bagus ardi saputro 3 1,2 universitas islam sultan agung 3 universitas pgri semarang article info abstract article history: received oct 28, 2019 revised jan 29, 2020 accepted jan 30, 2020 elementary school teachers in indonesia are required to master many subjects to be taught to their students. it is undeniable that the teachers’ mastery of knowledge (material) in some subjects inadequate. therefore, it is worth to argue that there was a misconception in mathematics teaching in elementary schools. this research was designed using a qualitative approach. the participants of this study were 30 elementary school teachers in semarang city area, central java province, indonesia. the research data were obtained through questionnaires, and interviews. the purpose of the study was to discuss the types and causes of the misconception of mathematics teaching in elementary schools. alternative solutions were also presented to problem-solving so that misconceptions do not occur anymore in mathematics teaching. the findings show that, teachers evenly experience types of misconceptions: (1) pre-conception, (2) under-generalization, (3) over-generalization, (4) modelling error, (5) prototyping error; and (6) process-object error in teaching mathematics in elementary schools. some misconceptions have taken root and are difficult to remove, called "ontological misconceptions" because of teachers' years of belief that the knowledge they received was true when in fact it was not quite right. keywords: elementary schools, misconception, ontology, teaching mathematics copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: imam kusmaryono, department of mathematics education, universitas islam sultan agung jl. kaligawe raya no.km. 4, terboyo kulon, semarang, central java 50112, indonesia email: kusmaryono@unissula.ac.id how to cite: kusmaryono, i., basir, m. a., & saputro, b. a. (2020). ontological misconception in mathematics teaching in elementary schools. infinity, 9(1), 15-30. 1. introduction teaching math is a difficult task under any circumstances. this is because of the complexity, characteristics and nature of mathematics itself. when beginning to study mathematics, students learn it themselves and or learn from others, especially with their teachers (skott, 2019). often, in mathematics learning misconceptions occur that hinder students' cognitive development. therefore the teacher must provide a careful explanation followed by opportunities that create opportunities for students to understand and absorb ideas that are presented clearly, so students become proficient in mathematics (sullivan, clarke, clarke, farrell, & gerrard, 2013). mailto:kusmaryono@unissula.ac.id kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 16 the results of the survey in the last ten years conducted by the program for international student assessment (pisa) and the survey by trends in the international mathematics and science study (timss) stated that student achievement in learning mathematics in indonesia is still at a lower level compared to several countries surveyed in the world (oecd, 2019). talking about the low performance of indonesian students in the field of mathematics, can not be separated from the existence of misconceptions in teaching. many research have focused on analyzing students' misconceptions in learning mathematics (aliustaoğlu, tuna, & biber, 2018; gooding & metz, 2011; mohyuddin & khalil, 2016; sarwadi & shahrill, 2014). however, the authors have not discussed yet about teacher mistakes in mathematics learning in school. then a questionarised: did the teacher involve as a causal factor of misconception in mathematics learning? findings in this research are very important to analyze the misconception in mathematics teaching by teachers in elementary schools in indonesia and analytical alternatives for problem solving to get rid of misconception a teacher has a key role and position in the entire education process. the teacher is the main factor of students' learning success. moreover, in elementary schools, teachers are required to master teaching materials and to develop teaching methods in accordance with the subjects taught (anwar, 2012). elementary school teachers have the most heavy responsibilities in their professional duties compared to grade teachers middle school and high school level. an elementary school teacher in indonesia is required to master many subjects, including language, mathematics, geography, history, cultural arts and skills. therefore, it is undeniable that their mastery of knowledge (material) in some subjects is not edequate. on one hand, teachers master subjects and are proficient in the field of language learning, but on the other hand, mastery of the material by the teacher is inadequate and are not proficient in mathematics learning. if the teachers do not have mathematical skills in teaching, it will hinder the achievement of learning goals, and affect students' positive dispositions towards mathematics learning (kusmaryono, suyitno, dwijanto, & dwidayati, 2019). the results of observations of mathematics learning of elementary school teachers in the central semarang area, there are still many learning misconceptions. they lack mastery and are not proficient in mathematics. therefore this research is very important to do, considering elementary school teachers are the first people to instill knowledge of mathematical concepts in formal education. mathematical proficiency is a skillfull quality that shows skills, competencies, knowledge, beliefs, and fluency in working on and teaching mathematics and being problem solvers who are proficient with high productive dispositions (groves, 2012; kilpatrick, swafford, & findell, 2001). it is important for teachers to understand, that mathematical proficiency in teaching will have implications in learning that misconceptions will not occur, so that the teacher can become a facilitator who encourages students to become constructors of "constructive knowledge" for themselves students (kistner, rakoczy, otto, & klieme, 2015). the misconception of mathematics by teachers in a teaching process in elementary schools can result in misconceptions or misunderstanding of a sustainable basis which lead to higher education level. this is because the characteristics of mathematics learning materials are interrelated and continuous with other materials. to learn one of the mathematics topics at the advanced level must be based on reasoning from basic knowledge or prior prerequisite knowledge. if someone experiences a conceptual error (misconception) of mathematics in lower classes learning and is not immediately volume 9, no 1, february 2020, pp. 15-30 17 addressed, it will have an impact on the learning of mathematics in high classes (flevares & schiff, 2014). misconceptions include understanding or thinking which is not based on true information. misconceptions occur because of errors in transferring concepts from information obtained into a framework. so, the concept understood may not be in accordance with the actual concept. teacher naturally forms ideas from everyday experience, but not all ideas developed are true in connection with evidence in a given discipline. in addition, some mathematical concepts in different content areas are very difficult to understand. even teachers sometimes can have misconceptions about materials (burgoon, heddle, & duran, 2017). for them, it may be a very abstract concept, counter intuitive or quite complex. therefore, changing a teacher's framework is the key to improving mathematics teaching for the better (skott, 2019). this paper outlines some of the misconceptions of teaching mathematics in elementary schools. in addition, it also provides alternative solutions to the problem, so that conceptual errors (misconceptions) do not occur anymore in mathematics teaching. basically, every teacher has the potential to successfully carry out his/her duties as a reliable learning agent. teacher’s success can be clearly seen from the teaching skills and students’ success in following the process and achieving learning goals (oecd, 2019). 2. method 2.1 research design this research used a qualitative approach. in this research, the hypothesis was not determined to be tested because the researcher wanted to get research findings that flowed and described the results of systematic observations (creswell, 2014; mcmillan & schumacher, 2014). this research was conducted during the active period of learning activities in elementary schools, precisely in january february 2019. 2.2 participants the participants of this research were 30 elementary school teachers of first to sixth grades, representing 10 elementary schools in central semarang sub-district, semarang city, central java province, indonesia. the teachers have had teaching experiences in primary schools for 4 to 20 years. 2.3 procedure at the beginning of the research, observations of the mathematics teaching process were conducted in several elementary schools. then, teachers completed questionnaires in the form of mathematical questions with answers written in the questionnaires. responses of answers from questionnaires were identified and analyzed in terms of types of errors, then grouped into types of misconceptions: (1) pre-conception, (2) under-generalization, (3) over-generalization, (4) modelling error, (5) prototyping error; or (6) error processes (ben-hur, 2006; diyanahesa, kusairi, & latifah, 2017; saputri & widyaningrum, 2016). based on the misconception data, teacher representatives were then selected through a purposive snowball technique to get the subjects interviewed (naderifar, goli, & ghaljaie, 2017). the following is the flow of research implementation in figure 1. kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 18 figure 1. research flowcart 2.4 instruments instruments for retrieving research data include questionnaires and a list of interview questions. the following is the flow of research implementation in figure 1. the instrument was validated by experts in the field of mathematics learning, namely dyana wijayanti, ph.d.the questionnaires contained six questions related to mathematics teaching materials and the teachers’ perspectives on mathematics teaching. the questionnaire instruments in the form of questions were designed to explore responses of teachers’ answers, types of errors and misconceptions in mathematics teaching in elementary schools. the list of interview questions was to reveal the causes of errors and misconceptions in mathematics teaching. the topics of mathematics teaching becoming the focus of this study were integers, flat geometry, rational numbers, and algebraic equations. following are some examples of questionnaires used in this research instrument (table 1). table 1. example of research questionnaire no. question capable not capable give answers and reasons 1 are you able to read the mathematical statement below? (a) 7 + (4) = 3 (b) -10 – (-6) = -4 2 are you able to prove that 1.252525 ... is a rational number? 3 are you able to solve problems ... 3 1 : 9 4  with procedures that you know about? 2.5 data collection, analysis, and triangulation the research data were collected through questionnaires and interviews. subject as informants were the first to sixth-grade teachers teaching mathematics in elementary schools. this qualitative research data analysis was described as an interactively connected cycle through the stages of data collection, data reduction, data presentation, and conclusion (miles & huberman, 2012; moleong, 2007). to ensure the validity of the data, the researcher used the triangulation theory and source triangulation (moleong, 2007). learning observation formulate the problem instrument preparation and validation questionnaire distribution data analysis and triangulation identification of misconceptions conclusion interview volume 9, no 1, february 2020, pp. 15-30 19 3. results and discussion 3.1. results in the early stage of the study, observations of mathematics teaching processes were carried out in several elementary schools. the observations were conducted when teachers gave explanations of how to overcome problems experienced by students. the results of the observations showed that the teacher's explanation was more on conceptual or procedural categories and/or both. successfully noted that the explanation from the teacher was sometimes illogical and not in accordance with the rules or mathematical principles. whereas, based on the results of the responses to the questionnaire responses, several misconceptions related to mathematics teaching in elementary schools were found. the following are the misconception of mathematics teaching found (table 2). table 2. types of misconceptions in mathematics teaching topic misconceptions types type 1 type 2 type 3 type 4 type 5 type 6 integers  ----- rational number -    - linear equation ---- - geometry field  ---- note: type 1=pre-conception type 2=under-generalization type 3=over-generalization type 4= modelling error type 5= process-object error type 6=prototyping error answer to problem 1 problem 1 is the problem related to the teachers’ understanding of symbol (+) and (-) as a sign of a count operation or integer name. most respondents (teachers) had the same answers when it came to reading math sentences. pay attention to the duplication of the respondents’ answers in figure 2a. misconceptions the alternative solution (a) 7 + (4) = 3 read: seven plus minus four equals to three (b) -10 (-6) = -4 read: minus ten substracted by minus six equals to minus four (a) 7 + (4) = 3 read: seven plus negative four equals to three (b) -10 – (-6) = -4 read: negative ten minus negative six equals to negative four figure 2a. respondent's answer (r.02) figure 2b. alternative solution to clarify the information from the respondent (r.02), let us consider the following excerpt from the interview. kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 20 researcher : do you understand the difference between of symbol (-) as a sign of a count operation and (-) as an integer name? respondent (r.02) : symbol (-) is read minus. so symbol (-) can be a count operation or number name. researcher : in an integer system, there is a positive number (4) and a negative number (-4), but there is no minus number. respondent (r.02) : oh yeah, it's different. now i understand symbol (-) as a sign of a count operation and (-) as a negative integer name answer to problem 2 the problem to prove that 1.252525 ... is a rational number is a very important thing to explain thoroughly. pay attention to the answer of the respondent in figure 3a. misconceptions the alternative solution answer: numbers 1.252525… . = 100 125 rational numbers so, 1.252525 ... = 100 125 it will be proven that 1.252525 ... is a rational number. for example: y = 1.252525 ... and 100y = 125.252525 ... then 100y = 125.252525… . y = 1.252525… . _ 99y = 124 y = 99 124 so 1.252525… . = 99 124 a rational number. figure 3a. respondent's answer (r.27) figure 3b. alternative solution the findings of the questionnaire analysis showed that the answers from the respondent were false. the following is an excerpt from the interview with one of the respondents (r.27). researcher : do you understand this number 1.252525 ... ? respondent (r.27) : number 1.252525 ... is an infinite number of repeated decimal places researcher : is the number 1.25 = 1.252525 ... ? respondent (r.27) : yes, 1.25  1.252525 … ... but this is difficult to prove in rational numbers. researcher : pay attention, please, to the solution presented in figure 3b. now, do you understand? respondent (r.27) : yes, i do. thanks for the explanation. volume 9, no 1, february 2020, pp. 15-30 21 answer to problem 3 the teaching of rational number division operations is always a serious concern in the procedural context. there are irregularities in the problem-solving process of all of the respondents' answers for solving this problem were correct and no need to question. but the problem was the respondent could not explain why the fraction division operation was changed to a multiplication operation and the divider is reversed (figure 4a). misconception the alternative solution 3 1 1 9 3 1 9 12 19 34 1 3 9 4 3 1 : 9 4  x x x there is a change in the distribution operation mark into a multiplication operation (a) 3 1 1 3 4 3:9 1:4 3 1 : 9 4  (b) 3 1 1 3 4 1 3:4 9:9 3:4 9 3 : 9 4 3 1 : 9 4  consistent and no changes in the operation marks figure 4a. respondent's answer figure 4b. alternative solution based on the finding of the questionnaires, it was found that the teacher’s answer to the problem was correct (figure 4a). however, the mathematical modelling presented as the solution to the problem could not precisely be explained with reasons given. the following interview excerpt is to strengthen this statement. re researcher : why 3 2 : 9 4 when you completed the division operation, did it turn into a multiplication operation and the dividing number becomes like this 2 3 9 4 x ? respondent (r.02) : i can't describe it correctly. i did a problem solving, as i understood. researcher : are you sure there is no other way to solve this problem? respondent (r.02) : i pretty am, there is no other way. all teachers solve this problem as i did. researcher : since when did you understand how to solve this? respondent (r.02) : since i studied in elementary school 25 years ago. i followed the teacher's instructions and i have been doing it until now. answer to problem 4 figure 5a is an example of an erroneous understanding of teaching turning ordinary fractions into decimal fractions. paying attention to the respondent's answer (r.11) that the decimal form of ¼ is 0.25 is correct (figure 5a). it was identified that the teaching process to get a 0.25 result kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 22 was deemed inappropriate. then the respondent confirmed (r.11) through the following interview. researcher : why do you always add zero (0) number to each number which is not divisible by four? respondent (r.11) : number 1, if added to zero (0) will be ten so that 10 can be divided by 4 researcher : supposedly, 1 + 0 = 1, it is not correct if 1 + 0 = 10? how do you explain this to students? respondent (r.11) : i learned from mathematics teaching at previous schools. if a number cannot be divided, then borrow zero (0) and the result of the division is zero points (decimal). researcher : are you not aware, that there has been a conceptual error in this learning? respondent (r.11) sorry, i can't explain correctly. i realized that there has been a teaching error, because all this time, i have only followed the books and habits applied and carried out by all the teachers at schools. below is shown the results of the subject's work in solving problem number 4. misconception the alternative solution change ordinary fractions 4 1 change ordinary fractions. the solution is by stacking as follows: 0.25 4 10 8 _ 20 20 _ 0 so, the decimal fraction of 4 1 is 0.25 the solution should be as follows: 4 1 = 100 100 4 1 x 4 1 = 4 100 x 100 1 4 1 = 25 x 100 1 4 1 = 100 25 4 1 = 0.25 so, the decimal fraction of 4 1 is 0.25 figure 5a. respondent's answer (r.11) figure 5b. alternative solution answer to problem 5 a process-object error was identified in the case (problem 5) of this study, namely the occurrence of an error in the completion process of a single variable linear equation (figure 6a). 1). borrow zero number 3). borrow zero number 2). written zero point volume 9, no 1, february 2020, pp. 15-30 23 misconception the alternative solution determine the value of x, so that 2x + 5 = 17 is correct 2x + 5 = 17 2x = 17 – 5 ??? 2x = 12 x = 2 12 x = 6 so, the solution 2x + 5 = 17 is x = 6 determine the value of x, so that 2x + 5 = 17 is correct 2x + 5 = 17 2x + 5 + (-5) = 17 + (-5) step 1 2x = 12 2x . 2 1 = 12 . 2 1  step 2 x = 2 12 x = 6 so, the solution 2x + 5 = 17 is x = 6 figure 6a. respondent's answer (r.08) figure 6b. alternative solution if we look at figure 6a, the result of the response answer is correct. however, the completion process in the second step on the right side displays a reduction operation with number 5. then the answer is confirmed through the interview below. researcher : is the completion process that you did right? respondent (r.08) : i'm sure, it is. value of x = 6 researcher : why is that in the second step 2x = 17 5, like this? respondent (r.08) : number positive 5 on the left segment is moved to the right segment to be negative (-5). answer to problem 6 the misconception problem arises when the teacher was confronted with a flat square image. the teacher was asked to show the name of the parallelogram. question: which form of a quadrilateral is the parallelogram? a b c d respondent's answer: model b is a parallelogram. model a, c, and d are not. figure 7. quadrilateral models the result of the respondent's answer stated that only one of the four images available was figure b (figure 7) considered a parallelogram. then the respondent confirmed (r.02) through the following interview. kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 24 researcher : why did you choose image b as a parallelogram? respondent (r.02) : because image b has a parallel hypotenuse researcher : why are image a, c or d not? respondent (r.02) : a is a rectangle, c is a square, and d is a cube. researcher : would you explain the definition of the parallelogram? respondent (r.02) : the parallelogram is a quadrilateral that has two pairs of sides facing the same length, there is an inclined side, and with equal angles. 3.2. discussion problem 1: pre-conception it was identified that respondents experienced pre-conception, namely problems in reading integer symbols. they were not able to distinguish between symbol (+) or (-) as a count operation or integer name. pre-conception is an initial mistake before someone understands the concept correctly (diyanahesa et al., 2017). based on the interview excerpts with the respondent (r.02), it can be said that the teacher failed to give an interpretation and interpreted the minus sign (-) as an operation to calculate the subtraction and negative in (-4) as the name of the number four negative. according to cockburn and littler's findings, integer material is one of the topics that is difficult to teach in embedding integer concepts (cockburn & littler, 2008). problem 2: under-generalization under-generalization is a more specific part of pre-conception. undergeneralization is expressed as a limited understanding and ability to apply the concepts (saputri & widyaningrum, 2016). this limited understanding explains various circumstances regarding teacher’s knowledge during all mathematical ideas develop. cases in rational and irrational numbers may be one of the most problematic in mathematics teaching in elementary school. many teachers only understand rational numbers as ordinary fractions, decimal fractions, and percent. in fact, fraction interpretation as a part-whole relationship is only a sub concept or one way of understanding rational numbers. the following under-generalization is identified from the response of the teacher's answer. teacher's mastery of the concept of rational numbers has not developed perfectly, the teacher only understands in a limited way. the alternative solution shown in figure 2b is the right step as a problem-solving instruction. then the instructions on the number system must be able to answer the problem of under-generalization because there is an assumption that certain characteristics in the number system inhibit general understanding (ben-hur, 2006). problem 3: over-generalization over-generalization is a case of misconception, where the application of concepts is not understood and the rules applied are considered irrelevant. figure 3a is an example of an erroneous understanding of teaching in turning ordinary fractions into decimal fractions. based on the interview excerpt, it was indicated that the respondent (r.11) had misrepresented an illogical interpretation which caused a false understanding. techniques for solving a mathematical problem can vary in ways, but the interpretation must generally volume 9, no 1, february 2020, pp. 15-30 25 be explained or understood by students (others). the solution in figure 3b shows that the strategy was chosen because equals to 1. in accordance with the algebraic law that all numbers, if multiplied by 1 are fixed, so that they are obtained . through interviews with respondents (r.11), information was obtained that there had been an error (misconception) on mathematics teaching. during this time, the teaching of mathematics conducted by teachers only following books and habits that had been valid for many years. so it can be interpreted that there has been a rooted misconception that the concept of teaching believed to be true turns out that the concept of teaching is false (ontological misconception) (ben-hur, 2006). ontological misconceptions in teaching mathematics occur because of the lack of mathematical knowledge from elementary school teachers. problem 4: modelling error modelling errors were identified when students (teachers) only imitated examples of wrong work from representations of rational number counting operations. in teaching rational number division operations, the teacher failed to give reasons through mathematical modelling displayed. an example of problems . mathematical modelling presented as a solution to the problem could not be explained precisely with given reasons. apparently, the way of the respondent’s (r.02) completion was obtained from their teacher while studying at the elementary level. they answered that the work process was obtained because of the teacher’s beliefs and doctrines that had to be followed. a doctrine that they had just to accept without reasons because they assumed that mathematics is an exact science and the teacher never went wrong. the method of completion was replicated by students without knowing the reasons for the steps (figure 4a). such misconceptions are grouped as modelling errors. compare it to the alternative solution in figure 4b, it appears that the proposed alternative solution is very logical and consistent in accordance with mathematical principles. some teachers’ answers in the questionnaire illustrate how limited understanding undermines the conception of key mathematical ideas. there is an opinion stating that maybe when the teachers experience a modelling error, the teachers have their own version of the model in the situation (blazar & kraft, 2017). so, it can be interpreted that in this case, there is also a deep-rooted misconception, that is, the teaching concept which was believed to be true turns out that the teaching concept is false (ontological misconception) (ben-hur, 2006). problem 5: process-object error process-object errors are identified in the case (problem 5), namely the occurrence of a process error completion from a single variable linear equation. if we look at figure 5a, the final result of the respondent's answer was correct. however, the completion process in the second step on the right side appeared a reduction operation with number 5. confirmation was carried out through interviews, some teachers were very confident and believed that the process of solving a single variable linear equation was completed as in figure 5a. they believed that the positive number on the left side, if moved to the right side, would change to a negative number. so it can be concluded that they do not understand the laws of algebra. the alternative solution in figure 5b is the best process for solving a single variable linear equation. the first step, the two segments get the same kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 26 treatment, which is added to the same number (-5), so that it still has the same value. the second step, multiplying the two segments with the same number (½). problem 6: prototyping error in the case (problem 6), the respondents refused to recognize that rectangles, squares, and rhombus are parallelograms. they did not understand the definition of parallelograms so it can be classified in the type of pre-conception. there were few respondents who could explain the definition of a parallelogram, that parallelogram is a quadrilateral which has two pairs of parallel equal sides and the opposite angles are equal. but in their minds, they still considered that image a, c, and d were not parallelograms. this misconception is classified in prototyping error. the teachers only understood the eternity of forms through a standard example of a parallelogram. the teacher considered the standard example of a concept to be the only type of example. the teacher did not understand the definition of a parallelogram, but only did representation through standard visual images. based on the explanation of the research findings discussed, it can be said that the things we have learned are sometimes not helpful in learning new concepts or theories. this happens when a new concept or theory is inconsistent with the material previously studied. thus, it is very common for students, teachers, and adults to have misconceptions in different domains (content knowledge fields). teachers evenly experience types of misconceptions: (1) pre-conception, (2) under-generalization, (3) over-generalization, (4) modelling error, (5) prototyping error; and (6) process-object error in teaching mathematics in elementary schools (ben-hur, 2006; ryan & williams, 2007). misconceptions in teaching mathematics in elementary schools occur for several reasons. teachers generally do not realize that the knowledge they have is incorrect. the teachers interpret new experiences through this erroneous understanding, thus disrupting the ability to understand new information correctly. understanding incorrect mathematical concepts for years has been stable, permanent and rooted (desstya, prasetyo, susila, suyanta, & irwanto, 2019; hughes, lyddy, & lambe, 2013). the stable, permanent and rooted misconceptions are called "ontological misconceptions," in teachers’ thinking. the ontological misconceptions relate to ontological beliefs, that is, beliefs about the category and nature of the world (burgoon et al., 2011). citing the opinion of harisman et al that teaching experience (duration of teaching) is not a determinant of teacher professionalism, but the level of education and experience attending training is a factor that influences teacher proficiency in problem solving (o'leary, fitzpatrick, & hallett, 2017). so, it should be argued that the misconceptions that students have actually originated from their teacher "ontological misconception" in mathematics teaching in elementary schools. based on the research findings, to eliminate errors and misconceptions in mathematics teaching in elementary schools, it is recommended: (1) teachers always improve mathematical skills in terms of understanding learning theory, and mastering the core material of each subject of mathematics; (2) the mathematics ability to change the framework in mathematics teaching can be improved through workshops, seminars, discussions with mathematical experts and teacher working groups; (3) applying mathematical concepts in daily life, especially the use of reasons and thought to solve life problems in society so as to support changes in logical and critical thinking. volume 9, no 1, february 2020, pp. 15-30 27 4. conclusion the findings show that, teachers evenly experience types of misconceptions: (1) pre-conception, (2) under-generalization, (3) over-generalization, (4) modelling error, (5) prototyping error; and (6) process-object error in teaching mathematics in elementary schools. the findings of this research reveal that the math skills of elementary school teachers need to be improved. various errors and misconceptions are oriented to conceptual and procedural errors in mathematics teaching. the misconceptions have been stable, permanent and rooted in "ontological misconception," in teacher thinking. the causes of misconceptions are (1) the teachers do not realize that the mathematical knowledge they have got because of teachers' years of belief that the knowledge they received was true when in fact it was not quite right.; (2) the mathematical knowledge possessed by the teachers have been accepted as rigid doctrines without any reasons to deny it for years. (3) the teachers’ confidence in the knowledge they receive, is stable, permanent and rooted in "ontological misconception," in the teacher's thinking. (4) the teachers interpret new experiences through incorrect understanding, thus inhibiting the entry of new information correctly. misconceptions tend to be very resistant to teaching and difficult to improve. therefore, learning requires replacing or reorganizing the teacher’s knowledge radically. through math skills training, misconceptions can be replaced or eliminated by changing the framework of teaching mathematics. acknowledgements the authors would like to thank for the support to elementary school teachers in semarang tengah sub-district and lppm universitas islam sultan agung who have helped smooth the research and funding assistance. references aliustaoğlu, f., tuna, a., & biber, a. ç. 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(2013). processes and priorities in planning mathematics teaching. mathematics education research journal, 25(4), 457-480. https://doi.org/10.1007/s13394-012-0066-z https://books.google.co.id/books?hl=id&lr=&id=lvhj4mjoflac&oi=fnd&pg=pp1&dq=ryan,+j.,+%26+william,+j.+(2007).+children%e2%80%99s+mathematics+4-15%3b+learning+from+errors+and+msconceptions.+london:+open+university+press.+241.&ots=fbaobcnss8&sig=q82iryducbeevzn4m4bm-o7bcxw&redir_esc=y#v=onepage&q&f=false https://books.google.co.id/books?hl=id&lr=&id=lvhj4mjoflac&oi=fnd&pg=pp1&dq=ryan,+j.,+%26+william,+j.+(2007).+children%e2%80%99s+mathematics+4-15%3b+learning+from+errors+and+msconceptions.+london:+open+university+press.+241.&ots=fbaobcnss8&sig=q82iryducbeevzn4m4bm-o7bcxw&redir_esc=y#v=onepage&q&f=false https://books.google.co.id/books?hl=id&lr=&id=lvhj4mjoflac&oi=fnd&pg=pp1&dq=ryan,+j.,+%26+william,+j.+(2007).+children%e2%80%99s+mathematics+4-15%3b+learning+from+errors+and+msconceptions.+london:+open+university+press.+241.&ots=fbaobcnss8&sig=q82iryducbeevzn4m4bm-o7bcxw&redir_esc=y#v=onepage&q&f=false https://www.learntechlib.org/p/208698/ https://www.learntechlib.org/p/208698/ https://www.learntechlib.org/p/208698/ https://doi.org/10.5899/2014/metr-00051 https://doi.org/10.5899/2014/metr-00051 https://doi.org/10.5899/2014/metr-00051 https://doi.org/10.1007/s10857-019-09446-z https://doi.org/10.1007/s10857-019-09446-z https://doi.org/10.1007/s10857-019-09446-z https://doi.org/10.1007/s13394-012-0066-z https://doi.org/10.1007/s13394-012-0066-z https://doi.org/10.1007/s13394-012-0066-z https://doi.org/10.1007/s13394-012-0066-z kusmaryono, basir, & saputro, ontological misconception in mathematics teaching … 30 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p161-174 161 prospective teachers understanding fraction division using rectangle representation muhammad ghiyats ristiana*1,2, dadang juandi1, sulistiawati3 1universitas pendidikan indonesia, indonesia 2institut keguruan dan ilmu pendidikan siliwangi, indonesia 3sekolah tinggi keguruan dan ilmu pendidikan surya, indonesia article info abstract article history: received jan 1, 2021 revised jan 15, 2021 accepted jan 16, 2021 fraction division is one of the most difficult subjects in elementary school. not only elementary students but many prospective teachers don’t understand the fraction division concept yet—most of them using a keep-change-flip algorithm to solve fraction division problems. a study using rectangle representation was conducted by us to prospective teachers. this study aims to see whether this rectangle representation will make prospective teachers understand or not. to do so, we made a mixed-method study with 80 prospective teachers as participants. the results show that 53,75% of prospective teachers use the keep-change-flip algorithm without understanding the concept of fraction division, and just 15% of prospective teachers understand fraction division. we assume that most prospective teachers still can’t imagine how fraction division works in a real-life context. they remember what they used to do to finish the fraction division problem that their teacher has introduced in primary school. based on the results, we conclude that the study with rectangle representation still needs an improvement, whether the teacher’s explanation or the rectangle media. keywords: fraction division, keep-change-flip algorithm, prospective teachers, rectangle representation copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: muhammad ghiyats ristiana, department of mathematics education, universitas pendidikan indonesia, jl. dr. setiabudhi no. 229, bandung, west java 40154, indonesia. email: mgristiana@upi.edu how to cite: ristiana, m. g., juandi, d., & sulistiawati, s. (2021). prospective teachers understanding fraction division using rectangle representation. infinity, 10(2), 161-174. 1. introduction fraction division is one of important subjects to learn, this concept can be utilized by students to solve their problems. for instance, when we want to divide 1/2 piece of cake to 3 people, and many more. as we know, formally fraction division problem usually solved by keep-change-flip algorithm. this was an easy problem to solve, but its difficult to understand how this algorithm can solve the fraction division problems. common core state standards initiative (ccssi) states that fraction subject began to be taught from third grade in elementary school (bentley & bossé, 2018). https://doi.org/10.22460/infinity.v10i2.p161-174 ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 162 however, fraction operation subjects still quite difficult in elementary school students, especially fraction division (fitri & prahmana, 2019; purnomo et al., 2019; widada et al., 2020; yeo, 2019). they still assuming that if a number divided by another, it will be smaller than before (ervin, 2017; yeo, 2019). besides that, they still using keep-change-flip algorithm to solve the fraction problems without understanding the meaning of fraction division and how that algorithm valid mathematically (alenazi, 2016; bentley & bossé, 2018; purnomo et al., 2019; sahin et al., 2020; whitehead & walkowiak, 2017; yeo, 2019). many things that can affect this problem, one of them is teacher habits that tends to give keep-change-flip algorithm directly to be memorized, remembered and implemented without telling them the concept of fraction division and how that algorithm valid mathematically (apsari et al., 2020; bentley & bossé, 2018; haji, 2013; purnomo et al., 2019; whitehead & walkowiak, 2017). fraction division also have the different meaning, so students still not properly understand the meaning of fraction division. fraction division known as measurement concept usually done with repeated substraction until the number that are divided become zero (purnomo et al., 2019; stohlmann et al., 2020; widada et al., 2020). furthermore, fraction division also known as equal share, for instance there is a half of cake that will be divided into a quarter of cake, how many person that will get that cake (purnomo et al., 2019; stohlmann et al., 2020). there are many ways to understand about the fraction division problems, and maybe if we are a creative person, it can be a new method to understand the fraction division problems. there are some ways to understand fraction division problems (adu-gyamfi et al., 2019) (see table 1). table 1. fraction division problems concept concept example in verbal representation algorithm measurement division needed 𝑐 𝑑 𝑚2 fabric to make one cloth. how many clothes that can make if just have 𝑎 𝑏 𝑚2 fabric? 𝑎 𝑏 ÷ 𝑐 𝑑 = 𝑎𝑑 𝑏𝑑 ÷ 𝑏𝑐 𝑏𝑑 = 𝑎𝑑 ÷ 𝑏𝑐 partitive division aldo has 𝑎 𝑑 chocolate that will be share with 𝑐 friends. how many chocolates for each friend? 𝑎 𝑏 ÷ 𝑐 = 𝑎 ÷ 𝑐 𝑏 𝑐 people bought 𝑎 𝑏 𝑘𝑔 chocolates that will be shared equal. how many chocolates for each person? 𝑎 𝑏 ÷ 𝑐 = 𝑎 𝑏 × 𝑐 determination of unit rate lanaya use 𝑎 𝑏 part of corned beef to make 𝑐 𝑑 servings of fried rice. how many servings that she can make if she uses a whole of corned beef? 𝑎 𝑏 𝑐 𝑑 = 𝑎 𝑏 × 𝑑 𝑐 𝑐 𝑑 × 𝑑 𝑐 = 𝑎𝑑 𝑏𝑐 inverse of multiplication from survey conducted to 𝑎 students, 𝑐 𝑑 students like mathematics subject and the others like biology subject. how many parts of students that like biology subject? 𝑎 ÷ 𝑐 𝑑 = 𝑑 𝑐 × 𝑎 to introduce the fraction concept, especially for fraction division, a study was present by klemer, rapoport, and lev-zamir (2019) that see how teacher tell students about fraction concept, especially fraction division concept. there are 3 ways to teach conducted by the volume 10, no 2, september 2021, pp. 161-174 163 participant of klemer et al. (2019) state that first, they illustrate fraction fivision with picture as grouping. checking how many divisors that are in the divider or checking how many second fraction that are in first fraction; second, gradually explanation. they tend to explain a real problem about fraction, make an equation where divisor as a whole, division operation integer with fraction, fraction with integer, fraction with fraction with same denominator, and fraction with fraction with different denominator; third, presenting students’ assumptions till using keep-change-flip algorithm. they will give a problem verbally about fraction division which make it possible to students to solve the problems in a various way. figure 1. the kob’s experiential learning cycle to make students understand about the fraction division, some media needed to be conducted to support their knowledge about this. a teacher should think how to make students understand about what have been learned. therefore, we try to make a rectangle representation media that may can make students easier to understand fraction division. this is based on kob’s experiential learning cycle that we can conducted a lesson from concrete experience until active experimentation (konak et al., 2014) (see figure 1). concrete experience that students should feel by themselves what they’ve learned during the lesson. concrete thing should be conducted especially in elementary school students. they tend to understand with the concrete media rather than the abstract one. so, students will be understood what the teacher said. then reflective observation where students doing some activity such as discussion with their friends, asking each other questions to reflective their experience. asking a question not always to be focused on the teacher, they can ask their friends to have their friend’s opinion about their understanding. the third cycle is abstract conceptualization where students can make a theoretical models and generalization from what they’ve learned. this model which is called abstraction from a ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 164 concept. mathematics will become abstract later, they can’t deny it. the last cycle is active experimentation, where they are planning and trying out what they’ve learned. there are two strategies, giving students a new task or combine some topics that connected with same activities. it is intended to see whether students are ready to have the next material or not, and whether the students are already understanding about the material that has been taught. we refer to the concrete experience first for making the students understand until they can active experimentation with the fraction division. as a teacher, we should be aware of the students answer about fraction division. so, a teacher must understand about the fraction division concept for reducing misconception about that topic (stohlmann et al., 2020). a teacher should introduce representation form that can be usefull for students, ask students to draw some fraction concept and proof that, and teacher should design how they do an assessment for students to see their understanding about fraction division. before teaching some materials, teachers should have a very good understanding about that concepts. so it is with fraction materials, especially fraction division which is known as a very difficult concept to understand among the elementary school students (purnomo et al., 2019; widada et al., 2020; yeo, 2019). therefore, teachers should know many different representations about fraction division to teach well about that. however, many prospective teachers even teachers still not understand about this fraction division concept (alenazi, 2016; whitehead & walkowiak, 2017). they still use keep-change-flip algorithm but not understanding about the fraction division concept (sahin et al., 2020). based on our preliminary studies, in one of university in cimahi, indonesia prospective teachers still not understand about fraction division. so, we decide to make an algorithm that may help prospective teachers understand about the fraction division. the aim of this study is to see whether prospective teachers understand about fraction division after they learn with this rectangle representation or not. fraction division can be represented by water filling from the big bottle to smaller one. suppose 1500 ml bottle considered as a whole, then 500 ml bottle considered as 1 3 part, 250 ml bottle considered as 1 6 part, etc. there is one 1500 ml bottle and one 500 ml bottle that filled with water. all of the water will be poured to 300 ml water. how many 300 ml bottle that needed? figure 2. rectangle which represents the size of each bottle we know that 1500 ml bottle as a whole, 500 ml bottle as 1 3 part, and 300 ml bottle as 1 5 part. so, we have 1 1 3 water that will be poured into 1 5 part bottle. mathematically, we can write 1 1 3 ÷ 1 5 and we can use keep-change-flip algorithm to solve that problem. 1 1 3 ÷ 1 5 = 4 3 ÷ 1 5 = 4 3 × 5 1 = 20 3 = 6 2 3 volume 10, no 2, september 2021, pp. 161-174 165 as we know that there are still many students think the result of division will be smaller than a number that will be divided. however, in this case the result is bigger than a number that will be divided and they may confused why the result is bigger (ervin, 2017; yeo, 2019). from this we try to make a rectangle which represent the size of each bottle (see figure 2). the answer of that problem is 6 2 3 , the six 300 ml bottle can be seen on that rectangle representation, but how can we have 2 3 ? as we know that we have 2000 ml water that will be poured into 300 ml bottle. so, we have to prepare seven 300 ml bottles to accommodate all of the water, because it will be 2100 ml. all of water can be poured into seven 300 ml bottles, but the last 300 ml bottle will not full and leaving 100 ml. from this case we can know that 100 ml from 300 ml is 1 3 part. so, we know that the last 300 ml bottle is 2 3 fulfill with water. given this rectangular representation will be easier to count, it’s just the last bottle will need more effort to count. rectangle representation with a number will be easier to count, however if we know only the fraction maybe it will be difficult to solve. from that problem we try to make a rectangle representation steps. figure 3 represents the problem above but without number. figure 3. rectangle which represents the problem figure 4 shows that there are seven 1 5 part will be poured with the 4 3 water. we know that there will be an empty space in the last 1 5 part that should we know to complete solving that problem. we can take the difference between white and blue rectangle, we called that the missing piece. figure 4. the missing piece shortly, we can write the algorithm of this rectangle representation step by step are as follows: a. known 𝐴 and 𝐵 as a fraction and we have a problem 𝐴 ÷ 𝐵. ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 166 b. first, we have to know whether 𝐴 < 𝐵 or 𝐴 > 𝐵. c. draw a rectangle that represents of each fraction. d. duplicate the smaller rectangle until the size is equal or more to the bigger rectangle. e. look for the difference between the smaller rectangle and bigger rectangle, we can call that the missing piece. f. do the repeated substraction of all the smaller rectangle that have been duplicated with the missing piece, now we know how many parts of missing piece that fulfilled all of the smaller rectangle. g. now put the bigger rectangle into all of the smaller rectangle and count it as a fraction with a whole as the smaller fraction. we try to apply that rectangle representation into a fraction division learning on prospective teachers. the aim of this study is to see whether this representation can make prospective teachers understand about the fraction division or not. 2. method the method used in this study is mixed method, quantitative with random sampling then qualitative descriptive method with case study design. eighty prospective teachers participated in this study to see their understanding about fraction division. the answer of that students will be categorized into 4 answer (apsari et al., 2020), are as follows: (a) no answer; (b) keep-change-flip; (c) not complete explanation; and (d) mathematically and example explanation. we used test and interview as the instrument of this study. before we conducted the instrument, we apply the rectangle representation to a fraction division lesson. after that, we conducted the test to all of participant that have been attend on that lesson. the test use to know how far hey understand about fraction division, and the interview is to know deeper about their answer and their understanding in fraction division. the instrument of the test are as follows (see figure 5): 1. why the division should change to multiplication and the second fraction should change to its invers? 2. the result of 2 3 ÷ 3 4 is … 3. how you explain this fraction division concept to students? figure 5. the instruments test first question is to see how prospective teachers using their knowledge to solve fraction division, second question is to see whether prospective teachers know why keepchange-flip algorithm valid mathematically, and last question is to see how they will tell students about fraction division concept so the students will not wrong again about the meaning of fraction division operation. the interview conducted after we analyzed their answers, and we just take one person for each category to know why they answer like that. we interviewed them with unstructured interview design. we are chasing their answer until we get what we want. after we interviewed them, we try to combine the answer between test and interview results. volume 10, no 2, september 2021, pp. 161-174 167 3. results and discussion 3.1. results after a lesson using rectangle representation was done, we conducted a test using the instrument above to all eighty participants. after finishing the test, we analyzed the answer of the prospective teachers and make it into 4 categorized. the following table 2 is the percentage of answers from 80 prospective teachers with 4 categorized: table 2. percentage of answers from prospective teachers into 4 categorized category the number of prospective teachers percentage no answer 3 3.75% keep-change-flip 43 53.75% not complete explanation 22 27.50% mathematically and example explanation 12 15.00% table 2 shows that 3.75% prospective teachers have no answer, they can’t answer correctly, whether they don’t understand the problem or even they don’t know how to solve it. also, we know that 53.75% prospective teachers prefer using keep-change-flip algorithm to solve the problem, but they still can’t understand about the fraction division. it can be seen from their answers that they just use the algorithm without giving the proper explanation about the fraction division. 27.50% prospective teachers answered correctly and give the explanation, but the explanation is not complete, so we can’t understand the whole of their answer. 15.00% prospective teachers answered correctly and can give a proper explanation with example and mathematically explanation. after we categorized their answer, then we take one from each category that will be interview. we want to know more about their answers, and why they answer like that. the interviewed is unstructured interview, we just refer to their test answer to know deeper about their answer. figure 6. the prospective teacher answer that represents “no answer” category ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 168 figure 6 show that the prospective teacher answer is wrong. he used keep-changeflip algorithm, but not have done correctly so the answer is not correct. three of the no answer category almost answer same as this. based on figure 6, prospective teachers who answer that said that “why the operation become multiplication because before it counts, the denominator also switched so the fraction still have an equal number. we don’t know why they didn’t know the concept of fraction division. whether the lesson is not suitable with him or indeed he can’t understand about the fraction division. therefore, we try to interview him to know deeper about his answer and to see whether he understand about fraction division or not. after we interviewed him, we know that he experienced the misconception about fraction division. he explains that the denominator is added by the numerator so the operation will be 1 5 ÷ 1 7 and because 3 is in both fractions so the first fraction is multiplied by 3 so the operation will be 3 5 ÷ 1 7 . he knows that fraction division can be done using keepchange-flip algorithm, so the operation will be 3 5 × 7 1 and the answer as in the picture above. we don’t understand why he answer like this because this thing is not accordance with fraction division concept. we conclude that he still not understands about fraction division and how to solve it even with keep-change-flip algorithm. after that interviewed, we try to explain his mistake and give the proper answer. so, he will not wrong again if he wants to solve fraction division problems later. also, we ask about how he will teach students about fraction division. he answers that he will give the students keep-change-flip algorithm after being told by us about the correct solution for fraction division problems. and he won’t tell any more, just tell the students about keep-change-flip algorithm. we conclude that one of the prospective teachers can’t teach fraction division properly. he needs to improve their knowledge about fraction division problems. so, he will be ready to teach this topic and make students understand properly about fraction division. most of prospective teachers in this study answer using keep-change-flip algorithm without giving their reasons why they answer using that algorithm. there are some of prospective teachers that give their reasons about why they answer using keep-change-flip algorithm. however, as we can see in figure 7 that the reasons are not explain why keepchange-flip algorithm can be used and valid mathematically, but they explain that using this algorithm is easier than solve the problem directly. we’re interested on their reason that tell us about easier using keep-change-flip algorithm than solve the problem directly, what solve the problem directly mean. so, we decided to interview one of them to know deeper about their reasons of their answer and to see whether they understand about fraction division properly or they still don’t understand about fraction division properly. volume 10, no 2, september 2021, pp. 161-174 169 figure 7. the prospective teacher answer that represents “keep-change-flip” category after we interviewed one of them to know the reason on their answer (see figure 7), they still give the reason like the test answer. we ask him what the meaning of using the keep-change-flip algorithm is easier than solve the problem directly. from his answer we know that solve the fraction division problem directly is with repeated substraction and it can’t because the first fraction is smaller than the second fraction. moreover, he just knows about keep-change-flip algorithm to solve the fraction division problem. he doesn’t know the other way to solve the fraction division problems. from this interview, we conclude that he understands about fraction division, but he doesn’t know why keep-change-flip algorithm can be used to solve the fraction division problems. also, he just knows that division is a repeated substraction and doesn’t know that fraction division can be done with other way. he said to teach students about fraction division is just directly give the keep-changeflip algorithm. he doesn’t understand the meaning of fraction division and the reason why keep-change-algorithm work with fraction division. that’s why he prefer just to tell students how to use keep-change-algorithm without telling them either the meaning of fraction division or the reason why keep-change-flip algorithm work. we turn to prospective teachers that categorized in not complete explanation. there are a quarter of all the prospective teachers that answered correctly and have an explanation, but the explanation is not completed yet. the prospective teachers can answer correctly, but ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 170 there is a missing explanation that should be completed more. he explains that it should be reversed so the denominator can be one and can answer why using a keep-change-flip algorithm is valid mathematically. according to our knowledge, we conclude that prospective teacher answers still not completed yet. after analyzing it, we conducted an interview to know deeper about his understanding of fraction division (see figure 8). figure 8. the prospective teacher answer that represents “not complete explanation” category after been interviewed, he can explain his answer with a good understanding about fraction division. he used keep-change-flip algorithm and he can proof why that algorithm valid mathematically and always used on fraction division problems. also, he explains that division is a repeated substraction or equal sharing, but they still not understand if the problem is fraction divided by the fraction. he also said that if the lesson of fraction division using rectangle representation is repeated, maybe he will now better the meaning of fraction division. to teach students about fraction division problems, he will start with the concrete representation that students know. but we know that his answer on the test is not representing a concrete problem, that’s why we categorized him to the not complete explanation. whereas he can explain the meaning of fraction division problems. after that, he explains a way to get the result of fraction division problems by telling them that 𝑎 𝑏 ÷ 𝑐 𝑑 is equal to 𝑎 𝑏 𝑐 𝑑 and the denominator of that new fraction should be 1, so that’s why he multiplied the numerator and denominator of the new fraction with the inverse of the denominator (see figure 9). volume 10, no 2, september 2021, pp. 161-174 171 figure 9. the process of solving dividing the fraction by invers method different from the others, there are only 15% prospective teachers participated in this study that answer as expected by us. but at least there are some prospective teachers that can understand about fraction division with the rectangle representation lesson. they can explain how to get the answer without using the keep-change-flip algorithm. they use some rectangle representation but not our rectangle representation that we conducted before. they prefer using vertical and horizontal rectangle representation, it might be easier according to them. they just need to combine the vertical and horizontal rectangle representation and will get how many parts they will get (see figure 10). we conclude that this way can be used to solve the fraction division problems and we see that they can understand about the fraction division concept. (a) (b) figure 10. the prospective teacher answer that represents “mathematically and example explanation” category after the explanation using rectangle representation, they write about the symbolic statement to solve this fraction division problem. they change the division into multiplication. as we know that if 𝐴 ÷ 𝐵 = 𝐶 then we can use 𝐵 × 𝐶 = 𝐴 to solve this fraction division problems. one of prospective teacher can proof why keep-change-flip algorithm used in fraction division problems and valid mathematically. he uses that concept but not all, he assuming what fraction that we can have to make the result of that fraction ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 172 division problem (see figure 10b). first, he knows that the result is from the rectangle representation (see figure 10). then he makes an equation 2 3 × … = 8 9 , and try to imagine what is the second fraction if he multiplied it by 2 3 he will get 8 9 . he can imagine it that the second fraction that make the result is 8 9 is 4 3 , because 2 × 4 = 8 and 3 × 3 = 9, so he got that 8 9 . after analyzing this answer, we conducted an interview to one of the prospective teachers to know deeper about their answer. after interviewed, he explains that the meaning of fraction division is when we equal share then count how many dividers that gets from the divided one. beside that, to proof the keep-change-flip algorithm he uses the concept 𝐴 ÷ 𝐵 = 𝐶 equivalent to 𝐵 × 𝐶 = 𝐴 (see figure 10b). so, he can conclude that keep-change-flip algorithm valid mathematically and can be used to solve fraction division problems. we asked him to explain how he will teach this fraction division problems to the students. he answers that he will tell the students step by step, from meaning of fraction until the operation, especially fraction division with different denominator. he will use some media to make easier for students to understand. for instance, he will use rectangle representation as he learned before from this class and will developed that media in order to make easier to understand. he tells us that the explanation of our rectangle representation a little bit difficult to understand, he should repeat again and again until he gets the whole meaning of fraction division through that rectangle representation media. these are the answer that represents for each category that we’ve been determined before. so, we can assume that some of prospective teachers can understand about fraction division problems using rectangle representation that we used before we conducted the test. and based on the interview, that the answer is diverse and many things that can influence their understanding about fraction division. 3.2. discussion based on the results, we can know that many prospective teachers do not understand fraction division. even we used rectangle representation before the test was conducted. prospective teachers use the keep-change-flip algorithm to solve the fraction division problems either without knowing whether that algorithm valid mathematically or understanding fraction division problems. they know if they are faced with fraction division problems, they use the keep-change-flip algorithm to solve the problems. it is not essential we understand the concept of fraction division or not. this finding is similar to many previous studies that still many prospective teachers and students that are using a keepchange-flip algorithm without understanding either fraction division concept or proofing keep-change-flip algorithm (apsari et al., 2020; bentley & bossé, 2018; purnomo et al., 2019; sahin et al., 2020; whitehead & walkowiak, 2017). most of the prospective teachers still just talk directly about keep-change-flip algorithm without making students understand either about the meaning of fraction division or the reason why keep-change-flip algorithm work with fraction division. this finding is in line with some of recent study that has been done (apsari et al., 2020; purnomo et al., 2019). but there are some prospective teachers that will explain the fraction division problems step by step. from representing the problems with a real-life condition, until the students get the abstract concept of fraction division problems. this finding is in line with recent study that teachers will teach students from the concrete one until the abstract (klemer et al., 2019). we assume that they still not understand the fraction division concept and keepchange-flip algorithm valid mathematically is because the rectangle representation media volume 10, no 2, september 2021, pp. 161-174 173 still not perfect and need to be developed until the audience will understand either about fraction division concept or keep-change-flip algorithm valid mathematically. different with the domino card media that can make students understand about fraction division (retnowati et al., 2018). 4. conclusion based on the results and discussion, there are still many prospective teachers who have not understood the problem of dividing fractions even though they have attended rectangular representation lessons. however, some know either the meaning of fraction division problems or why the keep-change algorithm works with fractional division problems. the problem of dividing fractions is a problem that is difficult to understand, even for teachers who have taught for a long time. the results showed that most of the prospective teachers still could not imagine how fractions would work. based on the experiences of researchers since elementary school, we used to solve the problem of dividing fractions with the keep-change-flip algorithm, and it was introduced directly without explaining the concept of fraction division by the teacher. so can cause most prospective teachers not to understand the concept of dividing fractions as a whole. most of the prospective teachers informed that they would teach students about the fraction division problem directly using the keep-change algorithm, as this is the easiest way to solve the fraction division problem. however, some prospective teachers will still teach their students in stages, starting from concrete questions to abstract questions about dividing fractions. therefore, the rectangular media representations still need to be developed, both media and their delivery from the teacher. the next researcher can use this rectangular representation as a reference to make a better media rectangular representation. references adu-gyamfi, k., schwartz, c. s., sinicrope, r., & bossé, m. j. (2019). making sense of fraction division: domain and representation knowledge of preservice elementary teachers on a fraction division task. mathematics education research journal, 31(4), 507-528. https://doi.org/10.1007/s13394-019-00265-2 alenazi, a. (2016). examining middle school pre-service teachers’ knowledge of fraction division interpretations. international journal of mathematical education in science and technology, 47(5), 696-716. https://doi.org/10.1080/0020739x.2015.1083127 apsari, r. a., sariyasa, s., indrawan, g., & maulyda, m. a. (2020). why should you reverse the order when dividing a fraction? a study of pre-service mathematics teachers’ pedagogical content knowledge in fractional concept. journal of physics: conference series, 1503(1), 012019. https://doi.org/10.1088/17426596/1503/1/012019 bentley, b., & bossé, m. j. (2018). college students' understanding of fraction operations. international electronic journal of mathematics education, 13(3), 233247. https://doi.org/10.12973/iejme/3881 ervin, h. k. (2017). fraction multiplication and division models: a practitioner reference paper. international journal of research in education and science, 3(1), 258-279. https://doi.org/10.1007/s13394-019-00265-2 https://doi.org/10.1080/0020739x.2015.1083127 https://doi.org/10.1088/1742-6596/1503/1/012019 https://doi.org/10.1088/1742-6596/1503/1/012019 https://doi.org/10.12973/iejme/3881 ristiana, juandi, & sulistiawati, prospective teachers understanding fraction division … 174 fitri, n. l., & prahmana, r. c. i. (2019). misconception in fraction for seventh-grade students. journal of physics: conference series, 1188(1), 012031. https://doi.org/10.1088/1742-6596/1188/1/012031 haji, s. (2013). pendekatan iceberg dalam pembelajaran pembagian pecahan di sekolah dasar. infinity journal, 2(1), 75-84. klemer, a., rapoport, s., & lev-zamir, h. (2019). the missing link in teachers’ knowledge about common fractions division. international journal of mathematical education in science and technology, 50(8), 1256-1272. https://doi.org/10.1080/0020739x.2018.1522677 konak, a., clark, t. k., & nasereddin, m. (2014). using kolb's experiential learning cycle to improve student learning in virtual computer laboratories. computers & education, 72, 11-22. https://doi.org/10.1016/j.compedu.2013.10.013 purnomo, y. w., widowati, c., & ulfah, s. (2019). incomprehension of the indonesian elementary school students on fraction division problem. infinity journal, 8(1), 5774. https://doi.org/10.22460/infinity.v8i1.p57-74 retnowati, r. p., kamsiyati, s., & matsuri, m. (2018). improving fraction operation skills of multiplication and division through the application of domino card to student of grade v sdn 02 pulosari academic year 2017/2018. social, humanities, and educational studies (shes): conference series, 1(1), 711-717. https://doi.org/10.20961/shes.v1i1.23626 sahin, n., gault, r., tapp, l., & dixon, j. k. (2020). pre-service teachers making sense of fraction division with remainders. international electronic journal of mathematics education, 15(1), em0552. https://doi.org/10.29333/iejme/5934 stohlmann, m., yang, y., huang, x., & olson, t. (2020). fourth to sixth grade teachers’ invented real world problems and pictorial representations for fraction division. international electronic journal of mathematics education, 15(1), em0557. https://doi.org/10.29333/iejme/5939 whitehead, a., & walkowiak, t. a. (2017). preservice elementary teachers’ understanding of operations for fraction multiplication and division. international journal for mathematics teaching and learning, 18(3), 293-317. widada, w., herawaty, d., lusiana, d., afriani, n. h., sospolita, n., jumri, r., & trinofita, b. (2020). how are the process of abstraction of the division of fraction numbers by elementary school students?. journal of physics: conference series, 1657(1), 012040. https://doi.org/10.1088/1742-6596/1657/1/012040 yeo, s. (2019). investigating children's informal knowledge and strategies: the case of fraction division. research in mathematical education, 22(4), 283-304. https://doi.org/10.7468/jksmed.2019.22.4.283 https://doi.org/10.1088/1742-6596/1188/1/012031 https://doi.org/10.1080/0020739x.2018.1522677 https://doi.org/10.1016/j.compedu.2013.10.013 https://doi.org/10.22460/infinity.v8i1.p57-74 https://doi.org/10.20961/shes.v1i1.23626 https://doi.org/10.29333/iejme/5934 https://doi.org/10.29333/iejme/5939 https://doi.org/10.1088/1742-6596/1657/1/012040 https://doi.org/10.7468/jksmed.2019.22.4.283 infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.218 121 learning media development approach with a rectangle problem posing based geogebra bagus ardi saputro mathematics education pgri semarang university, jalan didodadi east java, indonesia bagusardisaputro@yahoo.co.id received: june 28, 2016; accepted: july 18, 2016 abstract this study aims to develop learning media quadrilateral with problem posing approach based geogebra. 8 teachers from three different schools have stated that this media can be used to teach the nature the nature of the quadrilateral. after the learning is done using this media, this media can facilitate students in asking about the nature the nature of wake quadrilateral, facilitating students to learn the relationship between the type the type of wake rectangles that have the same properties, and provides the opportunity for teachers in the evaluation of mathematical communication current students ask and write. keywords: rectangular, problem posing, geogebra abstrak penelitian ini bertujuan untuk mengembangkan media pembelajaran segiempat dengan pendekatan problem posing berbasis geogebra. 8 guru dari 3 sekolah yang berbeda telah menyatakan bahwa media ini dapat digunakan untuk mengajarkan sifat – sifat segiempat. setelah pembelajaran dilakukan menggunakan media ini, media ini dapat memfasilitasi siswa dalam bertanya tentang sifat – sifat bangun segiempat, memfasilitasi siswa belajar hubungan antara jenis – jenis bangun segiempat yang mempunyai sifat yang sama, dan memberikan kesempatan kepada guru dalam melakukan evaluasi tentang komunikasi matematis siswa saat bertanya dan menulis. kata kunci: segi empat, problem posing, geogebra how to cite: saputro, b.a. (2016). learning media development approach with a rectangle problem posing based geogebra. infinity, 5 (2), 121-130 introduction quadrilateral is one of the study materials in the junior that is considered difficult by students (akhsani, sukestiyarno & wiyanto, 2012; rumiah, & darminto, 2015; disnawati, hartono & putri, 2015). this is shown by the low learning outcomes in the material quadrilateral (aryanti, zubaidah & nursangaji, 2013; rumiah, & darminto, 2015). flat material absorptive capacity in some areas such as in pacitan, banyumas, kebumen and also low (son, 2014; fadlilah, usodo & subanti, 2015; miftachudin, budiyono & riyadi, 2015). the problems arise because of several factors such as: (1) students tend to be passive in receiving a lesson, namely to listen and pay attention to the teacher (akhsani, sukestiyarno & wiyanto, 2012; setyaningsih, darminto, & purwoko, 2013; sasmi, aima & fitri, 2014; octavianti setiawan & trapsilasiwi, 2014), (2) the students used to memorize concepts, but can not yet understand the concept (yusuf, zulkardi & saleh, 2009; sasmi, aima & fitri, 2014; octavianti, setiawan & trapsilasiwi, 2014), (3) students rarely ask if any part of the material that is not yet mailto:bagusardisaputro@yahoo.co.id saputro, learning media development approach … 122 understood (putra, 2014), (4) students tend to be fearful and reluctant to answer questions provided by the teacher (putra, 2014), (5) a lot of students who do not understand the material prerequisites (putra, 2014 ), (6) when no homework or quizzes, students are less confident in their ability, they are lazy to think for themselves, the students enjoy asking her about material presented teacher (rumiah & darminto, 2015), (7) the student has not skillfully use tool the tool geometry such as the length and arc (sukayasa, 2014), (8) students are less able to identify the properties properties of the rectangle with the right, so it can not organize the linkages between wake by nature and nature (wulansari & rosyidi, 2013). the cause of the problem as: learning methods are used mostly still is direct / mechanistic (putra, 2014), teachers are just giving an example of completion of a problem by using a formula that is suitable and not yet using the form the form of other representations (aryanti, zubaidah & nursangaji, 2013), delivery of content quadrilateral only use images created on the board using chalk (akhsani, sukestiyarno & wiyanto, 2012; dwijayanti, 2012), teachers rarely pay attention to the potential, creativity and imagination of the students (yusuf, zulkardi & saleh, 2009; dwijayanti 2012, suyadi, 2015), the issue given problem has only one solution, so it does not facilitate the ability of divergent thinking (yusuf, zulkardi & saleh, 2009). efforts suggested in this problem are: education should provide experience to students to give examples examples that are not monotonous, and emphasis on nature the nature of the quadrilateral (wulansari & rosyidi, 2013), attention to student motivation (awaliyah, 2015) and attention to the selection models and methods appropriate to the material (setyaningsih, darminto & purwoko, 2013; ruminah & darminto, 2015; suryadi, 2015), trained from an early age to express ideas or opinions with many asking questions that could provoke the power of reason students (indahwati, 2015) , using the method of problem solving (hidayat, 2014), using open-ended questions (yusuf, zulkardi, & saleh, 2009). innovation in accordance with suggestions to solve these problems is the use of the modelbased learning problem posing geogebra. by using problem posing student achievement will be better (haji 2011; susanti, sukestiyarno, & sugiharti, 2012; robiah, 2013), understanding of mathematical concepts and understanding about the students will be better (herath, siroj & basir, 2010; haji, 2011). problem posing approach also provides an opportunity for students to convey (formulate), forming and asking a simple question or situation based on the information provided in order to solve a complex problem (haji 2011; herath, siroj, & basir, 2010). by using geogebra then (1) painting painting geometry generated faster than using a pencil, ruler and compass, (2) can be animated and moved by dragging on the object geometries that provide a visual experience, (3) be used as a feedback / evaluation make sure the paintings are made correctly, (4), enables teachers / students to investigate or show properties -sifat that apply to an object geometry (mahmudi, 2010), increasing the ability to think critically (ariawan, 2012), to facilitate students to try out, observe, reason and find the idea of an idea the idea of mathematical (saputro, prayito, & nursyahidah, 2015). therefore, this study sought to develop learning media rectangular-based approach to problem posing geogebra. method this study is one part of the implementation of the research methods research and development (r & d) is a test of learning media first. subjects were junior high school students from several schools in brebes central java province of indonesia. the instrument volume 5, no. 2, september 2016 pp 121-130 123 used in this study a quadrilateral instructional design approach based geogebra problem posing, interview guides, observation sheets incorporating sound recording device and image recording. interviews were conducted to the junior high school math teacher who was and is teaching material quadrilateral. results and discussion results development of instructional media quadrilateral with problem posing approach based geogebra produce three media with the same initial appearance is as shown in figure 1. figure 1. media problem posing media created by selecting properties special properties so wake quadrilateral has a special name. property that is used to construct one corner is the elbow the elbow, a pair of opposite sides are parallel, and a pair of side coincides same length. of these three properties, they invented the three wake quadrilateral. every waking quadrilateral is a situation that can raise many questions. questions questions that the students will be selected by the teacher to be discussed in the classroom. selected questions are questions that provide an understanding of the nature the nature of the quadrilateral wake. the media also comes with an information sheet about the nature the nature of all wake quadrilateral and student worksheet as media usage instructions. illustrations contained in the information sheet on purpose to be served by special properties possessed by a square, rectangular, trapezoid, parallelogram, kiteand rhombus. so that students easily understand the peculiarities of each type of the quadrilateral. student worksheets are designed so that students understand the wake like what they discover. then give the freedom to explore. exploration concludes with the question the question that will be posed to the class. discussion a. design validation responses math teacher secondary school to study media quadrilateral, namely (1) teacher teacher young mathematician with teaching experience of less than 6 years tend to reject the use of this media, it is because they think students they are incapable of learning with media and approach problem posing. they would suggest that the media is to be used in the classroom and the high caliber students who are already accustomed to using the computer as a media of learning. (2) teacher a teacher of mathematics with teaching experience of more than 13 years tend to be happy and want to use this media, only a few schools they do not have a computer lab facilities so that most of them feel sad not been able to use this media in a math class. however the teacher the teacher agrees that the instructional media saputro, learning media development approach … 124 quadrilateral with geogebra-based approach to problem posing can be used to teach the nature the nature of the quadrilateral wake. b. trial media learning in the pilot study, the size of which is always fixed for the first media is a corner that has a 90 angle measure. all groups responded that the angle a is always fixed at 90. but for the next question is what are the rectangles that can be formed when shifting points a, b, c, and d, each group has a different answer. the answer from each group are presented in table 1. table .1. recap answer question no. 2 rectangular types group a b c d e square      rectangle      kite  trapezoid     rhombus parallelogram   description:  = formed when teachers see students' answers that have been in the recap answer each group presented on teble 1. then the teacher can understand that some squares are not formed when the first media used. therefore, by using the lcd screen, the teacher with students can try, if true rectangle is indeed not formed. once tested, the trapezoid turns can be formed easily. but because the group d do not understand the nature of your trapezoid that has a pair of opposite sides parallel, then the group did not get a trapezoid. kite can also be formed. kite formed is the kite with one of its corners is 90. when discussing the wake kite. by shifting point c towards point a, it will obtain the square wake. students look at the size of the ab = da and bc = cd, which is a condition called quadrilateral kite. figure 2. square is a kite volume 5, no. 2, september 2016 pp 121-130 125 communication between teachers and students teacher: kite assume the character has two pairs of adjacent sides equal in length. square also has two pairs of adjacent sides equal in length. so the square can be referred to as kite. student: i do not agree the teacher! teacher: why do you disagree? student: it's square instead of kite. teacher: quadrilateral also has two adjacent sides of the same length. so there is no problem we say that the square is the kite. from the question no 3 obtained 28 questions. question this question consists of 15 questions about the quadrilateral and 13 questions instead of rectangular. the question is not about the quadrilateral consists of 8 questions about the technical use of media created, and 5 questions about the matter to another. questions about the quadrilateral consisting of nine questions about forming a quadrilateral and 6 questions about the nature the nature of the quadrilateral. all questions asked by the students are presented in table 2. table 2. recap questions from group 5 questions about the technical use of media 1. a corner why can not shift when making a parallelogram? 2. why can shift the angle b but could just up and down? 3. why the corner a has only 90 size? 4. why a corner can not be moved? 5. why is the angle b can not be in the right direction? 6. why can not the angle b to the left? 7. why quadrilateral retains 90 size? 8. why a corner can not be shifted? questions about other materials 1. if the angle b is pulled to the angle a will be arbitrary triangle. 2. what is the side that is owned by a circle? 3. what is the degree possessed angle isosceles triangle? 4. why is angle elbow length 90? 5. how many sides a triangle? questions about forming a quadrilateral 1. why kite difficult to set up? 2. why can not be shaped rhombus? 3. why kite can not be formed? but why the square can be formed? 4. why is a rectangle in shape? 5. why rhombus is also difficult in shape? 6. why can trapezoid shaped? 7. why parallelogram is very difficult to set up? 8. why does not form a trapezoid? 9. why angles b and c if it is pulled to the top will be a rectangle? saputro, learning media development approach … 126 questions about the nature the nature of the quadrilateral 1. why wake the same square with rhombus? 2. why has the size 90 square corners? 3. why the rectangles are facing the same length? 4. why does a trapezoid have a pair of parallel sides? 5. why rhombus has four sides of equal length? 6. why kite have two sides of the same length? when teachers get questions that relate the other material. teachers can, say that, we are learning about the rectangle, so we can not answer the question it. later we discuss the question after this study was completed. meanwhile, when the students started asking questions about the properties of the quadrilateral. here is your chance to answer the teacher and explain the properties properties of the quadrilateral and the relationship between the type the type of quadrilateral. one interesting question is why the building of the student square equal to a rhombus? answer this question is because a square has the same properties as the rhombus which has four sides of equal length. teachers can use the media to demonstrate what a second is to create a square with sides of 5 units. then shift the point d to get 90 angle and shift the point c to get side the side length 5 units. it concluded that a square is a rhombus whose angle 90. such a process can certainly be done for other types of quadrilateral. figure 3. a square is a rhombus most of the questions asked by the students have the same characteristics, namely using the question word "why". this question word indicates the student has a high curiosity. this allows students organize the relationship between each type of rectangular by nature nature. the obstacles encountered by teachers for use student learning media are not used to using a computer mouse is in use. so that students are less adept at shifting point. students also often do so make the scroll to be great. this is an excellent input for the revision process of this learning media is to turn off all geogebra tool unless the tool transfer. media first, second and third will be one to make it more efficient. the findings that are useful for teachers is the teacher can correct the students' mathematical communication is not right when the students express their opinions by speaking and writing. at the end of learning, the teacher asks the students to write down their thoughts on the learning that has been done. the result is all the group was pleased to learn the instructional media quadrilateral with problem posing approach based geogebra. some of those reasons are: volume 5, no. 2, september 2016 pp 121-130 127 "glad to study with a friend for help each other and feel easy because it uses leptop". "this type of learning is actually fun and easy to understand. because even though the material is solid, it is very easy for us to understand ". "i think the fun way of teaching". "i am pleased to learn together together to discuss about mathematics". "we are pleased to learn together because it is easier and fun, although a bit complicated". they also write the difficulties and feelings they experienced during the learning takes place. difficulties they include: "the difficulty for us when we shift and make the question" "not happy because it is rather difficult because sedikt scramble and noisy" "the bad news is our own class commotion" "that makes it difficult to learn is by clicking on his laptop" "the explanation is less obvious" student difficulties in learning are invaluable advice in using this media. some things that might be considered is the number of students per group, students' skills in using computers, classroom management and clear communication from teacher to student. conclusion from the trial learning in the classroom, can be obtained by several things, among others (1) instructional media quadrilateral with problem posing approach based geogebra can facilitate students to ask questions and express opinions, (2) can be used to demonstrate visually the relationship between the type the type of quadrilateral has the same properties, (3) and can be used by teachers in the evaluation of mathematical reasoning, mathematical communication when students ask and write. references akhsani, l., sukestiyarno & wiyanto (2012). pengembangan perangkat pembelajaran matematika dengan metode circ berbasis membaca berbantuan cd interaktif materi segiempat kelas vii. unnes journal of mathematics education research, 1(1). ariawan, i. p. w. 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(2009). pengembangan soal-soal open-ended pada pokok bahasan segitiga dan segiempat di smp. jurnal pendidikan matematika, 3(2), 48-56. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p197-212 197 profile of students' justifications of mathematical argumentation sukirwan1, dedi muhtadi2*, hairul saleh3, warsito3 1universitas sultan ageng tirtayasa, indonesia 2universitas siliwangi, indonesia 3universitas muhammadiyah tangerang, indonesia article info abstract article history: received apr 10, 2020 revised sep 12, 2020 accepted sep 13, 2020 this study investigates the aspects that influence students' justification of the four types of arguments constructed by students, namely: inductive, algebraic, visual, and perceptual. a grounded theory type qualitative approach was chosen to investigate the emergence of the four types of arguments and how the characteristics of students from each type justify the arguments constructed. four people from 75 students were involved in the interview after previously getting a test of mathematical argumentation. the results of the study found that three factors influenced students' justification for mathematical arguments, namely: students' understanding of claims, treatment given, and facts found in arguments. claims influence the way students construct arguments, but facts in arguments are the primary consideration for students in choosing convincing arguments compared to representations. also, factor treatment turns out to change students' decisions in choosing arguments, and these changes tend to lead to more formal arguments. keywords: mathematical argumentation, type of argument, justifying to argument, claim copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: dedi muhtadi, department of mathematics education, universitas siliwangi, jl. siliwangi no.24, tasikmalaya, west java 46115, indonesia email: dedimuhtadi@unsil.ac.id how to cite: sukirwan, s., muhtadi, d., saleh, h., & warsito, w. (2020). profile of students' justifications of mathematical argumentation. infinity, 9(2), 197-212. 1. introduction the national council of teacher mathematics (nctm, 2000; 2014) recommends proof as a fundamental mathematical aspect that should be studied at all levels of education. unfortunately, the clear proof is only being studied at the high school level. not surprisingly, when students are faced with proof, students experience many difficulties, as if the proof is a new learning and separate from the knowledge gained by previous students. it has triggered a variety of criticisms from many circles, which then encourage experts to think hard about how the proof should be taught to students (sukirwan et al., 2017). in this case, the proof should be a process, and human activity (stylianides & stylianides, 2006), meaning that every effort produced for the construction of proof cannot be ignored. these efforts then https://doi.org/10.22460/infinity.v9i2.p197-212 sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 198 encourage the importance of mathematical arguments that are considered capable of bridging between formal argumentation and proof (hidayat et al., 2018; pedemonte, 2008). rumsey & langrall (2016) reveals that argumentation is a process of social discourse to find new mathematical ideas to convince the truth of a claim. the form of argumentation is an argument, namely the reason or several reasons offered in rejecting or accepting propositions or claims (douek, 1999). wood (1999) calls argumentation as one type of social interaction, meaning that there is an interactive process in the exchange of discourse. in this case, argumentation is not only oriented towards the product produced but how to create a mathematical discourse process. thus the factors that influence the mathematical discourse process will determine the mathematical arguments produced. what will be constructed in an argument can come from claims. claims can be viewed as mathematical statements constructed to provide confidence to the audience based on data (erduran et al., 2004). data is the foundation on which arguments are based on facts that are relevant to claims. to accept or reject claims warrant (aberdein, 2009), which describes data into a collection of premises until a conclusion is obtained (spector & park, 2012). the extent of the warrant's power in deciphering data depends on the perspective used, both empirically and theoretically, even formal or informal (freeman, 2005). therefore, a review of the warrant has broad implications and is the basis for accommodating each of the arguments produced. viholainen (2011) once revealed that warrant could take various types of statements, either in the form of explanations or general statements in the form of formal or informal. in this context, the warrant is very open to formal and informal arguments. war flexibility allows students to justify, conjecturing, and generalizing (lin, 2018), meaning that students have the opportunity to construct arguments without being fixed on rigid mathematical rules. as such, students also have the opportunity to raise criticism of their arguments, thus opening the space to more formal arguments. several studies on argumentation have been carried out and become a trend in reforming international mathematics education (inglis et al., 2007; oecd, 2015). the study took different formats and constructs. at least there are three types of argumentation studies, namely: structure of argumentation, the taxonomy of proof schemes, and types of argumentation. studies on the structure of argumentation study the schema of argumentation constructed by students, identified as data, warrant, backing, qualifier capital, rebbutal, and conclusion (toulmin, 2003; zarębski, 2009). studies on the taxonomy of proof schemes identify arguments constructed by students based on abstraction levels (bergqvist, 2005; harel & sowder, 1998; stylianides & stylianides, 2009; varghese, 2011). both of these studies have the same characteristics, namely seeing argumentation as a product. behind that, both are still very strict on axiomatic systems so that if applied to different situations, there are chances that certain stages will not be fulfilled. studies on the type of argumentation identify emerging mathematical representations (healy & hoyles, 2000). although both are oriented towards argumentation as a product, this study appears more flexible and open. this study also accommodates the slightest amount of students' work in constructing arguments, so that product argument is not fixed on rigid argumentation schemes. liuа et al. (2016) even use mathematical representation to see the justification of students for argumentation. the result is that students have different views on convincing arguments. also, the tendency of students to argue with inductive patterns shows that students' habits in using arithmetic calculations, and routine algorithms still have a significant influence on student argumentation. from various studies of argumentation, students' discourse in producing arguments seems to be the most critical part of the study of argumentation. in this regard, two frameworks can be built, namely: how to accommodate every argument constructed by volume 9, no 2, september 2020, pp. 197-212 199 students and how to encourage students towards more formal arguments to bridge students towards learning scientific proof. these two frameworks open the space for the importance of a review of argumentation as a process, which today is an essential part of mathematics learning (lin, 2018; rumsey & langrall, 2016). studies on the type of argumentation and justification of students for argumentation basically can accommodate these two frameworks, but the fundamental thing that is not less important is understanding the factors that influence students' justification of argumentation. these results are ideally expected to provide input on the pedagogical actions needed for the process of mathematical argumentation. 2. method this study involved 75 eighth grade students who came from 2 different schools in tangerang city, banten province in 2019. from each school, one class was taken which was considered the most appropriate as the research sample compared to other classes because it was considered more mature in terms of research. think and have adapted to the school environment for longer. piaget (liuа et al., 2016), revealed that students at this level are in a critical cognitive phase where they can start to be involved in abstract and logical thinking. besides, activities that are not disturbed by preparation for final school exams are also a consideration in this study. a qualitative approach is used to examine students' arguments that arise from the results of mathematical argumentation tests (creswell, 2009). in this phase, students' arguments have not been identified, only coding arg 1, arg 2, arg 3, and so on. to identify these arguments, grounded theory is selected through the stages of open coding, selective coding, and theoretical coding (jones & alony, 2011). the open coding stage is the stage of analyzing the arguments that arise based on aspects of visualization, representation, mathematical expression, how to conclude, presenting context, and so on. meanwhile, at the selective coding stage, several student samples were selected to be followed up in interviews. theoretical sampling is carried out based on the need for supporting data to determine the similarities and differences in information that support theory formation (creswell, 2009). this stage determines the identification and justification of students' mathematical arguments constructed in the theoretical coding stage. guidelines for identification of students' full mathematical arguments can be seen in table 1. table 1. giving code to emerging arguments coding identity meaning arg 1 argument 1 the argument is stated by several examples (generally numeric) that support the validity of the claims submitted arg 2 argument 2 the argument is expressed from the context of symbolic representations which are then represented again to support the claims submitted arg 3 argument 3 arguments are expressed in graphs and images to provide an explanation of the claims/conjectures submitted arg 4 argument 4 an argument is expressed with a context known/imagined and supported by the conjecture through a connection ... ... ... arg l other arguments the argument that appears and is different from arguments 1, 2, 3, 4, .... etc. sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 200 data was collected through 2 main stages, namely: test mathematical arguments and interviews. in the first stage, the test is given by involving all students to construct arguments based on the statements and claims presented. the student's task is to provide an explanation of the truth of the claim or deny it. students can agree or not to the claims submitted by including reasons or proof that support students' assessment of claims. the test of mathematical argumentation is a test of mathematical abilities designed explicitly in the form of statements and claims. the statement contains information about claims data that can be developed into premises. claims contain conjectures or statements that must be verified. the substance of the complete mathematical argumentation test can be seen in table 2. table 2. mathematical argumentation test material statement claim surface and volume of cuboid there are two cuboids with different lengths, widths, and heights of each it. after the volume is calculated, it turns out that the two cuboids have an equal volume firman stated that although the volume of the two cuboids is equal, the surface must be different the volume of a rectangular pyramid in the qrst.uvwx cube there is a quadrilateral p.qrst as shown in the picture below. dudu suspects that the volume of the p.qrst pyramid is two times larger than the volume of p.qtxu! in the second stage, interviews were conducted involving some students selected based on theoretical samples. this sampling is carried out based on the consideration of representations of different types of arguments, the complexity of student answers, and other matters that arise and need to be further confirmed. while the purpose of the interview is to confirm the students' answers so that there is the relevance between the analysis of the answers predicted with the answers to the answers intended by the actual students. also, to get further information about students 'beliefs about argumentation, in-depth interviews were conducted on aspects related to students' understanding of the mathematical concepts used, comparing the types of arguments constructed, and the possibility of students to maintain arguments. 3. result and discussion 3.1. profile of students' mathematical arguments in the mathematical argumentation test, students' different arguments are grouped into argument 1, argument 2, argument 3, argument 4, and other arguments. the complete grouping results of these arguments can be seen in table 3. table 3. results of the mathematical argument test coding name conjecture 1 conjecture 2 n % n % arg 1 argument 1 26 34.67 16 21.33 arg 2 argument 2 10 13,33 12 16.00 arg 3 argument 3 3 4.00 9 12.00 volume 9, no 2, september 2020, pp. 197-212 201 coding name conjecture 1 conjecture 2 n % n % arg 4 argument 4 2 2.67 0 0.00 arg l other arguments 2 2.67 0 0.00 na not appear 32 42.67 38 50.67 in table 3, it appears that the arguments constructed by students are quite varied. in conjecture 1, there are 26 student arguments grouped into argument 1. whereas in conjecture 2 there are 16 student arguments grouped into argument 2. argument 1 is the argument that appears the most among other arguments. even so, the incidence of this argument is still low when compared to students who do not argue (na). there were 32 students or 42.67% of students in conjecture 1 whose arguments did not appear, while 38 students, or 50.67% of students in conjecture 2 had no arguments. this shows that in general students still have difficulty making claims so that they become valid arguments. a total of 10 students in conjecture 1 were identified as argument 2, and as many as 12 students in conjecture 2 were identified as argument 2. this number is less when compared to argument 1, but still more when compared to arguments 3 and 4. even in conjecture 2 argument 4 does not appear at all. in conjecture 1, 2 students are identified differently with arguments 1, 2, 3, and 4. to explore the different constructs of each of these arguments, the following is presented the students' arguments from each type of argument identified as well as the mathematical expressions constructed by students. 3.1.1. argument 1 in argument 1, students use several case examples to prove the conjecture. an example of a student's answer to the two conjectures identified as argument 1, can be seen in figure 1. figure 1. yumna's and puteri's arguments in conjecture 1 and 2, type 1 sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 202 yumna and putri have the same pattern in constructing arguments. both start the argument by displaying numeric data from units of known mathematical object elements. in conjecture 1, yumna presents 4 case examples with two primary case examples. the first case example is specifying two cuboids with the same volume, which is 60 cm3 with the size of the ribs of each cuboid {(3,2,10), (6,2,5)}. yumna found that the surface area of the two cuboids was different, namely: 112 cm2 for the first cuboid, and 104 cm2 for the second cuboid. this result shows the fact that the claim proved correct. the second case example is specifying two cuboids with the same volume, which is 48 cm3. yumna sets the size of the ribs of the two cuboids, respectively {(12,2,2), (4,3,4)}. once calculated, the surface area of the two cuboids turns out to be different, namely: 104 cm2 for the first cuboid, and 90 cm2 for the second cuboid. the second fact shows that the claim is proven correct. yumna then submitted a conjecture that the claim would be proven correct for another case with a volume of 56 cm3, 64 cm3, etc. because claims are proven correct for each case sample taken, the claim is generally valid. so the conjecture proved correct. in conjecture 2, putri presents 2 case examples to show that the comparison of the volume of pyramid p.qrst with the volume of p.qtxu's pyramid is 2: 1. for the first case, suppose that the ribs of the qrst.uvwx cube is 6 cm. the p.qrst and p.qtxu pyramid are inside the qrst.uvwx cube where p, qrst, and qtxu are located on the sides of the cube. putri calculated the volume of each pyramid so that the volume of the p.qrst pyramid was obtained at 72 cm3, and the volume of the p.qtxu pyramid was 36 cm3. the results of this calculation show that the comparison of the volume of the pyramid of p.qrst with the volume of the p.qtxu pyramid is 2: 1. for the second case, suppose that the ribs of the qrst.uvwx cube is 4 cm. the volume of the p.qrst pyramid is 21.33 cm3, and the volume of the p.qtxu pyramid is 10.67 cm3. the results of this calculation also show that the comparison of the volume of the p.qrst pyramid with the p.qtxu pyramid volume is 2: 1. based on the facts of the two case examples, putri concludes that the comparison of the volume of pyramid p.qrst with the volume of p.qtxu's pyramid is 2: 1. 3.1.2. argument 2 in argument 2, symbolic representations appear by presenting arbitrary elements of mathematical objects. the elements of this mathematical object are known as data from claims. data is then described in a mathematical model involving algebraic operations. if the results of the algebraic operation indicate that the claim can be proven, then the conjecture is correct. however, if there is only one case that shows a denial of claims, then the conjecture is wrong. sometimes proof of claims through direct algebraic operations cannot be made. in this condition, the truth or denial of an invisible claim is indicated by the results of the algebraic operation at the deadlock. to overcome this, the claim statement can be changed into a contradictory statement, so that its denial can indicate the truth of the claim. the following are examples of student arguments that are identified as arguments 2 (see figure 2). volume 9, no 2, september 2020, pp. 197-212 203 figure 2. arguments of rifa and yusuf in conjecture 1 and 2, type 2 in principle, the argument constructed by rifa is identical to the argument that joseph constructed. it is just that the arguments constructed by rifa do not arrive at a claim, so the conclusion taken is incorrect. this is evident when rifa concludes l1 ≠ l2 without showing that 2 {(ab) + (ac) + (bc)} ≠ 2 {(pq) + (pr) + (qr)} or {(ab) + ( ac) + (bc)} ≠ {(pq) + (pr) + (qr). the l1 ≠ l2 statement is an argument that will be proven, based on the previous set of arguments. because the l1 and l2 statements simultaneously do not lead to proof of the inequality of the surface area of the cuboid, then direct proof becomes difficult to express explicitly. alternatively, contradictory proof can be used by assuming l1 = l2. related to this, rifa seems to have not understood the statement, so that at that stage, she suffered a deadlock. although the arguments rifa did were not complete, the algebraic features in rifa's argument were apparent (see figure 2). this fact also shows that algebraic arguments are more complex than previous arguments. besides, the use of formal arguments has begun to appear with deductive principles applied. in conjecture 3, yusuf begins the construction of the data by declaring any pqrs.tuvw cube ribs. i am taking the r symbol to declare any rib of the pqrs.tuvw cube is the starting point in the use of symbolic representations. this statement was then represented again through the use of the quadrilateral pyramid formula so that the comparison of the volume of pyramid p.qrst with the volume of the p.qtxu pyramid was obtained by 2: 1. joseph's argument is evident with direct proof, even though the context is still elementary. 3.1.3. argument 3 argument 3 contains images, tables, or graphs that are re-represented through mathematical models based on the relationship between elements in the image or the relationship between known elements in images, tables, or graphs with elements outside of images, tables, or graphs. argument 3 can contain more flexible arguments by relying on active imagination. therefore, the review in the context of the problem in argument 3 is very open, even though the formal notation is still used. examples of student answers to argument three are presented in figure 3. sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 204 figure 3. irsan and nuraida's arguments in conjecture 1 and 2, type 3 based on figure 3, there is a different context between irsan's argument and nuraida's argument. although both rely on visual representation, irsan tries to link the context of the cuboid to the context of the unit cube. in irsan's argument, the cuboid partition in the unit cube the unit cube represents the volume of the block in the unit cube (block). the small cuboid partition in the picture next refers to half a unit cube or a quarter of a unit cube, so that the volume of the cuboid is still expressed in cubes (blocks). this argument seems to be somewhat similar to an inductive argument, but the number of partitions in irsan's argument only shows the unit volume of that partition. thus the partition can be done arbitrarily where the volume of the cuboid is expressed as a unit of partition volume. irsan then found that the total area of the partition on the surface of the two cuboids was different. this leads to the conclusion that the conjecture is true. meanwhile, nuraida utilizes internal relations between the elements contained in the main image. nuraida uses indirect relationships (outside the context in question) by utilizing all known elements in the main image. the complicated relationship is constructed by utilizing the conservation of volume law in which the volume of the pqrs. tuvw cube is the same as the volume of the p.qrst, p.qtxu, p.qrvu, p.rswv, and p.stwx as the constituent elements. because p is right in the middle of the uvwx field, the p.qtxu, p.qrvu, p.rswv, and p.stwx pyramid are congruent. if the volume of the p.qrst pyramid is 1/3 the volume of the qrst cube. uvwx has been known before, then the comparison of the volume of the pyramid of p.qrst with the volume of p.qtxu's pyramid is 2: 1. 3.1.4. argument 4 argument 4 is the least minimal argument providing information about the relationship between data and claims. this argument even requires further explanation to explain that the facts presented can be logically accepted. this argument relies on the context known in daily life as rationally acceptable as a fact that supports or denies claims. in this case, students must be good at choosing the right context so that the arguments put forward are accepted as the consensus. this work is not easy for students because not all volume 9, no 2, september 2020, pp. 197-212 205 mathematical concepts can be implied in the context of everyday life. therefore, it is not surprising that the arguments that appear in this type are the fewest, not even appearing in the second conjecture. examples of student work results on this type of argument can be seen in figure 4. figure 4. mega's argument on conjecture 1, type 4 in figure 4, mega takes the context of a cuboid-shaped object. two objects with different shapes, different surface areas. although not entirely true, perceptually, if the shapes of the two objects are different, then the plane of the side the corresponding side of the shape is different. consequently, the area of the side-plane plane of the corresponding side is also different. of course, if the corresponding pair of ribs is the same length, and it is seen that the two objects have different shapes, then the surface area of the two objects can be the same. this context does not appear to be in the perception of mega's argument. in the next statement, if the two objects are different, can the volume be the same? this statement is difficult to perceive because taking two objects whose volume is the same is difficult. however, when a certain amount of rice is poured into the first object so that it exactly meets the object, then poured it back into the second container and exactly fulfills the object, then the two objects of the volume are the same, even though the two objects are different in shape. 3.1.5. other arguments early detection of other arguments shows a different pattern in the construction of the argument. this argument arises in conjecture 1. examples of student work in this argument can be examined in figure 5. figure 5. yuki's argument at conjecture 1 for the other argument sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 206 in figure 5, yuki constructs an argument by presenting blocks in a unit cube (block). a total of 48 unit cubes are arranged in 2 different arrangements to form a cuboid. each arrangement consists of 24 unit cubes. thus, it is ensured that the two-cuboid arrangements have the same volume. the surface area of the two cuboids is then calculated based on the number of surfaces of the unit cube that appears. yuki found a difference in the number of surface units seen between the two units of the cube. this result leads to the conclusion that yuki took that conjecture 1 is correct. yuki's argument at first glance is similar to irsan's argument. it is just that yuki's argument is constructed based on the number of unit cubes defined from each cuboid arrangement, while irsan's argument is constructed based on partitions performed on two cuboids with the same volume. even though the partitions are counted, the partitions in irsan's argument are arbitrary. therefore, the partitions in irsan's argument can be different in shape. irsan's argument investigates the relationship to the parts of the image according to the characteristics of argument 3. 3.2. classification of mathematical arguments classification of emerging mathematical arguments is carried out by analyzing the mathematical characteristics of each type of argument as shown in table 1. in argument 1, students use several examples to prove claims. the student then concludes that on a more general basis the examples presented would indicate a true claim. these ways of proving such a claim appear identical to the inductive way of proof. in argument 2, students use a more general way by presenting the argument in symbolic representations. the presentation of mathematical expressions in the form of variables indicates that the use of algebra is an option for students to show claims. in argument 3, presenting the image is the first step chosen by students to show the truth of the claim. mathematical expressions are expressed based on the visualization of images where the proof of the truth about the claim is more real. in argument 4, students use a context that is constructed based on previous learning experiences. in this case, the mathematical expression is not presented in a formal form but uses non-formal rules that are perceived as logical. meanwhile, other arguments that have emerged seem to still use the rules of the previous argument, so that further new categories are not created. based on the patterns and general characteristics that appear in arguments 1, 2, 3, and 4; these arguments are further classified as inductive arguments, algebraic arguments, visual arguments, and perceptual arguments. this method of classification is the same as that of liu (2013) where the mathematical representations that appear in each argument that students construct are classified as inductive, algebraic, visual, and perceptual. the results of the analysis of these categories of arguments are further illustrated schematically in figure 6. figure 6. argument grouping scheme based on type of argument that appears in each category arg 1 arg 2 arg 3 arg 4 arg l induktif aljabar visual perseptua l volume 9, no 2, september 2020, pp. 197-212 207 referring to the variations that arise from each argument, three main characteristics must be further confirmed, namely: (1) student decisions in generating convincing arguments, (2) completeness of arguments, and (3) and mathematical concepts understood in every argument. these three characteristics become the essential foundation for knowing the factors that influence students' justification for the constructs of their arguments. 3.3. students' justification of mathematical arguments the deepening of the core categories is done by in-depth interviews with theoretical samples that represent the emergence of each argument. the interview refers to the three main characteristics that will be confirmed. the first characteristic is related to students' decisions in making convincing arguments. in constructing arguments, students should not necessarily use certain mathematical representations. students must have a reason to decide that the argument they construct can explain the claim correctly. the following interview quotes provide an overview of students' decisions in using certain types of arguments. ....................... researcher : "why did yumna choose this method?" [while showing the results of yumna's work on conjecture 1] yumna : "at the first test, i chose argument one, sir?" researcher : "what way did yumna think about being able to work on a problem like this?" yumna : ............. [looks confused] researcher : "what did yumna think about the cuboid in the question?" yumna : "o ... yes. i remember because the size of the cuboid is unknown, then i set the size first.” ....................... the results of interviews with yumna illustrate that the arguments he constructed were inspired by arguments that had been previously constructed. students are initially asked to look at existing arguments as part of the initial treatment before the test argument is given. yumna chooses inductive arguments based on the belief that this type of argument is most effective in explaining claims. from this explanation, yumna obtained information that the initial treatment had a strong influence on the students' decision to choose the argument they constructed. to complete information about a convincing choice of arguments, interviews were conducted with other students, as illustrated in the results of the following interview. ....................... researcher : "is mega thinking about two different shapes?" mega : "i imagine two containers in the form of cuboids; the shape is different." researcher : "what kind of container, for example?" mega : "the main thing is made of plastic." researcher : "where is mega, sure that the two containers are different?" mega : "yes, from the ribs, sir. for example, one container six, four, two, one another four, four, three." researcher : "why doesn't mega write like that?" mega : "that, the argument is different, right, sir." researcher : "okay. is the argument more convincing? " mega : "actually, yes, sir." researcher : "why not choose that argument?" mega : "at that time, i understood the argument four more, which in that example." ....................... sukirwan, muhtadi, saleh, & warsito, profile of students' justifications of mathematical … 208 the quote from the interview with mega illustrates that treatment influences students' beliefs about arguments. this belief can also change when more convincing arguments are found. in other words, students are not consistent with the choice of arguments and tend to change along with the treatment given. the characteristics of the next student's answer are the completeness of the argument. the characteristics of this answer are also related to mathematical concepts that are understood in every argument constructed by students. understanding of mathematical concepts is the main thing that influences student arguments, as illustrated in the following interview excerpt. ....................... researcher : "what is rifa's opinion about this conjecture 1?" rifa : "it's a general problem, sir." researcher : "i mean." rifa : "the sizes are unknown, so the proof is also arbitrary." researcher : "why aren't the ribs replaced with numbers?" rifa : "if the numbers only apply to certain blocks?" researcher : "okay, is rifa's answer, complete?" rifa : "i'm stuck here, sir" [while pointing at the answer sheet] researcher : "why not suppose that lone is not equal to l two." rifa : "don't understand, sir?" ....................... rifa understands that conjecture is a general statement. rifa considers that claims cannot be shown by a particular case, because the example only applies to that problem. it seems that rifa is not affected by the treatment given. this indicates that students in the formal category tend to defend their arguments. even so, rifa is not complete in constructing his argument. rifa does not seem to understand the ways to prove a statement, for example, by counterexample. rifa's opinion on the argument shows that students' understanding of claims influences the way students argue. students will try to prove the argument if the claim is understood correctly. understanding of claims will then influence the mathematical concepts described in the premises until a conclusion is obtained. in addition to understanding the claims, the facts in the argument also influence students' justification of arguments. this was revealed from the results of the following interview. ....................... researcher : "why did the putri choose this method to solve the problem?" puteri : "that's what i understand, sir." researcher : "are other arguments not understood? suppose this argument "[refers to an algebraic argument] puteri : "understood, too, sir. but if the ribs are determined it will be clearer." researcher : "what about the other arguments?" puteri : "understood that too. but the important thing is proven, right, sir." ....................... women's opinion shows that facts that show the truth of claims are one of the factors that influence the argument. the putri does not care about the representation that appears in certain arguments. the important thing is how the argument can show the truth of the claim. this is by the findings of previous research that the fact in the argument has a greater volume 9, no 2, september 2020, pp. 197-212 209 influence than the mathematical representation. the results of the interviews with the four students showed that the tendency of students to choose inductive arguments was more open than other arguments. this is relevant to the research of liuа et al. (2016) where the inductive way is the easiest to understand to explain the claim. another case with bergqvist (2005) where teachers in sweden underestimate students to use non-formal methods rather than formal methods. the findings of this study are relatively the same as berqvist's research, where the facts about the use of formal mathematics were chosen by students more than non-formal mathematics. it is interesting to explore in this study that students are always interested in changing their arguments into more formal arguments. argumentation may become a bridge for more formal mathematics learning, especially mathematical proof. as recommended by pedemonte (2008) about the cognitive unity hypothesis that allows students to learn mathematical evidence through mathematical argumentation. 4. conclusion based on the results of the research described in the result and discussion, it can be stated that the mathematical arguments constructed by students include four types of arguments, namely: inductive, algebraic, visual, and perceptual. the inductive argument presents several examples of cases which are then generalized. this argument also relies on numerical representation in strengthening the facts in a case example. the algebraic argument presents symbolic representation by specifying several elements of mathematical objects that are known arbitrarily. formal notation in this argument has also been seen by giving rise to a deductive approach. the visual argument presents images, tables, or graphs that are represented again in mathematical or visual models. this argument also relies on the relationship between elements in images or between images and contexts outside the image so that a mathematical model is obtained. the perceptual argument presents a known context that logically can be accepted as a fact that supports or denies claims. but because perceptions of the context can be different, the information in context sometimes needs to be explained further. three factors influence the justification of students for mathematical arguments, namely: students' understanding of proven claims, treatment given, and facts found in the argument. giving treatments is the main factor influencing justification. treatment even affects students' decisions in choosing arguments. these changes tend to lead to more formal arguments. behind that, students are also always interested in changing their mathematical arguments into arguments that are considered the most relevant for proving claims and leading to formal arguments. acknowledgments the authors would like to thank the teachers and students who participated in this research. references aberdein, a. 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(2021). identification of junior high school students’ error types in understanding concept about relation and function. infinity, 10(2), 175-190. 1. introduction indonesian 2013 curriculum says that relation and functions is a compulsory topic for junior high school students. in the attachment of education and culture ministry regulation no. 24 of 2016 states that the basic competencies that must be achieved in the material of relations and functions in the aspect of knowledge are to describe and to state relations and functions using various representations (words, tables, graphs, diagrams, and equations), and in the aspects of skills, the material covers solving problems related to relations and functions by using various representations (mecri, 2016b). relations are defined as relationships, and a function is defined as a process that connects each element from one set (domain) to exactly one element from another set (codomain) (as’ari et al., 2017; blanco et al., 2014). functions are not simple concepts, at least three systems of representation are used to represent the concept of functions in high school. they are tables (including ordered pairs), graphs, and formulas or equations. the characteristics of the https://doi.org/10.22460/infinity.v10i2.p175-190 fitrianna & rosjanuardi, identification of junior high school students’ error types … 176 concept of function make it able to be studied from two perspectives, as a part of mathematics and as an object in learning mathematics (blanco et al., 2014). based on studies regarding the analysis of students' abilities in relation and function material, the following results are obtained: 1) students experience epistemological obstacles related to the concept of relations and functions, 2) students experience misconceptions, 3) students have great difficulty in stating the definition of functions, solving problems in different contexts, and solving problems related to function (irawati et al., 2014; istiqomah, 2015). the results of interviews conducted by researchers with mathematics teachers at one of the tsanawiyah madrasah in kabupaten bandung barat, showed that the results of students' daily tests on relation and function material obtained an average value of 70.2, which is below the specified minimum completeness criteria, 72. hence, in this study, the types of errors made by students and students' conceptions of relations and functions will be identified. education and culture ministry regulation no. 21 of 2016 concerning content standards in primary and secondary education, states that one of the objectives of mathematics is to understand mathematical concepts, explain the relationship between concepts, and apply concepts or algorithms in a flexible, accurate, efficient, and precise manner in problem solving (mecri, 2016a). concepts are ideas that can be used or allow someone to classify an object (bell, 1981). abstract concepts are understood as two different forms, structurally as an object and operationally as a process. one is said to have an object of the concept if he/she is able to show the properties of the concept, while someone is said to have had a conceptual process if that he/she can discuss the concept using a mathematical object (sfard, 1991). a concept can be learned through definition (simon, 2017). when a student understands the definition of a concept presented in textbooks and classroom learning, the student will form an image of the concept in his mind (viholainen, 2008). concept images consist of all cognitive structures in an individual's mind that are associated with a particular concept, while concept definitions are the words used for a concept. initially a person has a mental picture of a given concept, all visual representations such as graphics, symbols of the concept, as well as a collection of properties related to the concept. the combination of these characteristics with mental images is called concept image (tall & vinner, 1981). furthermore, the concept image can be used as a guide which has been estimated to be the starting point for the emergence of students’ learning difficulties in certain materials and students' conceptions related to a concept. zetterberg (1966) provides an explaination that there are three components that make up a concept, they are symbols, objects and conceptions. conception is a model of explaining learners about a certain concept (simon, 2017). another explanation of conception is a form of internal representation of the concept, which is owned by students and becomes an element of a student's knowledge (sfard, 1991). bell (1993) states that several reasons for the importance of identifying students’ conceptions are: 1) students' conceptions are often not in accordance with scientific conceptions or the conceptions of experts, 2) students’ conceptions can help or hinder understanding of other concepts causing students’ difficulties in learning. students' difficulties in solving math problems can be seen from the mistakes made by students. this error can be seen from the identification of the student's work in doing the test (kariadinata et al., 2019). johari and shahrill (2020) show that it is very important to know the causes of common mistakes made by students. therefore, the teacher is able to facilitate students in reducing their errors in terms of analyzing the problem and reasoning about every step taken to solve it. analysis of students 'difficulties in this study was to reveal in depth the types of errors occurred, the factors that caused these errors, and students' conceptions regarding the material of relations and functions. volume 10, no 2, september 2021, pp. 175-190 177 the objectives of this study are to: 1) identify the types of errors made by students in relation and function material, and 2) identify students' conceptions regarding the concepts of relations and functions. 2. method this research is a descriptive study with a qualitative approach. this research involved 26 eighth grade students at one of the tsanawiyah madrasah, in kabupaten bandung barat. the research instrument used is in the form of a 6-question-diagnostic test of the relations and functions material (see table 1). before being given the test instrument, students had received online learning through the whatsapp group by the mathematics teacher, but in this study it would be limited to analyzing student errors seen from the results of student work and interviews. eight students were selected as interview subjects based on consideration of the types of errors made and the need to confirm students' answers. the interview questions are open-ended, and the questions are arranged based on basic competencies and indicators in the relationship and function material. table 1. distribution of indicators and question numbers basic competences indicators number 3.3. describing and expressing relations and functions using various representations (words, tables, graphs, diagrams, and equations) presenting relations with arrows, cartesian diagrams, and ordered pairs 1a 1b 1c showing examples of functions and not functions 2a 2b 2c 2d specifying the domain, codomain and result area of the function 3a 3b 3c expressing a function in an equation formula 6a 6b 6c 4.3. solving problems related to relations and functions by using various representations solving problems related to relations and functions 4 solve problems related to relations and functions in the form of equation formulas 5 the data that has been collected, then analyzed of students’ answer who made mistakes based on the types of errors and indicators as follows. the types of errors and indicators made by students are as follows (kiat, 2005): 1) errors in understanding questions (m) are the errors that occur in translating questions are indicated by errors in interpreting the language of the questions, 2) conceptual error type 1 (k1) is an error that occurs because students do not understand the concepts involved in the problem, 3) conceptual error type 2 (k2) is an error arising from the inability of students to determine the relationships involved in the problem, 4) procedural errors (p) are errors due to students' inability to manipulate or fitrianna & rosjanuardi, identification of junior high school students’ error types … 178 algorithms even though students already understand the concept behind the problem, 5) technical errors (t) are the errors due to carelessness. 3. results and discussion after checking the students' answers, the results obtained in the form of recapitulation of students' answers to each question (see table 2). the recapitulation of student’s answer is used to see the percentage of students who answered correctly, wrongly or not. table 2. recapitulation of student’s answer number true false no answer 1a 54% 23% 23% 1b 4% 62% 35% 1c 15% 42% 42% 2a 46% 46% 8% 2b 8% 85% 8% 2c 58% 35% 8% 2d 27% 35% 38% 3a 50% 31% 19% 3b 31% 50% 19% 3c 19% 62% 19% 4 77% 15% 8% 5 8% 92% 0% 6a 23% 46% 31% 6b 38% 31% 31% 6c 15% 42% 42% based on table 2, the students made mistakes in almost all the question numbers. the types of errors made by students were classified according to the types of errors that were made. volume 10, no 2, september 2021, pp. 175-190 179 figure 1. students’ types of errors recapitulation the types of errors made by students shown in figure 1 are quite diverse. to see the percentage of students who made each type of error (see table 3). table 3 is used to compare the number of students who make one type of error with another through the percentage. table 3. percentage of students’ error types error types understanding question conceptual procedural technical type 1 type 2 percentage 13% 62% 21% 2% 2% based on table 3, it can be seen that type one conceptual error (k1) and type two conceptual error (k2) were mostly committed by students. this is in accordance with research of hidayat and sariningsih (2018) that the mistakes often made by students in solving math problems, one of which is a conceptual error. the eight subjects selected for analysis on the results of their work, not all of them commit these types of errors. each question related to the causes of errors made by students and students’ conceptions related to indicators on the questions will be analyzed. 3.1. error analysis on question number 1 the first indicator for question number one is to present a relation with an arrow diagram, a cartesian diagram, and a set of consecutive pairs. the question can be seen in figure 2. figure 2. question number 1 for the first indicator 0 5 10 15 20 25 30 types of error m k1 k2 p t fitrianna & rosjanuardi, identification of junior high school students’ error types … 180 based on figure 1, it is found that the types of errors made by students on indicators presenting relations with arrow diagrams, cartesian diagrams, and consecutive sets of pairs are conceptual errors (k1 & k2). look at the answer given by s-3 (see figure 3), this student already had an idea of the concept of the arrow diagram but had not paired between set a and set b, after the interview it was known that she already understood the questions, but she did not understand the concept of relations and arrow diagram. this error includes the type of conceptual error (k1). thus, the conceptual error (k1), in this case, the student already had an image in the presentation of the relationship in various forms but she answered in another form that was not in accordance with the requested question, another form that was close to the arrow diagram. the reason was that she did not understand the arrow diagram and she answered by looking at the context of the sentences in the questions. s-19’s understanding was confused with another representation, the cartesian diagram. figure 3. the answer of s-3 on question number 1a based on the results of the interview, student’s assumptions about arrow diagrams were diagrams with arrows (see figure 4). thus, the conceptual error (k2) made by students was that students' understanding was confused with the concept of presenting one relationship with another. based on the results of the interview, student’s assumptions about arrow diagrams were diagrams with arrows on them, so that the students' conception of representations with arrow diagrams, consecutive sets of pairs and cartesian diagrams was still wrong. figure 4. the answer of s-19 on question number 1a volume 10, no 2, september 2021, pp. 175-190 181 3.2. error analysis on question number 2 the indicator for question number 2 showed examples of functions and not functions. the question can be seen in figure 5. figure 5. question number 2 for the second indicator based on figure 1, it was found that the types of errors made by students on indicators showing examples of functions and not functions are conceptual errors (k1 and k2) and technical errors. error types k1 and k2 will be analyzed. figure 6 show that s-3 didn’t answer accordingly to the concept of function. so, the conceptual error (k1) made was that the students had understood the problem, identifying a function and not a function but the student answered with another concept. s-3 gave the same answer pattern in picture 1-4. when the interview was conducted, the students already understood the problem, but the students' conception was a function is when there is a parallel straight line. figure 6. the answer of s-3 on question number 2 fitrianna & rosjanuardi, identification of junior high school students’ error types … 182 s-22's answer in figure 7 was correct, but for the reasons given it was still incorrect, because there was an empty domain. thus, the conceptual error (k2) was incorrect reasons although students had correctly answered. based on the results of the interview, the students' conception was that it is called a function if the domain and codomain were paired (nothing is empty). it is called not function because there were empty domains and codomains. the domain that the student meant was the one in b area. figure 7. the answer of s-22 on question number 2 3.3. error analysis on question number 3 indicator of question number 3 is determining the domain, domain and result area of the function. the question can be seen in figure 8. figure 8. question number 3 for the third indicator based on figure 1, it is found that the type of error made by students is conceptual error (k1 and k2). based on figure 9, the s-2’s answer shows that the students already understood the context of the question, but had not shown the conceptual image of the domain, codomain and result area. based on the results of the interview, the reason was that the students had not yet understood the concept of the domain, codomain and result area. so that the conceptual error (k1) made by students is that the students already understood the problem, determining the domain, codomain and result area, but did not understand the concept of domain, codomain, and result area. volume 10, no 2, september 2021, pp. 175-190 183 figure 9. the answer of s-2 on question number 3 the s-14’s answer in figure 10 shows an error in determining the result area. students already had an idea of the concept that the codomain is the set that will be paired with the domain, but in determining the result area, the students added up all members of the codomain that had pairs, 1 + 4 + 9 + 16 = 30. s-23 assumed that the result area was the final count or the final result so that all the numbers were added up. the conceptual error (k2) made by students was the answers in the results area were still incorrect although they had correct answers in the domain and codomains. students have a conception that the results area is summing up the elements in the paired domains. figure 10. the answer of s-14 on question number 3 3.4. error analysis on question number 4 the question indicator of number 4 is solving problems related to relations and functions. the question can be seen in figure 11. figure 11. question number 4 for the fourth indicator based on figure 1, it is found that the types of errors made by students are conceptual errors (k1) and technical errors. based on figure 12, s-21 had not answered correctly. the results of the interview stated that the students already understood the context of the questions, but this student did not understand what concepts were used and how to solve the fitrianna & rosjanuardi, identification of junior high school students’ error types … 184 questions. so that the conceptual error (k1) made was that he already understood what was known and asked, but did not understand the concepts used. figure 12. the answer of s-21 on question number 4 meanwhile, the s-26 made technical errors, this student already understood what was known and asked and carried out the calculation procedure, but were not careful in reading the questions (see figure 13). figure 13. the answer of s-26 on question number 4 3.5. error analysis on question number 5 indicator of question number 5 is solving problems related to relations and functions in the form of an equation. the question can be seen in figure 14. figure 14. question number 5 for the fifth indicator based on figure 1, the type of error made by students is a mistake in understanding the question (m). errors in understanding the questions made by s-22 was s/he did not volume 10, no 2, september 2021, pp. 175-190 185 understand the questions and concepts used, s/he only continued to compute from the table that has been given (see figure 15). figure 15. the answer of s-22 on question number 5 3.6. error analysis on question number 6 the indicator of question number 6 is to express the function in the equation formula and the graph of the function. the question can be seen in figure 16. figure 16. question number 6 for the sixth indicator based on figure 1, it is found that the types of errors made by students are conceptual errors (k1 and k2) and procedural errors (p). figure 17 show that s-14 already has a conceptual image of finding the value of the function by substituting the x value into the function, but the x variable remains in the final result and multiplying the value of the known variable by the second term. based on the interview, this student forgot to determine the procedure for determining the value of the function, as she remembered that the value in the brackets was multiplied, but the she forgot to multiply it by one (syllable) or both. so that at point a, the student made procedural errors, the students already understood the questions and concepts used but were wrong in carrying out the calculation procedure. in point b, the student made a conceptual error (k1), that is, the students already understood the questions but she was not correct enough to relate to the correct concept. in point c, students made conceptual errors (k2). in this case, students already understood the questions and concepts that must be used but had wrong understanding of the concept of the domain and result area. fitrianna & rosjanuardi, identification of junior high school students’ error types … 186 figure 17. the answer of s-14 on question number 6 the students s-25 can perform operations to determine the value of the function, but in point b students still cannot distinguish between codomains and result areas, and in point c in drawing function graphs there are still errors in writing the correct domain on the x-axis and the result area should be on the y-axis (see figure 18). based on the results of the interview, this student called the domain as the group that was on the right and the codomain as the group was on the left, the result area is the final result. as for the graph, s/he assumed that there were no certain conditions to place the domain and the result area. figure 18. the answer of s-25 on question number 6 based on the results of the answer analysis, students who made conceptual errors (k1), students had understood the questions well, but in answering it was not in accordance volume 10, no 2, september 2021, pp. 175-190 187 with the concept used. the following is the conceptual error type one (k1) on each question indicator: (a) in presenting the relationship with an arrow diagram, students presented it in another form that was close to the arrow diagram; (b) in identifying functions and not functions, students saw from the picture presented in the question, there were several parallel and crossed lines, so that students made mistakes in understanding the concept of function seen from parallel or cross arrows; (c) students did not understand the concept of domain, codomain and result area; and (d) in solving problems related to the concept of relations and functions, students did not understand what concepts used in solving problems. the following is the conceptual error type two (k2), students already understood the questions well and even they answered the questions correctly but they gave wrong reasons and were associated with concepts other than relations and functions: (a) in presenting relations with arrow diagrams, students' understanding was confused on the concept of presenting the form of relations and functions, one with another. the reason was that students had a different conception of the arrow diagram; (b) in identifying a function and not a function, students had a conception that it was called a function if the domain and codomain were paired (there was no one that was not paired). it was called not a function because there were domains and codomains that did not have a partner; (c) students had correctly answered for the domain and codomain, but the answer in the result area, the students had a conception for the result area that summing up the elements in the paired codomain; and (d) in solving problems related to the concept of relations and functions, they had used the requested concept, but in graphical presentation students did not pay attention to the location of the domain and result area. types of conceptual errors (k1) and (k2) are closely related to students' knowledge of the concept of relations, functions, domains, codomains, result areas and their presentation using arrow diagrams, sequential pairs and cartesian diagrams. for students who made conceptual errors (k1), they had a conceptual image that was asked for in the questions, but did not have the proper knowledge of the concepts, so they answered by paying attention to sentences in questions and other concepts outside the concept of relations and functions. this showed that the understanding of other concepts that were already owned by students affected students' understanding of concepts in the next material. these concepts were understood by students through visible characteristics. this is in accordance with (slavit, 1997), that a person understands the concept through various examples of functions and seeing its properties, students can understand the function as an object that has these properties. for students who made type two conceptual errors (k2), the reason was that they had a different conception of the correct concept. the wrong conception of students was the cause of mistakes made by students in relation to material and functions. the results of research by hatısaru and erbaş (2010) indicate that vocational secondary students have very weak perceptions of the concept of function. kamariah and marlissa (2016) also give similar results that students with the average ability had misconceptions, determining relations which are functions and determining certain values that fulfill a function. other error made by students was procedural errors, which are related to finding the value of the function of an equation. students had not been able to solve problems systematically which involved a thinking process. the cause of this error was students forgot how to perform the procedure for substituting variable values into equation functions. research results by hakim et al. (2020) show that the percentage of procedural errors committed by grade eight students at a junior high school in yogyakarta in the 2019/2020 school year in solving relationship and function problems is 93.7.%. in addition, the technical errors made were in the indicators of solving relationship and function problems. students were not careful in reading what questions were asked, even though the concepts fitrianna & rosjanuardi, identification of junior high school students’ error types … 188 and procedures were correct but the expected answers were still wrong. the students already had the concept description but there was inaccuracy from the students, resulting in errors. errors in understanding questions also occurred in indicators of solving problems of relations and functions, students did not understand what was known and asked about the questions and the use of concepts. students did not understand what was being asked so they did not have a conceptual description that will be used in solving the questions. the three types of errors were closely related to cognitive factors and students' conceptual images. this explanation is in line with tall and vinner (1981) that concept images consist of all cognitive structures in an individual's mind that are different from the formal concept definition and contain several factors that cause cognitive conflict. tall (1988) states that empirical research has emphasized that a person constructs a mental image of a concept in a way that may be inconsistent, and previous student experiences can influence the meaning of the phenomenon when students meet in a new context. 4. conclusion based on the results of the study, 62% of students did the first type conceptual and 21% of the students did the second type conceptual error. for students who make type one conceptual errors (k1), they have a conceptual image that is asked for in the question, but do not understand the correct concept, so that students solve the problem by paying attention to the sentence on the question and answer with other concepts that have nothing to do with the concept of relations and functions. for students who make type two conceptual errors (k2), students have a different conception of the correct concept. in procedural errors, students have not been able to solve problems systematically which involve a thinking process. in the misunderstanding of the questions, students do not know what is being asked so they do not have a conceptual description that will be used in solving the questions. then in technical errors, the concept description is already owned by students but there is inaccuracy from students, resulting in errors. incorrect students' conception regarding relation and function material are: (a) the arrow diagram is shown by a cartesian diagram and arrows are given on both axes; (b) it is called a function if the domain and codomain are paired (none of which is empty), it is called not function because there are empty domains and codomains; (c) the result area is summing up the elements in the paired codomain; (d) domain is the group that is on the right and the codomain is the group that is on the left, the result area is the final result; and (e) there are no rules for placing domains and codomain in drawing graphs of functions. references as’ari, a. r., tohir, m., valentino, e., imron, z., & taufiq, i. (2017). buku guru matematika (revisi). jakarta: pusat kurikulum dan perbukuan, balitbang, kemendikbud. bell, b. f. (1993). children’s science, constructivism and learning in science. victoria: deakin university pers. bell, f. h. (1981). teaching and learning mathematics (in secondary school). iowa: wnc brown comp. blanco, l. j., lizarazo, j. a. c., figueiredo, c. a., & contreras, l. c. (2014). the concept of function and his teaching and learning. far east journal of mathematical education, 12(1), 47-78. volume 10, no 2, september 2021, pp. 175-190 189 hakim, h., solechatun, s., & istiqomah, i. (2020). analisis kesalahan siswa dalam menyelesaikan soal uraian matematika kelas viii smp taman dewasa ibu pawiyatan. union: jurnal ilmiah pendidikan matematika, 8(1), 63-72. https://doi.org/10.30738/union.v8i1.7611 hatısaru, v., & erbaş, a. k. (2010). students’ perceptions of the concept of function: the case of turkish students attending vocational high school on industry. procediasocial and behavioral sciences, 2(2), 3921-3925. https://doi.org/10.1016/j.sbspro.2010.03.617 hidayat, w., & sariningsih, r. (2018). kemampuan pemecahan masalah matematis dan adversity quotient siswa smp melalui pembelajaran open ended. jnpm (jurnal nasional pendidikan matematika), 2(1), 109-118. irawati, r., indiati, i., & shodiqin, a. (2014). miskonsepsi peserta didik dalam menyelesaikan soal pada materi relasi dan fungsi kelas viii semester gasal smp negeri 4 kudus. mathematics and sciences forum 2014. istiqomah, d. n. (2015). learning obstacles terkait kemampuan problem solving pada konsep fungsi matematika smp. in prosiding seminar nasional matematika dan pendidikan matematika uny, 407-412. johari, p. m. a. r. p., & shahrill, m. (2020). the common errors in the learning of the simultaneous equations. infinity journal, 9(2), 263-274. https://doi.org/10.22460/infinity.v9i2.p263-274 kamariah, k., & marlissa, i. (2016). analisis kesalahan menyelesaikan soal relasi dan fungsi pada siswa kelas viii smp negeri buti merauke. magistra: jurnal keguruan dan ilmu pendidikan, 3(1), 30-42. kariadinata, r., yaniawati, r. p., sugilar, h., & riyandani, d. (2019). learning motivation and mathematical understanding of students of islamic junior high school through active knowledge sharing strategy. infinity journal, 8(1), 31-42. https://doi.org/10.22460/infinity.v8i1.p31-42 kiat, s. e. (2005). analysis of students’ difficulties in solving integration problems. the mathematics educator, 9(1), 39-59. ministry of education and culture of the republic indonesia [mecri]. (2016a). peraturan menteri pendidikan dan kebudayaan republik indonesia nomor 21 tahun 2016 tentang standar isi pendidikan dasar dan menengah. jakarta: kementerian pendidikan dan kebudayaan republik indonesia. ministry of education and culture of the republic indonesia [mecri]. (2016b). peraturan menteri pendidikan dan kebudayaan republik indonesia nomor 24 tahun 2016 tentang kompetensi inti dan kompetensi dasar pelajaran pada kurikulum 2013 pada pendidikan dasar dan pendidikan menengah. jakarta: kementerian pendidikan dan kebudayaan republik indonesia. sfard, a. (1991). on the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. educational studies in mathematics, 22(1), 1-36. https://doi.org/10.1007/bf00302715 simon, m. a. (2017). explicating mathematical concept and mathematical conception as theoretical constructs for mathematics education research. educational studies in mathematics, 94(2), 117-137. https://doi.org/10.1007/s10649-016-9728-1 https://doi.org/10.30738/union.v8i1.7611 https://doi.org/10.1016/j.sbspro.2010.03.617 https://doi.org/10.22460/infinity.v9i2.p263-274 https://doi.org/10.22460/infinity.v8i1.p31-42 https://doi.org/10.1007/bf00302715 https://doi.org/10.1007/s10649-016-9728-1 fitrianna & rosjanuardi, identification of junior high school students’ error types … 190 slavit, d. (1997). an alternate route to the reification of function. educational studies in mathematics, 33(3), 259-281. https://doi.org/10.1023/a:1002937032215 tall, d. (1988). concept image and concept definition. senior secondary mathematics education, 1983, 37–41. tall, d., & vinner, s. (1981). concept image and concept definition in mathematics with particular reference to limits and continuity. educational studies in mathematics, 12(2), 151-169. https://doi.org/10.1007/bf00305619 viholainen, a. (2008). incoherence of a concept image and erroneous conclusions in the case of differentiability. the mathematics enthusiast, 5(2), 231-248. zetterberg, h. l. (1966). on theory and verification in sociology. science and society, 30(1), 114-117. https://doi.org/10.1023/a:1002937032215 https://doi.org/10.1007/bf00305619 infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.220 141 the effectiveness of macros-based cognitive domain evaluation model in senior high school based on the curriculum 2013 djoko purnomo 1 , harjito 2 , rina dwi setyawati 3 , muhammad prayito 4 1,2,3,4 mathematics education pgri semarang university, jalan didodadi east java, indonesia 1 djokopurnomo@ikippgrismg.ac.id received: june 4, 2016; accepted: august 1, 2016 abstract the particular objective of this study was to investigate the effectiveness of implementation of learning device in the form of macros-based cognitive domain evaluation model via e-learning applied at 10 th grade of senior high school in the odd semester based on the curriculum 2013. the method of this study followed the procedures of r & d (research & development) developed by borg and gall. the results of the research and application development of macros-based evaluation model are effective which can be seen from (1) the results of students’ mastery learning, (2) students’ independence gives positive effect on learning outcomes, (3) the learning results of students who used macros-based learning evaluation model of cognitive domain are better rather than those in control class. based on the above results, it can be concluded that macros-based learning evaluation model of cognitive domain tested has met the quality standards. keywords: models, evaluation tool, curriculum abstrak tujuan khusus penelitian ini adalah mengetahui efektifitas penerapan perangkat pembelajaran berbentuk model evaluasi pembelajaran ranah kognitif berbasis program macros melalui e-learning pada sma kelas x semester gasal berdasar kurikulum 2013. metode penelitian yang digunakan dalam penelitian ini mengikuti prosedur r & d yang dikembangkan oleh borg and gall. hasil penelitian dan pengembangan aplikasi model evaluasi berbasis program macros efektif, yang dapat dilihat dari (1) hasil belajar mahasiswa tuntas, (2) kemandirian mahasiswa berpengaruh positif terhadap hasil belajar, (3) hasil belajar mahasiswa menggunakan alat evaluasi ranah kognitif berbasis macros lebih baik daripada hasil belajar peserta didik kelas kontrol. berdasarkan hasil diatas maka model evaluasi ranah kognitif berbasis program macros yang telah diujicobakan telah memenuhi standar kualitas. kata kunci: model, alat evaluasi, kurikulum how to cite: purnomo, d., harjito, setyawati r.d. & prayito, m. (2016). the effectiveness of macros-based cognitive domain evaluation model in senior high school based on the curriculum 2013. infinity, 5 (2), 141-146 introduction curriculum 2013 development is carried out on the basis of few key principles. first, competency standards are derived from necessity. second, content standards are derived from competency standards through their free subjects-core competencies. third, all subjects should contribute to the formation of attitudes, skills, and knowledge of learners. fourth, subjects are derived from the competency that needs to be achieved. fifth, all subjects are purnomo, harjito, setyawati & prayito, the effectiveness of macros-based cognitive … 142 bounded by core competencies. sixth, there is conformity of demands graduates, content, learning processes, and assessment. applications consistent on these principles become essential in realizing the successful implementation of curriculum 2013. based on the curriculum of 2013, the main problem in teaching high school (sma) students faced by the teachers is the change of students and teachers’ mindset in implementing learning activities. the transformation of ktsp into curriculum 2013 puts emphasize on development of the mindset improvement, reinforcement of curriculum management, deepening and expansion of the material, reinforcement of learning process, and adjustment of learning burden in order to ensure conformity between what is targeted and what is produced. curriculum development becomes very important in line with the continuity of the progress of science, technology, and arts & culture as well as transformation of society at local, national, regional, and global level in the future. various advances and changes lead to internal and external challenges in the field of education. therefore, the implementation of curriculum 2013 is a strategic step in dealing with globalization and the demands of indonesian society in the future. in line with transformation of curriculum 2013, then the learning process is more directed at active learning and teachers will be busy with the escort of the learning process, while conducting authentic assessments, both in the domain of cognitive, affective, and psychomotor. implementation of the assessment in every learning process at any time or certain period leads to the increase of correction burden for teachers in line with their teaching load consisting of 6 classes, with enrollment of about 6 x 30 = 180 students. in one semester there are about 8 subject matters; if any of the materials have three times competency tests, it means that there are 8 x 3 x 180 students = 4.320 students. in addition, teachers must also assess attitudes, performance, project, and portfolio. wood, cobb, and yackel (in turmudi, 2008) stated that mathematics should not be regarded as objective knowledge, but rather it has to be seen as an individuals’ active construction which is shared and understood by other people. so, in the learning process, there is the necessity of independence (self-regulated) to reconstruct knowledge and e-learning can be an alternative medium of learning to provide solutions to these problems. based on this point of view, we can develop macros program-based evaluation model of cognitive domain. the program of macros can be made through the system of network / lan in school laboratory or by preparing a cd / flash disk of evaluation that can be used by any student on a laptop/computer. method the method used in this study followed the procedures of r & d developed by borg and gall. the population was students of mathematics education at college in semarang. while the sample is students of sixth semester study program of mathematics education at the university of pgri semarang, walisongo state islamic university and the islamic university of sultan agung. data collection techniques used the method of documentation, test, of observation and questionnaire. volume 5, no. 2, september 2016 pp 141-146 143 results and discussion the validation of construction of macros-based evaluation model of cognitive domain covers main field testing and operational product revision; i.e. a tryout in class whose main purpose is to test the feasibility of the implementation of the evaluation model (second prototype). then, the third prototype is tested to the research subjects as a field test. the population in this tryout is the sixth semester students taking high school math courses. the sample was taken through purposive random sampling; namely, the class of mathematics education study program and the students from university of pgri. figure 1. photo of application of using macros program-based evaluation device mastery learning outcomes mastery learning test by using macros program-based evaluation device used one direction ttest, which is the right side. hypothesis used are as follows. h0 : 65 ha : 65 criteria for hypothesis testing i, reject h0 if sig <0.05 (5%) the calculations were performed using spss program and the results can be seen in table 1 and table 2 table 1. one-sample statistics n mean std dev std. error mean eksperimen1 28 71.5263 19.50985 3.16492 eksperimen2 28 68.5789 15.19536 2.46501 kontrol 29 66.1026 18.97338 3.03817 purnomo, harjito, setyawati & prayito, the effectiveness of macros-based cognitive … 144 table 2. one-sample test test value = 65 t df sig. (2tailled) mean difference 95% confidence interval of the difference lower upper eksperiment1 2.062 27 .046 6.52632 .1136 12.9390 eksperiment2 1.452 27 .155 3.57895 -1.4156 8.5735 kontrol -1.612 28 .115 -4.89744 -11.0479 1.2530 based on table 2, it was obtained that the value of sig. in class experimental 1 who used macros program-based evaluation device is 0.046. the value of sig. in class experimental 1 was compared to the significant level of 5%. it was obtained that sig value in class experimental 1 who used macros program-based evaluation device is less than 5%; it means that h0 is rejected. thus, the mean of the experimental class 1 who used macros programbased evaluation device is 71.53 and achieved mastery learning, more than 65. for the experimental class 2 who used macros program-based evaluation device, it was obtained that sig value is 0.155. sig. value of experimental class 2 was compared to the significant level of 5%. it was obtained that sig value in class experimental 1 who used macros program-based evaluation device is more than 5%; it means that h0 is accepted, so there is no significant difference. however, based on the table 4.4, the mean of class experimental 2 is 68.58 or more than 65 which is the value of the minimum criteria of mastery learning. as for the control class, it was obtained that the value of sig is 0,115. the value of sig. in control class was compared to the value of the significant level of 5%. it was obtained that the value of sig control class was less than 5%; it means that h0 is accepted, so there is no significant difference. based on table 1, the mean of control class is of 66.1026 or more than 65. so, the mean of class control is still above minimum criteria of mastery learning. effect of independence on learning outcomes to analyze the effect of evaluation model on learning outcomes, we can use linear regression and the results are shown in table 3 below. table 3. annova b sum of suares df mean square f sig. between groups 2673.650 1 1673.650 115.313 .000a within groups 555.125 27 14.609 total 3228.775 28 predictors: (constant), activity from the results above, it was obtained that the value of f = 115.313 and sig = 0,000 = 0%, which means that h0 is rejected which means linear regression equation. to measure the effect of independence on learning outcomes, we can be see table 4 below. volume 5, no. 2, september 2016 pp 141-146 145 table 4. model summary model r r square adjusted r square std. error of the estimate 1 .765a .732 .702 1.827 a. predictors: (constant), activity the amount of independence effect on learning outcomes can be seen from r square value in the table model summary which is 0.732 = 73.2%. this value indicates that mactro-based evaluation device affects learning outcomes by 73.2%. the comparison of learning outcomes between class experiment and control by using analysis of variance (anova), it was obtained that analysis of macros-based evaluation device is better than the students who did not use macros-based evaluation device. hypotheses used are as follows: h0 : 321   ha : at least there is an equal sign in equation h0. criteria for testing hypotheses ii, reject h0 if sig <0.05 (5%) based on the calculations using spss, the output was obtained presented in table 5. table 5. the results of anova spss output sum of suares df mean square f sig. between groups 2716.769 2 1358.385 4.190 .022 within groups 36306.327 83 324.164 total 39023.096 85 based on the table 5, it was obtained that the value of sig is 0.022. the value of sig was compared to the significant level of 5%. it was obtained that the value of sig. in the three classes is less than 5% or ho rejected. thus, there is a difference of mean between the learning applied to class experimental 1 subjected to macros-based evaluation model, class experimental 2 subjected to macros-based evaluation model and class control who is not subjected to macros-based evaluation model. therefore, it was followed by a further test using lsd. further test results can be seen from spss output in table 6. table 6. results of spss output post hoc (i) kelas (j) kelas mean difference (i-j) std. error sig. 95% confidence interval lower bound upper bound eksperimen 1 eksperimen 2 kontrol 2.94737 11.42375* 4.13053 4.10396 .477 .006 -5.2367 3.2923 11.1315 19.5552 eksperimen 2 eksperimen 1 kontrol -2.94737 8.47638* 4.13053 4.10396 .477 .041 -11.1315 .3449 5.2367 16.6079 kontrol eksperimen 1 eksperimen 2 -11.42375* -8.47638* 4.13053 4.10396 .006 .041 -19.5552 -16.6079 -3.2923 -.3449 *. the mean difference is significant at the .05 level purnomo, harjito, setyawati & prayito, the effectiveness of macros-based cognitive … 146 based on the table 6, it was obtained that the value of sig. between the class experimental 1 subjected to macros-based evaluation device and class experimental 2 subjected to macrosbased evaluation devices is 0.477. the value of sig. was compared to the significant level of 5%. it was obtained that the value of sig. between the two classes, namely the class experimental 1 subjected to macros-based evaluation device and class experimental 2 subjected to macros-based evaluation devices is more than 5%; it means that h0 is accepted. so, there is no difference significant mean between the class experimental 1subjected to macros-based evaluation device and class experimental 2 subjected to macros-based evaluation devices. the value of sig between class experimental 1 subjected to macros-based evaluation device and class control not subjected to macros-based evaluation device is 0.006. the value of sig. between these two classes compared to the significant level of 5%. the value of sig. that is obtained for the two classes, namely class experimental 1 subjected to macros-based evaluation device and class controls not subjected to macros-based evaluation device is less than 5% or h0 is rejected. so, there is a significant different mean between class experimental 1 subjected to macros-based evaluation device and class controls not subjected to macrosbased evaluation device. the value of sig. between the class experimental 2 subjected to macros-based evaluation device and class controls not subjected macros-based evaluation device is 0.041. the value of sig. of both classes was compared to the significant level of 5%. the value of sig obtained for the two classes, namely the experimental class 2 subjected to macros-based evaluation device and class controls not subjected to macros-based evaluation device is less than 5% or h0 is rejected. so, there is a significant different mean between class experimental 2 subjected to macros-based evaluation device and class control classes not subjected to macros-based evaluation device. conclusion based on the development process of macros-based evaluation model of cognitive domain, it can be concluded that the process and result of development had reached validity in the first year. then, in the second year, the effective application of macros-based evaluation device has been implemented, which can be seen from (1) the results of students mastery learning, (2) students’ independence gives positive effect on learning outcomes, (3) the results of students learning using macros-based evaluation device of cognitive domain is better than the result in the class control students. based on the above results, it can be concluded that macros-based learning evaluation model of cognitive domain tested has met the quality standards. references borg, w.r. and gall, m.d. (1983). educational research: an introduction. london: longman, inc. paechter, m. & maier, b. (2010). online or face-to-face? students' experiences and preferences in ict. the internet and higher education, 13 (4), pp 292-298. turmudi (2008). landasan filsafat dan teori pembelajaran matematika. jakarta: leuser cita pustaka. wijonarko (2011). efektifitas perangkat pembelajaran teori bilangan berbasis elearning pada mata kuliah teori bilangan. semarang. laporan penelitian. http://www.sciencedirect.com/science/article/pii/s1096751610000692 http://www.sciencedirect.com/science/article/pii/s1096751610000692 infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.217 109 increased capacity of the understanding of the concept and the ability to solve problems through the implementation of the model of teaching mathematics realistic based on cognitive conflict students dewi herawaty 1 , rusdi 2 1,2 mathematics education, bengkulu university, indonesia 1 dewiherawaty71@gmail.com, 2 rusdi.unib@gmail.com received: may 30, 2016; accepted: july 10, 2016 abstract this research aims at: 1) the influence of the implementation of the model of teaching mathematics realistic based on cognitive conflict students to the ability to understanding the concept and troubleshooting capabilities; 2) determine the larger capacity of the understanding of the concept through the implementation of the model of teaching mathematics realistic based on cognitive conflict junior secondary school students the city of bengkulu. 3) determine the great improvement of the ability to solve problems through the implementation of the model of teaching mathematics realistic based on cognitive conflict smp students bengkulu city.to achieve the goal of this research is to apply research design pseudo experiments with research design pretest-postest nonequivalent control group design, with the test instrument the ability to understanding the concept and test the troubleshooting capabilities. the data has been analyzed using the test gains. the results of this research is 1) the ability of understanding the concept and troubleshooting class experiment the given learning with pmr is better than with the ability to understanding the concept and troubleshooting control classes assigned to conventional mathematics lesson; 2) increase the ability of the understanding of the concept through the implementation of the model of teaching mathematics based on cognitive conflict smp students bengkulu city is significant with the index gain of 0,755 (highlevel); 3) increase the ability to solve problems through the implementation of the model of teaching mathematics based on cognitive conflict smp students bengkulu city is significant with the index gain of 0,500 level (is). keywords: model of teaching, conceptual understanding, troubleshooting, cognitive conflict abstrak penelitian ini bertujuan: 1) pengaruh penerapan model pembelajaran matematika realistik berdasarkankonflik kognitif siswa terhadap kemampuan pemahaman konsep dan kemampuan pemecahan masalah; 2) menentukan besar peningkatankemampuan pemahaman konsep melalui penerapan model pembelajaran matematika realistik berdasarkankonflik kognitif siswa smp kota bengkulu. 3) menentukan besar peningkatankemampuan pemecahan masalah melalui penerapan model pembelajaran matematika realistik berdasarkankonflik kognitif siswa smp kota bengkulu.untuk mencapai tujuan penelitian ini menerapkan desain penelitian eksperimen semu dengan desain penelitian pretest-postest nonequivalent control group design, dengan instrumen tes kemampuan pemahaman konsep dan tes kemampuan pemecahan masalah. data dianalisis dengan menggunakan uji-gain. hasil penelitian ini adalah 1) kemampuan pemahaman konsep dan pemecahan masalah kelas eksperimen yang diberi pembelajaran dengan pmr adalah lebih baik dibandingkan dengan kemampuan pemahaman konsep dan pemecahan masalah kelas kontrol yang diberi pembelajaran matematika konvensional; 2) peningkatan kemampuan pemahaman konsep melalui mailto:dewiherawaty71@gmail.com herawaty & rusdi, increased capacity of the understanding … 110 penerapan model pembelajaran matematika berdasarkankonflik kognitif siswa smp kota bengkulu adalah signifikan dengan indeks gain sebesar 0,755 (level tinggi); 3) peningkatan kemampuan pemecahan masalah melalui penerapan model pembelajaran matematika berdasarkankonflik kognitif siswa smp kota bengkulu adalah signifikan dengan indeks gain sebesar 0,500 (level sedang). kata kunci: model pembelajaran, pemahaman konsep, pemecahan masalah, konflik kognitif how to cite: herawaty, d. & rusdi (2016). increased capacity of the understanding of the concept and the ability to solve problems through the implementation of the model of teaching mathematics realistic based on cognitive conflict students. infinity, 5 (2), 109-120 introduction various government efforts, school principals and teachers in improving the quality of mathematics teaching is continuing. but the fact that found in schools, learning materials mathematics smp is still not meaningful for students. this is as a result of the learning process where the smp mathematics materials delivered very theoretic and the students are learning in mekanistik (widada, 2005). the students that will be experiencing cognitive conflict between the conception that he had with the strukturalistik mathematics materials and very theoretic. the difference is too far between the conception of students called from the long term memory and the concept of/principles of mathematics received in working memory students, resulting in imbalance in this leads to cognitive conflict for students in mathematics teaching. in addition, as part of the effort to accelerate the change in the understanding of the concept on the students then applied a strategy that can cause an imbalance in (disequiibrium) in the mind of the students or cognitive conflict (al-arief, 2012). according to asdar (2012) that cognitive conflict is a condition of awareness of the individual who is experiencing the imbalance. based on the initial survey conducted researchers against the students smp n 3 and junior secondary school students n 18 bengkulu city, obtained cognitive conflicts students in mathematics teaching. cognitive conflict between the students is as follows. 1) how 2/5 + 1/3? students answer 3/8, cognitive conflict occurs when the students give reasons (2+1)/(5+3). 2) simplify 3x 7y 6x + 3y 8? the students answered 3x 7y 6x + 3y 8 = 3x 6x 7y + 3y 8 = -3x 10y 8. cognitive conflict occurs when the students complete that -7y+3y = -10y. 3) simplify (15x3y + 5x)/5x. students answer 15x3y + 1. cognitive conflict experienced by students as follows (15x3y + 5x)/5x = 15x3y + 1. in addition, widada (2013) find a cognitive conflict in the form of overgeneralisasi done junior secondary school students in factoring square equation. a student overgeneralisasi revealed as in the interview footage as follows. interviewer : mas try whether you know how to complete the following equation x 2 3x + 2 = 0? students : ... from my teacher pak... similarities compute square roots can be resolved with factoring... ....... [students' can factoring square equation correctly]..... interviewer : ... okay ... well. if so try try you complete the square equation volume 5, no. 2, september 2016 pp 109-120 111 x 2 3x + 2 = 20? ............. [students working on paper in a specific time]................... interviewer : ... how the result? students : well pak.... square equation x 2 3x + 2 = 20, can be completed with factoring that the left so obtained: (x-2)(x-1) = 20, means equivalent with x 2 = 20 or x-1 = 20, means the solution is x = 22 or x = 21 and finished pak. interviewer : ... well, ... if so do you know what is common square? students : i ca bu teacher writes ax2 + bx + c = 0 are equivalent square, so pak definition. widada (2013) analyzing the preview above that overgeneralisasi occurs when the students asked to complete the square equation x2 3x + 2 = 20, without using the concept of the square students directly factoring equation the equation. the students only memorized the definition of square equation, but an understanding of the definitions are only as declaration of sentences that less meaningful for students. so that the students said that: "square equation x2 3x + 2 = 20, can be completed with factoring that the left so obtained: (x-2)(x-1) = 20, means equivalent with x 2 = 20 or x-1 = 20, means the solution is x = 22 or x = 21 and finished pak." if the students really understand the concept of the square equation, then the students will restore it in accordance with the definition of square equation, so that the equation x2 3x + 2 = 20 can be formed in the square equation becomes x2 3x 18 = 0. square equation x2 3x 18 = 0 this can be accomplished with factoring become (x-6)(x+3) = 0, so obtained x 6 = 0 or x+3 = 0 and the solution is x = 6 or x = minus 3. in mathematics teaching the students are seen as active information processing, so that students are able to perform the adaptation to school environment. adapatasi as intended by piaget, include assimilation, accommodation, disekuilibrasi, and ekuilibrasi/re-ekulibrasi. students who experience disekuilibrasi which duration long possible cognitive conflict occurs. therefore teachers should provide assistance in the form of scaffoldding zone based on the development of undistributed (as vygotsky theory (arends, 1997; arends, 2001; slavin, 1994)). mathematics teaching is the process of manipulating the condition that allows learners to do a logical activities in an effort to the discovery or the discovery of the principle of/mathematical concepts. for mathematics is a human activity (u.s. mathematics s human activity, freudenthal (in gravemeijer, 1995). therefore required a conscious effort that can condition the interactive activities between the components of the lesson the students of teachers learning resources), so make it easy for students to complete the task. furthermore it is expected that each student is able to perform internalisasi in processing system information. there are four teaching approaches in mathematics education based on the horizontal and vertical components matematisasi namely mekanistik, empiristik, strukturalistik and realistic (treffers, 1991). based on the opinion, realistic approach to approaches that can help smp students (which the majority are in the stage of development of concrete operational inteletual) in developing the ability to understand the concept and solve mathematics problems with easy. for according to gravemeijer (1994) in mathematics teaching realistic, there are three main principles namely, find back and progressive matematisasi, the phenomenon of journalists teamt, and build your own model. herawaty & rusdi, increased capacity of the understanding … 112 to take advantage of the real problem with mathematics teaching realistic based on cognitive conflict encourage the students are able to solve the problem with his own way. the ability to solve mathematics problems is the ability to search for and resulted in the settleme nt of a mathematics problem with various activities: 1) understand the problem that is given, 2) model the mathematics, 3) solving problems in accordance with the model of mathematics made, 3) interpret the settlement (hamalik, 2001). mathematics teaching realistic based on cognitive conflict encourage students to mengkonstruk a concept or find the concept of mathematics (treffers, 1991). therefore, learning outcomes can improve the ability of students in mmahami mathematical concepts. the ability of understanding the concept covers several indicators that are able to claim the concept that has been studied, able to classify objects based on met or whether or not the requirements that form the concept, able to apply the concepts in the algorithm, is able to give an example and non-examples from the concept that has been studied, able to present the concept in various forms of representation is able to associate the concept of internal and external mathematics), and able to develop the necessary conditions and the condition is quite a concept (kilpatrick, swafford, & findell (2001). in efforts to achieve the purpose of mathematics teaching and take advantage of cognitive conflict as a starting point mathematics teaching mathematics teaching realistic approach is needed (pmr). mathematics teaching realistic is basically the utilization of the reality and the environment is understood learners to facilitate the process of mathematics teaching to achieve the goal of mathematics education is better than in the past. according to soedjadi (2001) also explain what is with reality, and the things that real or concrete steps that can be observed or understood learners through imagine, while is with the environment is an environment where learners are good school environment, families and communities can be understood learners. the environment is also called the daily life, and glasersfeld (1992), said mathematics is "" reflects the real world through the process of empirical abstraction. mathematics teaching should be done with attention to all parts of the development of psychology students, and done with the humanist. according to the hendriana (2012), create mathematics that humanists in the learning is the initial capital to provide a stimulus early to the students so that the negative responses toward mathematics decreases. with enjoying the mathematics learning mathematics become a pride for the students so that it is expected that the habit of creative thinking become trained. one of the alternative learning approach that is expected to support the huanis mathematics and creativity of the students are metaphorical approach thinking. the writer is trying to examine about the implementation of the approach metaphorical thinking is on the power increase student creativity. based on the explanation above, the process of mathematics teaching realistic use contextual problems (contextual problems) as a starting point in learning mathematics. by applying the pmr, it is hoped that students have troubleshooting capabilities and have the ability to understanding the concept. to achieve this, then dotted with the reject mathematics problem realistic, students can easily complete tasks/questions about mathematics. based on the explanation above, the process of mathematics teaching realistic use contextual problems (contextual problems) as a starting point in learning mathematics. the characters in mathematics teaching realistic and take advantage of cognitive conflict (asdar, 2012; muhammad, 2012), the students are able to perform cognitive process to achieve the ability of understanding the concept and troubleshooting issues with good. volume 5, no. 2, september 2016 pp 109-120 113 in this article discussed about the influence of the implementation of the model of teaching mathematics realistic based on cognitive conflict against the ability of understanding the concept and troubleshooting capabilities junior high students. method to achieve the goal of this research is conducted research experiment facades with research design pretest-postest nonequivalent control group design, with the test instrument the ability to understanding the concept and test the troubleshooting capabilities. a sample of this research is 80 junior secondary school students in bengkulu city which is selected by the technique of cluster random sampling. this research instrument in the form of the test the ability of understanding the concept and test the troubleshooting capabilities. the data has been analyzed using kovarians multivariat analysis. the results of the test data capabilities of the understanding of the concept of the test and troubleshooting capabilities, analyzed by using the index score gains (g). if g>= 0.7 then n-gain is categorized as high; if 0.7 > g >= 0.3, then n-gain is categorized as being; and if 0.7 > g < 0.3, then n-gain categorized low (hake, 1999). results and discussion the results of research and this discussion is the analysis of the data has been seminar in activities semirata veterinary (bks west in unsri 22-24 may 2016 (dewi herawaty & rusdi, 2016). observation of the students during the learning process is done by the two observers for each time the meeting. the observation data obtained using a sheet of observation of 6 students as the sample observation. the students who made samples of observation randomly selected from 25 students in each school that made the research. the results of these observations data two teachers dirata averaged, then calculated with the precentage. there is 89,01% time used students to perform activity: 1) observe and understand contextual problem is given in the book the students; 2) working together to resolve the issue in the book the students/worksheets in iranian groups (pairs); 3) pour answers permaslahan into worksheets individually; 4) asked the teacher or a group of friends/another friend; 5) appreciate ideas friends; and 6) membandingan results of the solution to the problems with the results of the work of a friend. this shows that the average every aspect/category that is observed on the activity of the students during the learning process to four the rpp imposed including effective category. thus the activity of the students in the learning process with the implementation of the model of teaching mathematics realistic is effective. furthermore based on the results of the test data capabilities of the understanding of the concept of the test and troubleshooting capabilities, analyzed with kovarian analysis. kovariat variable in this research is the ability to start the students who obtained from the value of the pretes students whereas bound variable is the ability to understanding the concept and problem solving the students who obtained from the value of the postes. inferential analysis using kovarians analysis (anakova) with the steps as follows: regression model y = a + bx, with a and b is estimatasi for 1 and 2 from the equation y = 1 + 2x based on the results of the calculation of the class regression model experiment obtained similarities following regersi model. y = 75,59 + 0,34x herawaty & rusdi, increased capacity of the understanding … 114 based on the results of the calculation of the control class regression model regression model equation is obtained as follows. y = 51,23 + 0,45x the analysis to test the independency of the class regression model experiment is as follows, with significant rank  = 5 % diproleh f (0.95; 1 ; 43) = 4.08 which means f* > ( 0.95; 1; 43) then regression model coefficient is equal to zero. this means that the ability of the beginning students (x) has a significant impact on the ability of understanding the concept and problem solving students (y). to test the indepedensi analysis class regression model control, namely with significant ttaraf  = 5 % diproleh f (0.95; 1 ; 43) = 4.8 which means f* > ( 0.95 ; 1 ; 43) then regression model coefficient is equal to zero. this means that the ability of the beginning students (x) has a significant impact on the ability of understanding the concept and problem solving students (y). test the linieritas regression model based on the analysis of the class regression model lineritas experiment obtained that, with significant rank  = 5 % diproleh f (0.95; 4 ; 39) = 2,61 which means f* > f( 0.95 ; 4 ; 39) then class regression model coefficient experiment is linier. this means that the relationship between the ability of the beginning students with the ability to understanding the concept and problem solving students peda class experiment is linier. based on the analysis of the class regression model linieritas control that with significant rank  = 5 % diproleh f (0.95; 4 ; 39) = 2,61 which means f* > f( 0.95 ; 4 ; 39) then control class regression model coefficient is linier. this means that the relationship between the ability of the beginning students with the ability to understanding the concept and problem solving students peda control classes are linier. because the data on the class of experiments and control classes are linier then continued with the next test. test the similarity of two regression model based on the results of the test calculations common two class regression model experiment and control classes obtained regression model linier aggregated data as follows. y = 14,29 + 3,45x f* = 158,05 using the rank of significant  = 5 % diproleh f (0.95; 2 ; 86) = 3,15 which means f* > f( 0.95 ; 2 ; 86), then the ho was rejected. this means that the regression model class linier experiment and control classes are not the same. test the equality of two regression model because the two regression model is not the same, then will continue with the test regression coefficient consistency. based on the results of the test calculations parallel class regression model experiment and control classes obtained the results of the analysis as follows. a = 2662,45 b = 2346,42 f* = 5.46 volume 5, no. 2, september 2016 pp 109-120 115 using the rank of significant  = 5 % diproleh f (0.95; 1 ; 86) = 4.00 which means f* > f( 0.95 ; 1 ; 86), then the ho was rejected. this means that the regression model class linier experiment and control classes aligned. because all varians analysis are met and both the regression model is aligned it can be concluded that there is a difference between the ability of understanding the concept and problem solving the students in the classroom experiment the given learning with pmr, with the ability to understanding the concept and troubleshooting control classes assigned to conventional mathematics lesson for the subject of the same. class regression lines of the experiment and control classes aligned and planck regression line for the class experiment is greater than planck regression lines to control classes shows that there is a significant difference between the two regression model. regression lines are geometrically to the classroom experiments on the control class regression lines, which means that the ability of understanding the concept and troubleshooting class experiment the given learning with pmr is better than with the ability to understanding the concept and troubleshooting control classes assigned to conventional mathematics lesson. the result of the study of the capacity of the understanding of the concept and the ability to solve mathematics problems smp students bengkulu city 2016 (has been disseminated in jambi international seminar on education, 3-5 april 2016 by herawaty, dkk., 2016), which can be served one by one in the ability test score statistics diagram understanding the concept and test scores troubleshooting capabilities as follows. figure 1. average postes pretes score and the ability of understanding the concept based on the figure 1, average pretes score the ability of understanding the concept is of 7.82, and average scores postes the ability of understanding the concept is 77,44. this may indicate that the difference in average scores pretes and postes is 69,62, which means that the increase in the ability of understanding the concept of very high. furthermore can be served the graph average postes pretes score and troubleshooting capabilities in the picture below. 0,00 50,00 100,00 average postes average pretes 77,44 7,82 the ability of understanding the concept of herawaty & rusdi, increased capacity of the understanding … 116 figure 2. average postes pretes score and troubleshooting capabilities based on the figure 2, average scores pretes troubleshooting capabilities are 8,38, and average scores postes troubleshooting capabilities is 54,22. this may indicate that the difference in average scores pretes and postes is 45.84, which means that the increase in troubleshooting capabilities is very high. furthermore can be served the graph average gains the ability to score on the concept of understanding the picture below. figure 3. average and deviation stadart score gains the ability of conceptual understanding based on the figure 3, average gains pretes score the ability of understanding the concept is of 69,62, and standard deviation score postes the ability of understanding the concept is 9,57. this may indicate that the average capacity of the understanding of the concept of the students is very high. furthermore can be served the graph average and standard deviation score gains troubleshooting capabilities in the picture below. 0,00 10,00 20,00 30,00 40,00 50,00 60,00 average postes average pretes 54,22 8,38 troubleshooting capabilities 0,00 10,00 20,00 30,00 40,00 50,00 60,00 70,00 average gains sd 69,62 9,57 the ability of understanding the concept of volume 5, no. 2, september 2016 pp 109-120 117 figure 4. average and stadart deviation gain troubleshooting capabilities score based on the figure 4, average gain score pretes troubleshooting capabilities is 45,84, and standard deviation score postes troubleshooting capabilities is 7,23. this may indicate that the average increase in the ability of understanding the concept of problem solving the students is high enough. based on the figure 1 until with figure 4, can be calculated the index score gains as follows: (1) the index score gains the ability to understanding the concept of (gpk) = 0,755; (2) the index score gains troubleshooting capabilities (gpm) = 0,500. with the index score gains the ability to understanding the concept is the gpk = 0,755, n score gains the ability to understanding the concept is categorized as high. means that the ability of understanding the concept of mathematics smp students n 10 bengkulu city after being taught by applying the model of teaching mathematics realistic based on cognitive conflict students have increased high. the contribution of the implementation of the learning model are able to perform the horizontal matematisasi easily and to take advantage of the realistic approach, the students were able to achieve the concept and understand with both through the vertical matematisasi (as revealed by treffers, 1995). so also based on the index score gains troubleshooting capabilities (gpm) = 0,500, then nscore gains troubleshooting capabilities are categorized as being. means that the ability to solve mathematics problems smp students n 10 bengkulu city after being taught by applying the model of teaching mathematics realistic based on cognitive conflict students have increased high enough. with pemerapan mathematics teaching realistic based on cognitive conflict students, making it easier for students to understand contextual problems that was given to make it easy for students to make up the model of mathematical prowess, and make it easy for students to solve the problem are given by using a model that he had made. as the results of research widada (2013), in learning equation system linier two variables (spldv) in junior high school teachers often told with strukturalistik approach. teachers give definitions of spldv then gives an example and continued with the exercise. but there are some teachers who presents spldv through learning that more closely with the basic scheme students. based on this second teachers, obtained decompositions of genetic in memory system students is as follows. 0,00 10,00 20,00 30,00 40,00 50,00 average gains sd 45,84 7,23 troubleshooting capabilities herawaty & rusdi, increased capacity of the understanding … 118 teachers : [students without giving the concept of] try you complete the questions about "eat in the canteen" following. [teachers give activity sheet students (las) the contents as follows.] figure 5. problem eat in the canteen in las i [.... around 15 minutes the students are given time menyelesesaikan questions las i...] students : pak i will try to answer ... is jawaba i pak... [students providing answers in las i as figure 5] figure 6. one of model answers the students from the problem of food in the school canteen in las i students : .... my answer in las i [as picture 5 above] two items namely one meatball and one ice valuable intervention the same 13 thousand rupiah... i can scrub and the result as follows. [students pointed toward the settlement of such as figure 6 below) .... means one sliced meatballs price rp 8,000. figure 7. answers the students after the same removal rp 21,000, rp 13,000, rp 21,000, rp 13,000, rp 8,000, volume 5, no. 2, september 2016 pp 109-120 119 students : ... sir, then i can replace the price of the bowl meatballs into one of the picture... [students point in the answer in las i as figure 7] figure 8. students replaced the rp 8,000 to meatballs students : because 1 bowl meatballs i replace with rp 8,000 and whole rp negara which means one glass of ice him five thousand rupiah sir. based on the genetic decompositions students about spldv above shows that the students are given las based on issues that near with his thoughts very well and very good to take advantage of previous schema-him in processing system information. short-term memory as working memory works so good as do the process based on the procedures that he had. widada (2013) stated that based on the above quotation, the activities of the action against the objects of the physical and metal can dienkapsulasi, processed through ditematisasi interiorisasi and be mature about eleminasi scheme and substitution spldv without must be delivered down the facts about eleminasi, and substitution and principles. but in the genetic decompositions thus students find themselves eleminasi principles in accordance with expressions: ".... my answer in las i [as figure 5 above] two items namely one meatball and one ice valuable intervention the same 13 thousand rupiah... i can scrub and the result as follows. [students pointed toward the settlement of such as picture 6 below) .... means one sliced meatballs price rp 8,000." said "...removes ..." in the phrase kebermaknaan eleminasi principles. in this case the students have the understanding of the concept of/principles that are stored in the scheme of eleminasi good in the system memory although the name of the principle of ("eleminasi") is not yet he know, as well as the principle of substitution of oun have students understand with both in the system memory. conclusion based on explanation of research results in the conclusions of this research is 1a) the activity of the students in the learning process with the implementation of the model of teaching mathematics realistic is effective. 1b) the ability of understanding the concept and troubleshooting class experiment the given learning with pmr is better than with the ability to understanding the concept and troubleshooting control classes assigned to conventional mathematics lesson. 2) the index score gains the ability to understanding the concept is the gpk = 0,755, nscore gains the ability to understanding the concept is categorized as high. means that the ability of understanding the concept of mathematics smp students n 10 bengkulu city after being taught by applying the model of teaching mathematics realistic rp 13,000, rp 8,000, rp 5.000, herawaty & rusdi, increased capacity of the understanding … 120 based on cognitive conflict students have increased high. 3) the index score gains troubleshooting capabilities (gpm) = 0,500, then n-score gains troubleshooting capabilities are categorized as being. means that the ability to solve mathematics problems smp students bengkulu city after being taught by applying the model of teaching mathematics realistic based on cognitive conflict students have increased high enough. references al-arief, m.a. & suyono. 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(2013). beberapa dekomposisi genetik siswa dalam pembelajaran matematika. artikel dimuat dalam prosiding seminar nasional bimbingan dan konseling fkip universitas bengkulu. http://www.physics.indiana.du/ sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p1-16 1 students’ reflective abstraction ability on linear algebra problem solving and relationship with prerequisite knowledge rahayu kariadinata universitas islam negeri sunan gunung djati bandung, indonesia article info abstract article history: received aug 3, 2020 revised oct 18, 2020 accepted nov 9, 2020 this study aims to describe the achievement of the ability of students' reflective abstraction in solving linear algebra problems and the relationship with prerequisite knowledge. the important of this research because the characteristic of linear algebra requiring reflectif abstraction skill that must be support by the prerequisite knowledge. the reflective abstraction abilities studied in this study are level, i.e.1) recognition,2) representation, 3) structural abstraction, and 4) structural awareness. these stages are adjusted to polya's problem solving stages, namely: understanding the problem, devising a plan, carrying out the plan, and looking back. this type of research is descriptivequantitative. the subjects of this study were students of the mathematics education study program, faculty of tarbiyah and teacher training of uin sunan gunung djati bandung indonesia. collecting data through tests and interviews, data were analyzed with percentage and the pearson productmoment correlation.the results showed that the achievement level consisiting of ) recognition,2) representation, 3) structural abstraction, and 4) structural awareness of the students’ reflective abstraction abilities on linear algebra problem solving are very good, this can be seen from the percentage achieved at stages of the recognition,the representation,the structural abstraction, and the structural awareness which is associated with polya problem solving measures above an average of 73,31% (moderat category); there are relationship between students' reflective abstraction abilities and their prerequisite knowledge; and prerequisite knowledge influences the students’reflective abstraction abilities. keywords: prerequisite knowledge, problem solving, reflective abstraction copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: rahayu kariadinata, mathematics education study program, faculty of tarbiyah and teacher training, universitas islam negeri sunan gunung djati bandung jl. a.h. nasution no. 105a, bandung, jawa barat 40294, indonesia email: rahayu.kariadinata@uinsgd.ac.id how to cite: kariadinata, r. (2021). students’ reflective abstraction ability on linear algebra problem solving and relationship with prerequisite knowledge. infinity, 10(1), 1-16. 1. introduction linear algebra is a compulsory subject that is presented at the mathematics education study program uin sunan gunung djati bandung with the aim that students can master and be skilled in applying linear algebra concepts for solving problems in daily life, in various fields both in mathematics and scienceother sciences through linear equation https://doi.org/10.22460/infinity.v10i1.p1-16 kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 2 models, for example, in the field of information technology, linear algebra has become the foundation for algorithms, the application of vectors in the computer field is very clear in graphic design application programs, and in the economic and social fields through the matrix, linear algebra is also useful in terms of optimization. linear algebra represents, with calculus, the two main mathematical subjects taught in science universities (dorier, 2002) where linear algebra has been traditionally taught in a very theoretical way, this teaching tends now to be more orientated towards numerical computations, even if the structural approach remains important for the minority of students majoring in mathematics (dorier & sierpinska, 2001). linear algebra is a field of mathematical study that studies systems of linear equations and solutions, special properties of vector spaces and lineari transformations (kariadinata, 2019). the importance of the role of linear algebra above, students should be able to understand the concept of linear algebra adequately so that good learning outcomes are obtained, but in reality, based on the experience of teaching linear algebra courses and interviews with a number of students, it was found that most of the students felt difficulties in solving linear algebra problems. this is due to a change in students' thinking from elementary mathematical thinking to advanced mathematical. on one level, students' difficulties with linear algebra stem simply from their inexperience with proofs and proofbased theories. indeed, students' proofrelated difficulties (hillel, 2000). students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder (stewart & thomas, 2010). another fact is that mathematics education students tend to have a lack of confidence in communicating mathematical ideas, not being able to provide appropriate arguments in accordance with facts, principles and mathematical procedures, difficulties in proving theorems, reasons for fear of being wrong or not mastering the material properly the ability of symbol sense and student structure sense is still low because of a lack of conceptual knowledge and algebraic manipulation (kariadinata et al., 2017; sugilar et al., 2019). mathematics in higher education has characteristics towards a formal framework of axiomatic and verification systems, so students must have advanced mathematical thinking abilities. which is an ability that can train students to construct and create their own mathematical definition images. advanced mathematical thinking includes the ability of representation, abstraction, creative thinking, and mathematical proof. the main factor that becomes a barrier for students is a lack of understanding of rules, definitions, procedures, and concepts that are very abstract. panjaitan (2009) one of the characteristics of mathematics is to have abstract study objects, so that abstraction becomes an important part that cannot be separated from mathematics. in these conditions, students need the ability of abstraction which is an important part of learning mathematics. abstraction is an activity which is a mental process in forming a mathematical concept that involves relationships between mathematical structures or objects (wiryanto, 2014). the results of a study conducted by tall (1994) conclude that mathematical concept abstractions basically use abstractions that focus on objects and abstractions that focus on object operations. examples of abstractions that focus on objects are geometry, and examples of abstractions that focus on object operations are arithmetic and algebra. mitchelmore and white (2004) provide examples of abstract objects in mathematics in schools, for example on the topic rate of change in calculus. the most basic idea in calculus is the rate of change, leading to differential. this is a major reform movement in the last ten years related to meaningful ideas by initially exploring a series of rates of change to realistic situations. in this way students build intuitive ideas of rate of change before studying the topic abstractly. mathematical objects are constructed through the formation of relationships in such a way that they find new generalizations, evidence, or strategies for volume 10, no 1, february 2021, pp. 1-16 3 problem solving (schwarz et al., 2010). while, tall (1994) argues that abstraction is the process of drawing certain situations into concepts. which can be thought of (thinkable concept) through a construction. these concepts that can be thought of can then be used at a more complicated and complex level of thinking. there is a three-part theory of theory (tripartite theory) about abstraction proposed by piaget (gray & tall, 2007), namely: 1) empirical abstraction which focuses on the way students construct the meaning of objects, 2) pseudoempirical abstraction which focuses on the way students construct the meaning of the nature of the action on the object, 3) reflective abstraction that focuses on the idea of action and operations into thematic objects in thought or assimilation, which is related to the categorization of mental operations and abstraction of mental objects. reflective abstraction is an important mechanism that can explain how students build conceptual knowledge by giving reasons for decisions made (goodson-espy, 1998). reflective abstraction activities have a level as stated by cifarelli (1990), namely the first level is recognition, the second level is representation, the third level is structural abstraction, and the fourth level or the highest level is structural awareness. when students solve linear algebra problems, they must be aware of what is abstracted. here it needs to be seen whether students are able to express or demonstrate their awareness of problem solving activities, and provide reasons for decisions or conclusions obtained in problem solving (wiryanto, 2014). problem solving is as an attempt to find a way out of a difficulty, achieve a goal that is not easily achieved. the series of steps that students must take in solving mathematical problems as recommended by polya (1973), namely: (1) understanding the problem (this includes): a) what is known? b) what was asked? c) is the condition of the problem given sufficient or incomplete to look for what was asked ?; (2) devising a plan, (planning for completion) this includes: a) what theories can be used in this problem? b) should other elements be sought in order to take advantage of the problem or state it in another form? (3) carrying out the plan (solving the problem) this includes: a) implementing the settlement by checking each step whether it is correct or not?, b). prove that the chosen step is correct; and (4) looking back in this section emphasizes how to check the answers that have been obtained. the ability to solve the linear algebra problems associated with levels in reflective abstraction requires adequate prerequisite knowledge. this is because the characteristics of mathematical concepts that are hierarchical, meaning to be able to understand a particular concept must understand the previous concept. based on the experience of researchers teaching linear algebra courses, students have difficulty in certain concepts that require prerequisite knowledge. some concepts in linear algebra require prerequisite knowledge, namely in finding solutions to the set of systems of linear equations using elementary line operations. in the matter of linear independence, to prove whether a vector set is a set that is linear independence or linearly dependent, students must have knowledge of the homogeneous linear equations system and their solutions. the prerequisite knowledge has been given. gagne and briggs (riyanto, 2014) state that in the learning process there is a very important component, namely learning-prerequite sequences which are interpreted as a sequence of learning prerequisites or learning hierarchy, so that the presentation of certain material will not be carried out if the material which is a prerequisite has not been presented. some research results on prerequisite knowledge reveal that mastery of basic materials as a prerequisite in learning mathematics is still very low. research indicates that prior to learning algebra, students must have an understanding of numbers, ratios, proportions, the order of operations, equality, algebraic symbolism (including letter usage), algebraic equations and functions (welder, 2006). kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 4 likewise, various theories state that the importance of prerequisite knowledge will be linked to new knowledge, including ausubel et al., (1968) learning theory which is famous for meaningful learning. according to this theory, learning is said to be meaningful if the information students will learn is arranged in accordance with the cognitive structure (can be in the form of facts, concepts or generalizations that have been obtained or even understood beforehand by students) that students have so that students can associate information new with the cognitive structure it has. so in learning mathematics students must have prerequisite knowledge inherent in their minds. as stated by nur (2000) that prior knowledge is a compulsory ability possessed by students in the learning process. in this study the prerequisite knowledge that was studied was knowledge of the matrix algebra concept. some concepts in linear algebra require knowledge of the concept of matrix algebra. as the results of previous research conducted suryaningsih (2017), regarding the correlation of learning outcomes in elementary linear algebra for students of the mathematics education study program, fkip university of lambung mangkurat based on prerequisite subjects, conluded that there is a positive and significant correlation between learning outcomes in the matrix prerequisite course and learning outcomes for elementary linear algebra. other relevant research conducted mufidah et al. (2019) conducted a study on the analysis of understanding the concept of algebra in the elementary linear algebra course, concluding that the ability to understand students majoring in mathematics education at uin alauddin makassar class of 2016 in solving elementary linear algebra problems in terms of ability indicators mathematical understanding, students are quite capable of indicators: (1) understanding is able to restate a concept, (2) presenting a concept in various forms of mathematical representation, (3) using, utilizing and selecting certain procedures, and (4) applying concepts / algorithms to problem solving. however, students are classified as incapable of the ability to give examples and not examples. the difference between this study and the studies mentioned above is in the domain of thinking. learning outcomes and understanding can be seen separately with the ability of reflective abstraction which is a useful tool for learning high-level mathematical thinking, which will give rise to a theoretical basis that supports and contributes to teachers' understanding of thinking and how teachers can help students improve their abilities. reflective abstraction is a concept introduced by piaget to explain the construction of a person's mathematical logic structure in cognitive development when studying a concept. the result of reflective abstraction is a schema (mental structure) of knowledge at each stage of development and reflective abstraction brings together a corresponding scheme of action patterns (fuady & rahardjo, 2019). mastery of the concept of matrix algebra is one of the skills or skills expected to be achieved in linear algebra lectures through the ability of students to make connections between concepts, application of concepts and algorithms flexibly, accurately, efficiently, and precisely in problem solving. a study of the importance of learning algebra concludes that misunderstanding of concepts in algebra material will have an impact on other material and will certainly lead to learning difficulties that lead to low learning outcomes (irfan & anzora, 2017). this study aims to analyze the ability of students' reflective abstraction on linear algebra problem solving in basis material and the relation of prerequisite knowledge. 2. method this research is a descriptive study with a quantitative approach, which is a study that aims to explain a phenomenon by using meaningful numbers based on processing to draw conclusions. the method used in this research is the correlational method. the correlational method is one of the quantitative research methods based on the positivism volume 10, no 1, february 2021, pp. 1-16 5 philosophy which emphasizes objective phenomena and is studied quantitatively. this study will describe the ability of students' reflective abstraction in linear algebra problem solving and their relationship to prerequisite knowledge. the subjects of this study were 31 students in 3th semester of 2019/2020 academic year who were taking linear algebra in the mathematics education study program, faculty of tarbiyah and teacher training of uin sunan gunung djati bandung. data collection is done by providing linear algebra test on basis material. the results of student work were analyzed and then interviews were conducted. this is to reveal the description of students' abilities about the concept of basis through the levels of reflective abstraction according to cifarelli (1990), namely the level of recognition, representation, structural abstraction and structural awareness and problem-solving abilities based on (polya, 1973). the results of written and verbal answers (obtained during interviews) are then reviewed based on the reflective abstraction descriptor and problem solving. the instruments used in this study were the linear algebra problem solving test and interview guides. respondents who were interviewed were as many as research subjects, namely 31 students, this was based on considerations in order to obtain data that could support quantitative data. the analysis in this study emphasizes the results of linear algebra written tests and interview results. to find out the level of achievement of the reflective abstraction ability of students in linear algebra problem solving, percentage analysis is used. the five levels of students' reflective abstraction ability attainment are presented in table 1. table 1. criteria for achievement of students' reflective abstraction ability as a percentage (%) no level of achievement category of achievement 1 86 – 100% very high 2 74 – 85.99% high 3 62 – 73.99% moderat 4 50 – 61.99% low 5  49.99 % very low the relationship between variables in this study is illustrated as the following figure 1. figure 1. chart of research thinking framework whereas to determine the effect of the prerequisite ability on the reflective abstraction ability of students in solving linear algebra problems a product moment very high high moderat students’ process of linear algebra problem solving (polya, 1973) and reflective abstraction ability (cifarelli, 1990) level of achievement of students’ reflective abstraction ability low very low prerequisite knowledge relationship kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 6 correlation test was used. based on the definition of reflective abstraction and linear algebra problem solving for the basic concept, in this study descended descriptors as shown in table 2. table 2. problem solving descriptors based on polya and the level of reflective abstracts and their activities step of problem solving (polya, 1973) level of reflective abstraction (cifarelli, 1990) descriptor and activities understanding the problem recognition student read the problem given and organize the structure that must be solved representation student express the results of previous thoughts in the form of words; translating and transforming information or structure into mathematical models structural abstraction student develop a new strategy for a problem, which has not been used before and organizing the structure of mathematical problems structural awareness student give arguments or reasons correctly to the decisions made and reflects the decision obtained for the next activity. devising a plan recognition student recall previous activities related to the problem to be solved and identifying previous activities related to the current problem representation student express the results of previous thought in the form of mathematical symbols, words, graphs to help reconstruction; translating and transforming information into mathematical models structural abstraction student develop a new strategy for a problem and organize the structure of mathematical problems such as organizing, organizing and developing carrying out the plan recognition students can use alternative solutions and develop other strategies for problem solving representation students can plan correctly, represent mathematical models and solve problems. structural abstraction students can develop a new strategy for a problem, which has not been used before and reorganize the structure of mathematical problems structural awareness students can give arguments or reasons correctly to the decisions made and summarize their activities correctly during problem solving and connected in a structured way looking back recognition students can identify or recall the results obtained representation students can see the plan has made and represent mathematical models to solve the problems structural abstraction students can examine new strategies for a problem, which had not been used before and examine the structure of mathematical problems such as the preparation and organization of a problem modified from (wiryanto, 2014) volume 10, no 1, february 2021, pp. 1-16 7 3. results and discussion 3.1. results the results of this study will describe the achievement of the students’ reflective abstraction abilities on linear algebra problem solving obtained from test and interview results and the relationship between the reflective abstraction ability with prerequisite knowledge. following is the linear algebra test, basis material. let, v1 = (2,1,-1), v2 = (-1,5,1) dan v3 = (2,1,3). find out whether the set s = {v1, v2, v3} is the basis for r3 ? the following summarizes the results of tests and interviews about the achievement of students' reflective abstraction ability on linear algebra problem solving, basis material adjusted to polya's problem solving steps (polya, 1973) and the level of reflective abstraction from (cifarelli, 1990). in the step of problem solving polya (1973) namely understanding the problem on level of reflection abtraction cifarelli (1990) namely recognition, through interview 86.45% students understand the problem through reading and reorganize the structure of the problem by writing a basic definition that is write it : “ if v is any vector space and s = { v1, v2,…., vr } is a finite set of vectors in v, then s is called a basis for v if: s is linearly independent and s span v ; level representation, through interview and written test 30.22% and 78.89 % student can identify existing vectors then connect with one of the statements of the theorem : “ s is a set with two or more vectors is linearly independent if and only if at least one of the vectors is a linear combination of the remaining vectors” ; level structural abstraction, through interview and written test, 74.36% and 87.98% student understand that to prove s basis for v, it must be filled that s linearly independent and s span v; level structural awareness, through written test, 89.98% student can prove that s is linearly independent and span v can jointly prove it by showing that the coefficient matrix of a linear equations system that is formed from the description of linearly independent and stretching is a coefficient matrix, then the determinant value will be sought. in the step of problem solving polya (1973) namely devising a plan on level of reflection abtraction cifarelli (1990) namely recognition through written test, 93.54% students can write the term of form the equality of linearly independent and span v : a) k1 v1 + k2 v2 + k3 v3 = 0 b) z = k1 v1 + k2 v2 + k3 v3 ; level representation through written test, 85.67% students can represent the form of equations from: a) linearly independent : k1 v1 + k2 v2 + k3 v3 = 0  k1 (2,1,-1) + k2 (-1,5,1) + k3 (2,1,3) = (0,0,0)  (2k1, k1,k1) + (-k2, 5k2, k2) + (2k3, k3, 3 k3) = (0,0,0)  (2k1 k2 +2 k3 , k1 + 5k2 + k3 , -k1 + k2 +3 k3 ) = (0,0,0) b) span v : z = k1 v1 + k2 v2 + k3 v3  (z1, z2, z3) = k1 (2,1,-1) + k2 (-1,5,1) + k3 (2,1,3)  (z1, z2, z3) = (2k1, k1,k1) + (-k2, 5k2, k2) + (2k3, k3, 3 k3)  (z1, z2, z3) = (2k1 k2 +2 k3 , k1 + 5k2 + k3, -k1 + k2 +3 k3) level structural abstraction through written test, 85.69% students can express a system that matches the equation: k1 v1 + k2 v2 + k3 v3 = 0 that is : 2k1 k2 +2k3 = 0 k1 +5k2+ k3 3 = 0 ... (*) -k1 + k2 +3 k3 = 0 kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 8 states the system that matches the equation: z = k1 v1 + k2 v2 + k3 v3, that is 2k1 k2 +2 k3 = z1 k1 + 5k2 + k3 = z2 ..... (**) -k1 + k2 +3 k3 = z3 students can prove that s is linearly independent and span v simultaneously by showing that the coefficient matrix of the system (*) and (**) in the step of problem solving polya (1973) namely carrying out the plan on level of reflection abtraction cifarelli (1990) namely recognition through interview, 81.20% students can solve problems according to plan, use alternative solutions and develop other strategies to get a solution; level representation through written test, 85.69% students can plan correctly and represent mathematical models and solve the problems. representation of coefficient matrices of linear system form (*) and (**) a =           − − 311 151 212 , level structural abstraction through written test, 75.69% students can develop new strategies for a problem, which before they have not used, reorganize the structure of mathematical problems in the form of organizing, organizing and developing, based on the formed coefficient matrix, students look for the determinant value, that is: det (a) = 311 151 212 − − = 44 states the terms s linearly independent and s span v are associated with the determinant value through the theorem which states: “if a is an n x n matrix, then the following statements are equivalent. a) a is invertible b) ax=0 has only the trivial solution c) a is row equivalent to in d) ax=b is consistentfor every x x1 matrix b level structural awareness through written test, 45.69%, students can give arguments or reasons correctly to the decisions made, and are able to summarize their activities correctly during problem solving and connected in a structured way, can associate the determinant values obtained with the theorem above. the value of det (a) = 44, this value is not equal to zero (≠ 0), then the matrix can be reversed, in a homogeneous linear equation system if the coefficient matrix has a determinant value ≠ 0 then the system has a trivial solution, thus s is linearly independent, so too because the system is consistent, there are values of k1, k2 and k3, so s spans v. therefore, because both conditions have been fulfilled, s is the basis for v. in the step of problem solving polya (1973) namely looking back on level of reflection abtraction cifarelli (1990) namely recognition through interview, 76.56%, students can recall the results obtained: 1) the terms of linearly independent, the form of: k1 v1 + k2 v2 +k3 v3 = 0 have trivial solution, then: k1 = 0, k2 = 0, k3 = 0; 2) the terms of span v: there are any vectors in v for example z = (z1, z2, z3) can be expressed as a linear combination of z1, z2, z3; level representation through interview, 63.76% students have planned it correctly and look back at the form of mathematical models and problem solving. volume 10, no 1, february 2021, pp. 1-16 9 for linearly independent : if k1=0, k2 =0 and k3=0 are substituted into the equation:  k1 (2,1,-1) + k2 (-1,5,1) + k3 (2,1,3) = 0  0 (2,1,-1) + 0 (-1,5,1) + 0 (2,1,3) = (0,0,0)  (0,0,0) + (0,0,0) + (0,0,0) = (0,0,0) for span : based on the theorem that "a can be reversed and ax = b is consistent for each matrix size n x 1" so that there are values k1, k2 and k3 so that z = (z1, z2, z3) can be expressed as a linear combination of vectors v1 = (2.1, -1), v2 = (-1,5,1 ) and v3 = (2,1,3). level structural abstraction through interview, 34.31 % students can complete the terms of linearly independent and span by reducing the equations (*) and (**) one by one using elementary row operations to obtain a trivial solution, k1 = 0, k2 = 0, k3 = 0 and many solutions that are responsible (consistent) for values of k1, k2, k3 thus solving this basis problem in a different way, students can reorganize the structure by looking back to the trivial solution from a homogeneous linear equation system, if k1 = 0, k2 = 0, k3 = 0 and if there are other solutions (non-trivial) then the conditions for linearly independent are not met, so too an ax = b must be consistent , then the span requirements are met. the following description of the results of research on the achievement of the students' reflective abstraction on linear algebra problem solving on the basis material based on polya's problem solving step (polya, 1973) and the level of reflective abstraction (cifarelli, 1990). in the understanding of the problem (polya) step at the recognition level of 86.45% students can reorganize the structure and can remember the requirements of a basis for r3; at the representation level of 54.55% students can state the results of previous thinking in the form of words and relate the theorem to the completion of bases relating to linearly independent; at the structural abstaction level of 81.12% students can translate and transform information or structures into mathematical models; at the level of structural awareness of 89.98% of students can give arguments correctly to the decisions they make. this is supported by the results of the interview that students understand the problem through reading reorganize the structure of the problem by writing a basic definition. in the devising a plan (polya) step at the recognition level of 93.54% students can identify previous activities related to the problem to be solved; at the level of representation 85.67% of students can state the results of previous thinking in the form of mathematical symbols, words, graphics/form of equations to help reflection / reconstruction; at the structural abstaction level of 85.69% students can reorganize the structure of mathematical problems in the form of compiling, organizing and developing. in the step of carrying out the plan (polya) at the recognition level of 81.20% students can solve problems according to the plan and use and develop other strategies to get a solution; at the level of representation of 85.69% students have been able to represent in the form of mathematical models and solve the problem by writing the form of the coefficient matrix representation of the form of linear systems ; at the structural abstaction level of 75.69% students can develop new strategies for a problem, which have never been used before and reorganize the structure of mathematical problems in the form of compiling, organizing and developing, for example students can look for determinant values from the coefficient matrix that was written before, students can declare the requirements of s linearly kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 10 independent and s span v is associated with the determinant value through the theorem associated with it; at the level of structural awareness of 45.69% students can provide arguments or reasons correctly for the decisions made, and be able to summarize their activities correctly during problem solving and linked structurally, for example by knowing the determinant value is not equal to zero (≠ 0 ), students can relate to solving a homogeneous linear equation system, which has a trivial solution, so that s is linearly independent, so also because the system is consistent, then there are values k1, k2 and k3, so s spans v. in the looking back step (polya) at the recognition level of 76.56% students can identify or recall the results obtained, namely about the basic requirements namely s linearly independent and s span v; at the representation level of 63.76% students have examined the form of mathematical representation and examined its completion; at the structural abstaction level of 34.31% students can check the strategies used for a problem, namely strategies that simultaneously prove that s is linearly free and span r3 by showing the efficiency matrix is not equal to zero (≠ 0) based on the theorem related to it. based on the description above, in summary the average percentage of students' ability to achieve reflective abstraction in solving linear algebra problems in base material can be seen in table 3. table 3. average persentage of the students’ achievement of reflective abstraction ability on linear algebra on basis material step of problem solving (polya, 1973) level of reflective abstraction cifarelli (1990) persentage average persentage understanding the problem recognition 86.45 74.65 representation 1) 30.22 2) 78.89 structural abstraction 1) 87.98 2) 74.36 structural awareness 89.89 devising a plan recognition 93.54 88.30 representation 85.67 structural abstraction 85.69 carrying out the plan recognition 81.20 72.07 representation 85.69 structural abstraction 75.69 structural awareness 45.69 looking back recognition 76.56 58.21 representation 63.76 structural abstraction 34.31 overall average 73.31 based on table 3, it can be seen that the average percentage of students' reflective abstraction achievement on linear algebra problem solving in basis material is 73.31%, this is categorized as moderate. furthermore, the results of the descriptive statistical analysis are shown in table 4. volume 10, no 1, february 2021, pp. 1-16 11 tabel 4. descriptive statistics n minimum maximum mean std. deviation reflective abstraction ability 31 56 92 73.42 9.626 prerequisite knowledge 31 60 92 75.81 8.550 based on table 4 it can be seen that the average value of reflective abstration ability is 73.42 and the average value of prerequisite knowledge is 75.81. while the minimum value of reflective abstraction ability is 56 and for prerequisite knowledge is 60, while the maximum value of the two variables is the same, namely 92. furthermore, product moment correlation analysis is performed to see the relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge, the results of the analysis are shown in table 5. table 5. correlations prerequisite knowledge reflective abstraction ability reflective abstraction ability pearson correlation 0.688** 1 sig. (2-tailed) 0.000 n 31 31 prerequisite knowledge pearson correlation 1 0.688** sig. (2-tailed) 0.000 n 31 31 to examine the relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge, the following hypotheses are first made: h0 = there is no relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge h1 = there is a relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge decision: in the asymp column. sig = 0,000 this value is smaller than 0.05 then h0 is rejected and h1 is accepted, so it is concluded that there is a relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge. the magnitude of the correlation coefficient (relationship) is r = + 0.688, this shows a positive relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge and when viewed from the pearson correlation classification, then this value is in the range 0.60-0.799, i.e. strong category, so it can be concluded that there is a strong relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge. furthermore, to see how much the relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge, the coefficient of determination will be sought namely: d = (r)2 = (+ 0.688)2 = 0.47344 or 47.34%. this shows that 47.34% of the students' reflective abstraction on linear algebra problem solving kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 12 are influenced by their prerequisite knowledge, namely students' algebra matrix knowledge. while the remaining 52,66% is influenced by other factors not examined in this study. 3.2. discussion based on the results of research on the percentage of the students’ achievement of reflective abstraction ability on linear algebra on basis material at each polya step (polya, 1973) and the level of reflective abstraction from cifarelli (1990), the overall average was 73.31% (medium category), but on several aspects produced a variety of average presentations including some above 73.31% and some below 73.31%, for example, in the step of understanding the problem (polya) on representation level, only 30.22% of students who could construct the results of his thinking in the form of words. based on the results of interviews at the understanding the problem stage, as many as 86.45% students understand the problem through reading, and reorganize the structure of the problem by writing a basic definition, this shows that students understand the problems given. this relates to student learning experiences, especially in solving various problems, it requires adequate experience and learning environment among students, as the opinion of (dick et al., 2006) states that a student's task is to create a learning environment that is often termed "scenario problems" which reflects the existence of authentic or real learning experiences and can be applied in a situation. in the step of devising a plan (polya) at all levels of reflective abstraction, students show very good abilities. this can be seen from the average percentage of 88.30%, including students who can translate and transform information or structures into mathematical models and run alternative possible solution methods, and develop new strategies for a problem. this is supported by the results of the interview that students can solve problems according to plan, use alternative solutions and develop other strategies to get a solution. this indicates that students have been able to make an external representative related to the problem formula. in the transition from mathematical models to mathematical results students use their mathematical abilities. this is the fifth stage of the modeling stage (ferri, 2006). in the step of carrying out the plan (polya) at all levels, the students’ reflective abstraction show good ability, this can be seen from the average percentage of 72.07%, including students who have been able to solve problems according to plan, use alternatives and develop other strategies to get a solution . thus, students have been able to structure the logic in cognitive development. it can be said that students already have the ability of reflective abstraction, as stated by piaget reflective abstraction is a concept to explain the structure of one's mathematical logic in cognitive development when learning a concept (gray & tall, 2007). but in the looking back (polya) step, the average percentage at the level of reflective abstraction of students showed a low ability, this can be seen from the average percentage of 58.21%. based on the results of the interview most of the students could not develop a new strategy for a problem, which had not been used before (34.31%) and almost the majority of students did not carefully re-analyze every step of the completion that had been done. this is understandable because according to them they spend time when a careful reexamination of the results have been obtained. whereas the benefits obtained by looking back at what has been done, according polya (1985) that many benefits are obtained by taking the time to re-examine the work that has been done. doing this according to polya allows students to predict appropriate strategies that can be used to solve problems in the future. the things that are needed in the looking back step, namely: a) can students check the results?; b) can students check the argument ?; c) can students get different solutions?; d) can students see at a glance?; e) can students use the results or methods for some of these volume 10, no 1, february 2021, pp. 1-16 13 problems?. thus, it is clear that to solve the problems, students need the reflective abstraction ability, as goodson-espy argues (fuady & rahardjo, 2019) which says that the results of students' reflective mental abstraction are schemes used to understand things, find solutions, or solve problems. furthermore, in linear algebra problem solving on the basis material, students connect from a problem solving activity to the next problem solving, for example, in proving whether a vector set s = {v1, v2, ..., vn} is the basis for rn, students at least have to understand the basis requirements, then he will explore various problem solving activities about linearly independent and continue with problem solving activities about span of vector, but students who are creative and critical will solve them together proving that s is linearly independent and s span rn by showing the coefficient matrix of the two can be reversed (has an inverse) it is clear here that many students involve and connect problem solving activities, as piaget's opinion states that reflective abstraction is very important for higher mathematical logic thinking as occurs in logical thinking in students, students are able to solve new problems by using certain coordination of structures that have been built and reorganized (tall, 1991). furthermore, the students’ achievement of reflective abstraction ability on linear algebra on basis material that is inseparable from the prerequisite knowledge, namely matrix algebra in the material linear equation system and homogeneous linear equation system. the understanding ability of the completion with the homogeneous linear equation system will relate to the material linearly independent which is one of the requirements of a basis. based on the research results obtained that there is a positive relationship between the students' reflective abstraction on linear algebra problem solving with prerequisite knowledge, and the magnitude of the effect of prerequisite knowledge is 47.34% or in other words by 47.34% achievement of the students' reflective abstraction on linear algebra problem solving on basis material is influenced by its prerequisite knowledge. mastery of prerequisite knowledge is very important because it shows the readiness of students to take the next material lesson. advanced concepts are difficult to understand before understanding the previous concepts which are prerequisites. as according to understanding constructivism states that knowledge will be arranged or built up in the minds of students themselves as he seeks to organize his new experiences based on the cognitive framework that is already in his mind. bodner (1986) states, "knowledge is constructed as the learner strives to organize his or her experience in terms of pre-existing mental structures". thus, knowledge cannot be transferred from the teacher's brain to the student's brain. ausubel's meaningful learning theory states that meaningful learning will occur if students can associate new knowledge about a problem solving process that can be linked to prerequisite knowledge that students have learned. so according to ausubel's theory of meaningful learning a very important factor according to him is old knowledge where new knowledge will adjust. he suggested a meaningful learning that is a learning process where new knowledge or experience can be related to old knowledge (prerequisite knowledge) that is already mastered by students, or in other words, meaningful learning will only occur if there is other knowledge in the student's mind (cognitive structure) such that new experiences can be related to it. 4. conclusion based on the results obtained as follows that the percentage of the students’ achievement of reflective abstraction ability on linear algebra problem solving in basis material at : 1) the step of to the understanding the problem (polya) at each level of reflective kariadinata, students’ reflective abstraction ability on linear algebra problem solving … 14 abstraction is a high category; 2) the steps of to the devising a plan (polya) at each level of reflective abstraction of the high category; 3) the steps of the carrying out the plan (polya) at each level of the reflective abstraction of the moderat category ; 4) the steps of the looking back (polya) at each level of reflective abstraction are in the low category, but overall the percentage of the students’ achievement of reflective abstraction ability at the problem solving step (polya) at each level of the reflective abstraction are the moderate category; 5) there is a positive relationship between prerequisite knowledge and students 'reflection abstration abilities, this indicates that the prerequisite knowledge influences the students' reflection abstraction ability and is important in learning mathematics. based on the conclusions above, it is recommended that lecturers who teach algebra group courses always check students' prerequisite knowledge before teaching a material. this can be done by giving a prerequisite knowledge test or by looking at the grades that have been achieved on the material that is a prerequisite. the ability of reflective abstraction must continue to be developed among students so that they can focus more on ideas about actions and operations which are thematic objects of thought or assimilation, which are related to the categorization of mental operations and abstraction of mental objects. acknowledgments the author would like to thank for the support to the dean and vice deans, chair of the department of mathematics and natural sciences (mipa) and chair of the mathematics education study program, faculty of tarbiyah and teacher training, uin sunan gunung djati bandung, that has been support my research. references ausubel, d. p., novak, j. d., & hanesian, h. 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(1994). a versatile theory of visualisation and symbolisation in mathematics. invited plenary lecture at the ciaem conference, 1, 15-26. welder, r. m. (2006). prerequisite knowledge for the learning of algebra. conference on statistics, mathematics and related fields, honolulu, hawaii, 1-26. wiryanto, w. (2014). level-level abstraksi dalam pemecahan masalah matematika. jurnal pendidikan teknik elektro, 3(3), 569–578. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.377.6882&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.377.6882&rep=rep1&type=pdf https://d1wqtxts1xzle7.cloudfront.net/56703532/2006_welder_hicsmrf5_pp1642-1667.pdf?1527827833=&response-content-disposition=inline%3b+filename%3dprerequisite_knowledge_for_the_learning.pdf&expires=1605510679&signature=cmpth22me3u1oimwt2w99eksacwuh1me4oxdddmwpgqzxy9eodzrs75g-rkgjvicaimjs-jhm8u8hpnch2bvegm7nk~iqrgn374cvca6neuepjssxjixgul0aczywffqhw467w29zyxb1id3t06aaahkh2i8skx8f1zbbhptj5cwqdtgdq4sshsoqqtdlkk-360vx1ko71ely8hsm1yxqzgud5bladgdlrh5xdd5ydztim07j~w~esr7i9ty8hrqythkenx3-1b18wt5kdmietrmmc1hjpzajo48cf~yzobxf1dlcltbktv4cb1ibpcynm0gynbesmxewlblhnxydw__&key-pair-id=apkajlohf5ggslrbv4za https://d1wqtxts1xzle7.cloudfront.net/56703532/2006_welder_hicsmrf5_pp1642-1667.pdf?1527827833=&response-content-disposition=inline%3b+filename%3dprerequisite_knowledge_for_the_learning.pdf&expires=1605510679&signature=cmpth22me3u1oimwt2w99eksacwuh1me4oxdddmwpgqzxy9eodzrs75g-rkgjvicaimjs-jhm8u8hpnch2bvegm7nk~iqrgn374cvca6neuepjssxjixgul0aczywffqhw467w29zyxb1id3t06aaahkh2i8skx8f1zbbhptj5cwqdtgdq4sshsoqqtdlkk-360vx1ko71ely8hsm1yxqzgud5bladgdlrh5xdd5ydztim07j~w~esr7i9ty8hrqythkenx3-1b18wt5kdmietrmmc1hjpzajo48cf~yzobxf1dlcltbktv4cb1ibpcynm0gynbesmxewlblhnxydw__&key-pair-id=apkajlohf5ggslrbv4za infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.214 75 prospective teachers’ ability in mathematical problem-solving through reflective learning yunika lestaria ningsih 1 , rohana 2 1,2 department of mathematics education, pgri palembang university, south sumatera, indonesia 1 yunika@univpgri-palembang.ac.id, 2 rohana_pgri@yahoo.com received: may 30, 2016; accepted: august 22, 2016 abstract the research aims to determine the mathematical problem-solving ability of prospective teachers’ through reflective learning. reflective learning is a learning process that provides students the opportunity to examine and investigate the problems that is triggered by experience, analyzing of individual the experiences, and facilitate the learning of the experiences. these lessons are identified to improve mathematical ability students. by using a descriptive qualitative research. the subject of this study were students of mathematics education program in one of private universities in palembang, consisting of 34 students. this study was conducted in odd semester academic year of 2015/2016. the instruments in this study were mathematical problem-solving ability test, observation sheet, and interview guide. the data were analyzed descriptively. based on analysis of the data are found that the average mathematical problem-solving ability of students’ through reflective learning in good categories. keywords: reflective learning, mathematical problem-solving ability abstrak tujuan penelitian ini adalah untuk mengetahui kemampuan pemecahan masalah matematis mahasiswa calon guru melalui penerapan pembelajaran reflektif. pembelajaran reflektif adalah suatu proses pembelajaran yang memberikan kesempatan kepada pebelajar untuk menguji dan menyelidiki persoalan yang menarik perhatian yang dipicu oleh pengalaman, melakukan analisis atas pengalaman individual yang dialami dan memfasilitasi pembelajaran dari pengalaman tersebut. pembelajaran reflektif ini diidentifikasi dapat meningkatkan kemampuan matematis pebelajar. penelitian ini menggunakan metode penelitian deskriptif kualitatif. subjek penelitian ini adalah mahasiswa program studi pendidikan matematika fkip universitas pgri palembang semester genap tahun akademik 2015/2016 yang berjumlah 34 orang. data penelitian dikumpulkan melalui tes dan wawancara. data dianalisis secara deskriptif. berdasarkan hasil analisis data diketahui bahwa rata-rata kemampuan pemecahan masalah matematis mahasiswa setelah diterapkan pembelajaran reflektif termasuk dalam kategori baik. kata kunci: pembelajaran reflektif, kemampuan pemecahan masalah matematis how to cite: ningsih, y.l. & rohana (2016). prospective teachers’ ability in mathematical problem-solving through reflective learning. infinity, 5 (2), 75-82. ningsih & rohana, prospective teachers’ ability in mathematical problem-solving … 76 introduction according to the regulation of the minister of national education in indonesia number 20/2006, the purposes of mathematics learning are students able to: (1) have knowledge of mathematics (the concept, the relationship between concepts and algorithms), (2) use of reasoning, (3) solve the problems, (4) communicate ideas through symbols, tables, diagrams, or other media to clarify the situation or problem, and (5) have an attitude that appreciate the usefulness of mathematics. moreover, the purpose of learning mathematics is also formulated by the national council of teachers of mathematics (2000), namely that the students have the competence: (1) to solve the problem; (2) reasoning; (3) communication; (4) relate the idea; and (5) positive attitudes towards mathematics. based on explanation above is known that the problem-solving ability is very important in learning mathematics. arthur (2008) stated that problem-solving is a part of think. as part of the thinking, problem-solving exercises can improve high-level thinking ability that requires the modulation and control more over routine or basic skills. this opinion suggests that in problem-solving, the students not only require routine or basic skills, but also have a variety of other skills to manage all the thinking process in order to solve their problems. it means the students have control and sort ability all their knowledge, finally they can find the best way to solve it. the speed and accuracy in selecting and sorting relevant knowledge is crucial in problem-solving. as effendi's statement (2012) that through problem-solving ability, students can solve their various problems, both in the mathematical problems as well as problems in daily life. the mathematical problem-solving ability is not only important for students but also for prospective mathematics teachers. according to widjajanti (2010), a prospective mathematics teacher must know, understand, and can apply the process of mathematical problem-solving. because in the future they should guide the students to have ability of mathematical problem solving. but the facts, the learning process in college is too much emphasis on the doing aspect but less on the thinking aspect (fahinu, 2007). what is taught in the classroom more concerned to manipulative skilled. several results of research in indonesia that showed the low ability of mathematical problem-solving at the level of prospective teachers' of mathematics as proposed by widjajanti (2010), karlimah (2010), and prabawanto (2012).. therefore, as an effort to achieve the goal of learning mathematics, especially on improving mathematical problem-solving ability of students, researchers applied a reflective learning. according to insuasty and castillo (2010), the reflection should be a fundamental part for the development of teachers. it caused, teachers have an obligation to be able to evaluate and restructure capability of teaching in order to optimize the teaching-learning process. a reflective teacher too, must be able to be critical of his own teaching abilities so that students could get a dynamic learning experience, valuable and meaningful to their lives. furthermore zeichner and liston in radulescu (2013) stated that the concept of reflective learning as a means to develop the professional capabilities of teachers. it is caused by the concept of reflective learning consists of several processes, which generally aims to foster an attitude of exploration and investigation so as to raise the awareness of prospective teachers as well as being factors that affect the learning process of students. the application of this volume 5, no. 2, september 2016 pp 75-82 77 learning model can improve the ability of students’ mathematical thinking (lasmawati 2011; nainggolan, 2011; rohana, 2015). in the world of education, reflective learning has been developed by many education experts. so a lot of variety of reflective learning comes to us. one of reflective learning model is formulated by the international center for jesuit education (icaje) is ignatian pedagogical paradigm (sirajuddin, 2009). reflective learning that is based on the ignatian pedagogical paradigm has been applied to the jesuit schools worldwide (icaje, 1993). according drost (sirajuddin, 2009) concept of reflective thinking through reflective learning is the core of reflective learning. there are three major elements, namely experience, reflection (reflection), and action. figure 1. ignatian paradigm (icaje, 1993) to carry out these three elements, the supporting elements are required, pre-learning element (context) and post-learning element (evaluation). thus, the reflective learning includes five steps such as: 1) context; 2) experience; 3) (reflection); 4) action; and 5) evaluation. based on the explanation above, the purposes of this study is determine the mathematical problem-solving ability of prospective teacher through the reflective learning. the indicator of the students’ mathematical problem-solving abilitiy that were examined in this study include: (1) identify the adequacy of the data to solve the problem, (2) create a mathematical model and how to solve it, (3) select and implement strategies to solve mathematical problems and or outside mathematics , (4) explain and check the correctness of the answer. method the subjects of this study were students of mathematics education program of private universities in palembang, consisting of 34 students. this study was conducted in 4 th semester academic year of 2015/2016. the instruments in this study were mathematical problemsolving ability test, observation sheet, and interviewing guide. the test consisted of six questions the description has been prepared based on indicators of mathematical problemsolving ability and has been declared valid and reliable. the data of students’ mathematical problem-solving ability after their participation in reflective learning was collected. the data were analyzed descriptively. the students’ mathematical problem-solving ability is obtained by examining the test answer sheets according to the scoring rubric. then the data were analyzed descriptively to see the achievement of mathematical problem-solving ability of students in the lecture. on average the final value obtained is used to view the categories of mathematical problem-solving ability ningsih & rohana, prospective teachers’ ability in mathematical problem-solving … 78 of students. interviews were conducted orally to students with different levels of mathematical skills. interview data were analyzed descriptively and used as supporting data test mathematical problem-solving ability of students. results and discussion results the learning activities are carried out in class 4a by the number of students as many as 34 people, divided into 7 groups with heterogeneous mathematical skills. the research was conducted as many as five sessions, with one test at the last meeting. mathematical statistics 1 material studied in this research is limited on combinatorial analysis, probability and conditional probability. researchers applied reflective learning at every meeting. the reflective learning includes five steps are as follows: (1) context. lecturers presented the topic of learning as new knowledge that will be discussed, and then do the debriefing to check the prerequisite knowledge and skills possessed by students. lecturers are also directs students through the questions that triggered the students to relate their prior knowledge of the topic to be discussed. (2) experience. lecturer raises new issues related to the topic will be discussed and presented in the student worksheet (mfi). it is intended to stimulate the students understand their own thinking process. in this step, students examine the problems that arised, seeks to sharpen the problem and identify strategies for problem-solving, using these strategies to solve problems, and to determine the factors that allegedly led to the emergence of problems. (3) reflection. lecturer direct students to find a variety of information (collect data to support), formulated the layout and boundary issues, as well as the settlement of possible problems. lecturers take a part as the facilitator and mediator to provide scaffolding for groups in need through reflection questions. (4) action. lecturers provide opportunities for students to present their answers and sharing in class discussions. answers submitted student is not the result of an agreement because a student group may disagree with the group. furthermore, lecturers steer students to be able to implement the settlement proceeds obtained (newly acquired knowledge of students) in other situations. in this case, the lecturer can give new problems as the continued problems related to issues that have been resolved by the students, that students should always modify the understanding that has been incorporated (prior knowledge) in order to solve new problems. (5) evaluation. to determine individual student achievement to the topic that has been studied, lecturer evaluate by asking students to answer some questions. in addition, the lecturers also ask students to write a reflective journal as a training tool for students to be able to assess and monitor the success of the learning process. volume 5, no. 2, september 2016 pp 75-82 79 to obtain a picture of the quality of mathematical problem-solving ability of students, test data were analyzed descriptively. summary of test results of mathematical problem-solving ability of students is presented in table 1. table 1. the students’ mathematical problem-solving ability test result postest score frequency category 81 – 100 61 – 80 41 – 60 7 20 7 excellent good fairly from table 1, the result of mathematical problem-solving ability test students show the excellent categories equal to 20.59% (7 people), good categories equal to 50.82% (20 people), and fairly categories equal to 20.59% (7 people). while the test average score is 69.5 in good categories. the test results for each indicator mathematical problem-solving ability of students can be seen in table 2. table 2. the score of mathematical problem-solving ability for each indicator no. the indicator of students’ mathematical problem-solving ability average score category 1. identify the adequacy of the data to solve the problem 74,26 good 2. create a mathematical model and how to solve it. 66,91 good 3. select and implement strategies to solve mathematical problems and or outside mathematics 60,29 fairly 4. explain and check the correctness of the answer 70,59 good average 69,49 good discussion results of this study have shown that the average mathematical problem-solving ability of students through the application of reflective learning falls into good categories. these findings reinforce and complement the results of previous research on reflective learning, among other research conducted by nainggolan (2011), lasmanawati (2011), and rohana (2015) found that learning can improve the mathematical thinking ability. in each steps of reflective learning, students are given the opportunity to play an active role in the learning process and are involved in considering the success of their learning. for example, in step experience, students are faced with a problem based questions that they are working in groups. students are trained and familiarized reflective thinking mathematically through questions based on those problems. social interaction through group discussions such as asking each other, respond or criticize answers friend, giving students the opportunity to have a very big role in the effort to understand the concepts, develop procedures, found the principles and apply the concepts, procedures, and principles in solving a given problem , ningsih & rohana, prospective teachers’ ability in mathematical problem-solving … 80 in addition, according hmelo & ferrari (song, koszalka, and grabowski, 2005) lecturer's role as a facilitator in the reflective learning by giving instructions or scaffolding through questions reflections give to students to practice solving mathematical problems. the same thing also expressed wahyudin (2008) that teachers have an important role in helping to empower the development of habits of reflective thinking by asking questions like: "before we go, whether we believe already understand this?", "why we think this is true ? ". these questions make students tend to learn responsibility to reflect on their own work and make the adjustments necessary when solving problems. based on the test results of mathematical problem-solving ability, it is known that the lowest average scores of students with enough catagory located on the third indicator, the ability to select and implement strategies to solve mathematical problems and or outside mathematics. this is the example of questions and answers of students to measure theirs ability in third indicators. figure 2. the result anwers for the third indicator to solve the problem on this indicator, the student should be able to determine the number of ways or the structure menu that can be eaten by dina. students can use some solving dina is enjoying the original food in palembang city. the menus which are available in ‘wak aba’ store are pempek, model, kemplang, and many kind of drinking. pempek is including pempek telur, pempek kulit, pempek adaan, and pempek krupuk. model is including model ikan and model gandum. kemplang is consisting of fried kempang and baked kemplang. the drinking is consisting of peanut’s ice and cincau ice. determine in two different ways the illustration of dina’s choices to get all kind of the food and drink. volume 5, no. 2, september 2016 pp 75-82 81 strategies such as: a tree diagram, sequential pair, placement rules or the rules of multiplication. figure 2 shows that the students used a tree diagram and placement rules to address these problems. based on these answers, the third indicator has been reached. based on analysis of student work on this indicator, six students answered correctly and completely. students who answered correctly but incomplete on this indicator only write one problem-solving strategies. this mistakes made by students when answering after further explored through an oral interview is a student did not interpret the question correctly. in addition, students who incorrectly answered caused due to poor understanding of the concept of algebra, so the problem of combinatorial analysis could not be solved correctly. conclusion based on the results of this study, concluded that the average score of students’ mathematical problem-solving ability of students is fall into good category. from the analysis of problemsolving abilities per indicator, known to students classified as either on (1), (2) and (4), while quite indicator (3). thus a good problem solvers, students are able to identify the adequacy of the data to solve the problem, is able to create a mathematical model of a problem and explain or check the correctness of the answer, but students are not quite capable of selecting and implementing a strategy to resolve the problem. references arthur, l. b. (2008). problem-solving. u.s.: wikimedia foundation, inc. [online]. available: http://en.wikipedia.org/wiki/problemsolving. [7 th april 2008]. effendi, l. a. (2012). pembelajaran matematika dengan penemuan terbimbing untuk meningkatkan kemampuan representasi dan pemecahan masalah matematis siswa smp. jurnal penelitian pendidikan, 13(2), 1 – 10. fahinu. (2007). meningkatkan kemampuan berpikir kritis dan kemandirian belajar matematika pada mahasiswa melalui pembelajaran generatif. bandung: dissertasion in mathematics education, school of post graduate studies, upi. icaje, the international centre for jesuit education in rome. (1993). ignatian pedagogy: a practical approach. [online]. available: http://www.rockhurst. edu/media/filer_private/ uploads/ignatian_pedagogy_apractical_approach.pdf. [27 th desember 2013]. insuasty, e.a. dan castillo, l.c.z. (2010). exploring reflective teaching through informed journal keeping and blog group discussion in the teaching practicum. profile: issues in teachers` professional development vol.12 no.2, october 2010. issn 1657-0790. bogotá, columbia. pages 87-105. karlimah. (2010). pengembangan kemampuan komunikasi dan pemecahan masalah serta disposisi matematis mahasiswa pgsd melalui pbm. bandung: dissertasion in mathematics education, school of post graduate studies, upi. lasmanawati, a. (2011). pengaruh pembelajaran menggunakan pendekatan proses berpikir reflektif terhadap peningkatan kemampuan koneksi dan berpikir kritis matematis siswa. bandung: thesis in mathematics education, school of post graduate studies, upi. http://en.wikipedia.org/wiki/problemsolving http://www.scielo.org.co/scielo.php?script=sci_serial&pid=1657-0790&lng=en&nrm=iso http://www.scielo.org.co/scielo.php?script=sci_serial&pid=1657-0790&lng=en&nrm=iso ningsih & rohana, prospective teachers’ ability in mathematical problem-solving … 82 nainggolan, l. (2011). model pembelajaran reflektif untuk meningkatkan pemahaman konsep dan kemampuan komunikasi matematis. bandung: thesis in mathematics education, school of post graduate studies, upi. prabawanto, s. (2012). peningkatan kemampuan pemecahan masalah, komunikasi, dan self-efficacy matematis mahasiswa melalui pembelajaran dengan pendekatan metacognitive scaffolding. bandung: disertation in mathematics education, school of post graduate studies, upi. radulescu, c. (2013). reinventing reflective learning methods in teacher education. procedia social and behavioral sciences, 78, 11 – 15. rohana. (2015). peningkatan kemampuan penalaran matematis mahasiswa calon guru melalui pembelajaran reflektif. infinity, 4(1), 105 – 119. sirajuddin. (2009). model pembelajaran reflektif: suatu model belajar berbasis pengalaman. in didaktika jurnal kependidikan, 4(2), 189-200. song, h.d., koszalka, t. a., dan grabowski, b. (2005). exploring instructional design factors prompting reflective thinking in young adolescents. in canadian journal of learning and technology, 31(2), 49-68. wahyudin (2003). peranan problem-solving. makalah seminar technical cooperation project for development of mathematics and science for primary and secondary education in indonesia, 25 th agustus 2003. widjajanti, d., b. (2010). analisis implementasi strategi perkuliahan kolaboratif berbasis masalah dalam mengembangkan kemampuan pemecahan masalah, kemampuan komunikasi matematis, dan keyakinan terhadap pembelajaran matematika. bandung: disertation in mathematics education, school of post graduate studies, upi. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.231 21 a relationship between problem solving ability and students’ mathematical thinking nita delima department of mathematics education, subang university, indonesia nitadelima85@yahoo.com received: september 18, 2016; accepted: november 24, 2016 abstract this research have a purpose to know is there an influence of problem solving abilty to students mathematical thinking, and to know how strong problem solving ability affect students mathematical thinking. this research used descriptive quantitative method, which a population is all of students that taking discrete mathematics courses both in department of information systems and department of mathematics education. based on the results of data analysis showed that there are an influence of problem solving ability to students mathematical thinking either at department of mathematics education or at department of information systems. in this study, it was found that the influence of problem solving ability to students mathematical thinking which take place at mathematics education department is stonger than at information system department. this is because, at mathematics education department, problem-solving activities more often performed in courses than at department of information system. almost 75% of existing courses in department of mathematics education involve problem solving at the implementation of courses, meanwhile, in the department of information systems, there are only 10% of these courses. as a result, mathematics education department students’ are better trained in problem solving than information system department students. so, to improve students’ mathematical thinking, its would be better, at firstly enhance the problem solving ability. keywords: students’ mathematical thinking, problem solving ability abstrak penelitian ini bertujuan untuk mengetahui apakah terdapat hubungan yang signifikan antara kemampuan mathematical thinking dengan problem solving mahasiswa program studi sistem informasi dan untuk mengetahui seberapa besar kemampuan problem solving mempengaruhi kemampuan mathematical thinking mahasiswa. populasi dalam penelitian ini adalah seluruh mahasiswa yang mengontrak mata kuliah matematika diskrit pada program studi sistem informasi dan program studi pendidikan matematika. berdasarkan hasil analisis data diperoleh bahwa terdapat pengaruh yang signifikan antara kemampuan problem solving dan mathematical thinking baik pada mahasiswa prodi sistem informasi maupun pendidikan matematika. pada penelitian ini, ditemukan fakta bahwa pengaruh kemampuan problem solving terhadap mathematical thinking lebih besar terjadi pada mahasiswa prodi pendidikan matematika. ini disebabkan, karena kegiatan memecahkan masalah dalam perkuliahan mahasiswa prodi pendidikan matematika lebih sering dilakukan dibandingkan dengan mahasiswa prodi sistem informasi. hampir 75% mata kuliah yang ada dalam program studi pendidikan matematika melibatkan problem solving dalam tujuan mata kuliahnya, sementara itu, pada program studi sistem informasi, hanya terdapat 10% saja mata kuliah tersebut. akibatnya mahasiswa pendidikan matematika lebih terlatih kemampuan problem solvingnya dibandingkan dengan mahasiswa prodi sistem informasi. kata kunci: kemampuan mathematical thinking siswa, kemampuan problem solving delima, a relationship between problem solving ability and students’ … 22 how to cite: delima, n. (2017). a relationship between problem solving ability and students’ mathematical thinking. infinity, 6 (1), 21-28. introduction indonesia have two main goals on mathematics education, there is: (1) formal goals, which emphasize the arrangement of reasoning and personality development of the child and (2) material goals, that emphasizes the application of mathematics and problem solving abili ty in mathematics (soedjadi, 1999). when learning of mathematics are doing in the classroom, a teacher are often stuck with the second goals of mathematics education, that make some differences perspective of learning mathematics, i.e. viewing mathematics as a process or product. when mathematics is viewed as a product, then in learning activity,a teachers only teach how to doing arithmetic, student is only given some exercises which routine problem, it caused some students lost of interest in doing mathematics itself. it would be different if mathematics is viewed as a process, the teachers will provide a challenge to stimulate the curiosity of students through the given problem, assist them in completing a problem by providing some stimulation, students is directed to think like a mathematician, and of course it is expected can resulting the meaningfull mathematical learning. the mathematics education goals, requires a thinking process of mathematics. it can be a dynamic process that includes the complexity of an idea the idea of mathematical owned and can expand the understanding of mathematics. mason and johnston-wilder (breen & o’shea, 2010) says that when a mathematician solved a mathematical problem, their doing some processes and action, like : exemplifying, specializing, completing, deleting, corrected, comparing, sorting, organizing, changing, varying, reversing, altering, generalizing, conjecturing, explaining, justifying, verifying, convincing and refuting, and they called that as mathematical thinking. stacey (2006) conclude that the ability to think mathematically and to use mathematical thinking to solve a problems is an important goal of schooling. in this point, mathematical thinking ability will support science, technology, economic life and development in an economy. thus, a student needs to have the ability of mathematical thinking, to enable them mastering anothers materials course. nctm (2000) recomended that in teaching and learning mathematics, there are five standards process that must be fulfilled, i.e. problem solving, reasoning and proof, connections, communication, and representation. nctm mention that standards process problem solving ability includes the ability to: 1) build new knowledge through problem solving ; 2) solve problems that arise in mathematics and in other contexts; 3) apply and adapt a variety of appropriate strategies to solve problems; 4) monitor and reflect on the process of mathematical problem solving. because problem solving makes it possible to structure knowledge and to bring into connection with the other knowledge, costa and kallick (scusa, 2008), said that the problem solving ability can affects the flexibility of their thinking process. based on these analysis, the authors have hypothesis that the problem solving ability can affect the mathematical thinking ability. therefore, the authors need to do research to prove these hypothesis. volume 6, no. 1, february 2017 pp 21-28 23 method this study is a quantitative descriptive research that conducted at the subang university. a population in this study were all of students that taking discrete mathematics courses as many as 32 student of information systems department and 20 students of mathematics education department, then it is taken to be a random sample as many as 20 students from information systems department and 13 students from mathematics education department. an instruments in this study is a test of problem solving ability and mathematical thinking. an indicator of problem solving ability test that is compiled based on the standard process from nctm (2000), which include the ability to: 1) build new knowledge through problem-solving; 2) solve problems that arise in mathematics and in other contexts; 3) apply and adapt a variety of appropriate strategies to solve problems; 4) monitor and reflect on the process of mathematical problem solving. here is a problem of the tests that is used in this study: the test that is used to measure the mathematical thinking ability have some indicators that is given by mason, burton & stacey (2010), i.e. (1) specializing : trying special cases, looking at some examples; (2) generalizing: looking for patterns and relationships;(3) conjecturing : predicting relationships and results; (4) convincing: finding and communicating reasons why something is true. here is a problem of the mathematical thinking tests that is applied in this study: consider the following map : an explorer wants to explore a number of routes between cities on the territory described by the map above. can you find the routes that only through each road exactly once for him? (give a reason.) if you can, please state which route can be passed by the explorer. meanwhile, the travelers would like to visit a number of cities in the region of the map above. is there any route so that he can be visiting each city exactly once? if there are exist, please state which route can be passed by the travelers. o m n q p l k somebody invested 10 million on a bank with an annual interest of 10%. determine the amount of investment after the end of the 1 st year until 5 th year. if an stated the amount of investment after the end of the n th year, determine an. can you find a pattern to determine the amount of the investment at an? explain whether the pattern generally accepted? delima, a relationship between problem solving ability and students’ … 24 results and discussion results the objectives of this study is to determine whether there is significant influence problem solving ability of students to mathematical thinking and to determine how strong the problem solving ability can affect students’ mathematical thinking. the statistical analysis used in this study is the bivariate correlation analysis. the correlation coefficient is used to see the relationship between these two variables is the pearson correlation coefficient. as for the calculation of the correlation coefficient performed with spss 16.0, while the results can be seen in the table below: table 1. pearson correlation coefficient information system department mathematics education department mathematical thinking problem solving mathematical thinking problem solving mathematical thinking pearson correlation 1 .453 * 1 .736 ** sig. (2-tailed) .045 .004 n 20 20 13 13 problem solving pearson correlation .453 * 1 .736 ** 1 sig. (2-tailed) .045 the table above showed that the pearson correlation coefficients between problem solving ability and students mathematical thinking from department of information system is at 0.453 with a positive sign, while the pearson correlation coefficients between problem solving ability and students’ mathematical thinking from department of mathematics education is at 0.736 with a positive sign. based on the criteria on the table below, an interpretation of the result above, that the strengtness of the relationship between problem solving ability and mathematical thinking from information system student is on moderate level, while the strengtness of the relationship between problem solving ability and students’mathematical thinking from mathematics education students is on strong level. table 2. the criteria of correlation coefficients correlation coefficients interpretation 0.000 – 0.019 very weak 0.200 – 0.399 weak 0.400 – 0.599 moderate volume 6, no. 1, february 2017 pp 21-28 25 correlation coefficients interpretation 0.600 – 0.799 strong 0.800 – 1.000 very strong (sugiyono, 2009: 184) to determine whether there is a significant influence of the problem solving ability to mathematical thinking, then it is taken a testing of statistical hypothesis as follows : and a tests carried out using spss 16.0, with the testing criteria as follows: is rejected if sig. (2-tailed) < 0.05, whereas is accepted if sig. (2-tailed) 0.05 in table 1 above, it is showed that a sig. (2-tailed) value of the pearson correlation coefficient from student in the department of information system is equal to 0.045, and based on the testing criteria is rejected. there are a significant correlation between problem solving ability and mathematical thinking in the department of information system. meanwhile, the sig. (2-tailed) value of the pearson correlation coefficient from department of mathematics education is for 0.004, and based on the testing criteria is rejected. in the other word, we says that there are a significant correlation between problem solving ability and mathematical thinking in department of mathematics education. then to see a degree of strengtness of the relationship between the two variables or to find out how strong problem solving ability can contribute to the students’ mathematical thinking, the author use the coefficient of determination to describe it. the coefficient of determination can be calculated using the formula: (sugiyono, 2009) as for the guidelines to provide interpretation of kd values, it can be seen in the following table: table 3. criteria of kd values kd values interpretation 0% 19.9% very weak 20% 39.9% weak 40% 59.9% moderate 60% 79.9% strong 80% 100% very strong (sugiyono, 2009) based on the calculations, kd between problem solving ability and mathematical thinking from department of information system student is 20.5%. this means that the problem solving ability contributed to the mathematical thinking of students as much as 20.5% , while the rest is influenced by other factors that not examined by the authors. according to table 3, the delima, a relationship between problem solving ability and students’ … 26 interpretation of kd values, that is the problem solving ability influence the students’ mathematical thinking from department of information system at the low level. meanwhile, kd between problem solving ability and students’ mathematical thinking from department of mathematics education is 54.2%, this means that the problem solving ability contributed to the mathematical thinking of students as much as 54.2%, while the rest is influenced by other factors not examined by the authors. according to table 3, the interpretation of kd values, that is the problem solving ability influence the students’ mathematical thinking from department of mathematics education at the moderate level. discussion in line with arguement that is proposed by costa & kallick (scusa, 2008) and ersoy & guner (2015), that the problem solving ability of a person can affects the flexibility of their thinking processes, in this research was found the phenomenon that the problem solving ability of students, both in mathematics education department and information system department, can affect their mathematical thinking. the ability of mathematical thinking itself is a thinking process that is owned by a person who is doing activities of thinking either in mathematics or outside mathematics. its reinforced by watson’s (2001) opinion that, mathematical thinking is an ordinary ways of thinking and problem-solving which play a specially important part in mathematics. schorr (2000) also said that, problem solving ability is not just simply recalling algorithms, rules, or procedures, but it is the way the student thinks about problems. so, if student were engaged in problem-solving experiences, it can built their own mathematical thinking. in this study, it was found that the influence of the problem solving ability to the students’ mathematical thinking which is stronger, take place in students’ mathematics education department, its because, in this department, a problem solving activities more often performed than information system department. in line with stacey (2005) that, it would be better if problem solving is now more often treated as a teaching method. henningsen and stein (1997); ishida (1997) ; weber (2005) also said that in order to develop student capacities in mathemathical thinking ability, it must followed with classroom environments which student are able to engage actively in problem solving task. nearly 75% of existing courses in mathematics education department involve a problem solving at the courses, meanwhile, in the information systems department, there are only 10% of these courses. as a result, students’ problem solving ability is better in mathematics education than information system department. thus, it is clear that the influence of the problem solving ability to the students’ mathematical thinking is stronger in students’ mathematics education department than students of department of information systems. conclusion based on the results of data analysis is showed that, there are a significant correlation between problem solving ability and students’ mathematical thinking both in the student of information system department and mathematics education department. at the information systems department, problem solving ability contributed to the mathematical thinking of students as much as 20.5%, or in other words, the problem solving ability influence the students’ mathematical thinking at the low level. meanwhile, at the department of mathematics education, problem solving ability contributed to the mathematical thinking of students as much as 54.2%, or in other words the problem solving ability influence the students’ mathematical thinking at the moderate level. this is because, at mathematics volume 6, no. 1, february 2017 pp 21-28 27 education department, problem solving activities more often performed in courses than at department of information system. almost 75% of existing courses in department of mathematics education involve problem solving to the objective of courses, meanwhile, in the department of information systems, its just 10%. as a result, mathematics education department students is better in problem solving ability than information system department students. so, to improve students’ mathematical thinking, its would be better, to enhance the problem solving ability at first. references breen, s., & o'shea, a. (2010). mathematical thinking and task design. irish mathematical society bulletin(66), 39-49. ersoy, e., & guner, p. (2015). the place of problem solving and mathematical thinking in the mathematical teaching. the online journal of new horizons in educationjanuary, 5(1). henningsen, m., & stein, m. k. (1997). mathematical tasks and student cognition: classroom-based factors that support and iinhibit high-level mathematical thinking and reasoning. journal for research in mathematics education, 524-549. ishida, j. (1997). the teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process. school science and mathematics, 97(3), 155-162. mason, j., burton, l., & stacey, k. (2010). thinking mathematically secon edition. london: pearson education limited. nctm. (2000). principles and standards for school mathematics. reston, va: the national council of teachers of mathematics inc. schorr, r. y. (2000). impact at the student level: a study of the effects of a teacher development intervention on students' mathematical thinking. the journal of mathematical behavior, 19(2), 209-231. scussa, t. (2008). five processes of mathematical thinking. retrieved 2016, from http://digitalcommons.unl.edu/mathmidsummative/38 soedjadi, r. (1999). kiat pendidikan matematika di indonesia. jakarta: direktorat pendidikan tinggi departemen pendidikan nasional. stacey, k. (2005). the place of problem solving in contemporary mathematics curriculum documents. the journal of mathematical behavior, 24(3), 341-350. stacey, k. (2006). what is mathematical thinking and why is it important.progress report of the apec project: collaborative studies on innovations for teaching and learning mathematics in different cultures (ii)—lesson study focusing on mathematical thinking. sugiyono. (2009). metode penelitian bisnis (pendekatan kuantitatif, kualitatif, dan r & d). bandung: alfabeta. watson, a. (2001). instances of mathematical thinking among low attaining students in an ordinary secondary classroom. the journal of mathematical behavior, 20(4), 461475. delima, a relationship between problem solving ability and students’ … 28 weber, k. (2005). problem-solving, proving, and learning: the relationship between problem-solving processes and learning opportunities in the activity of proof construction. the journal of mathematical behavior, 24(3), 351-360. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.222 1 the effectiveness of guided discovery learning to teach integral calculus for the mathematics students of mathematics education widya dharma university yuliana 1 , tasari 2 , septiana wijayanti 3 1,2,3 department of mathematics education widya dharma university, klaten, indonesia 1 pakyulikids@unwidha.ac.id, 2 tasarilutfi@gmail.com, 3 septiana.wijaya@unwidha.ac.id received: october 7, 2016; accepted: november 8, 2016 abstract the objectives of this research are (1) to develop guided discovery learning in integral calculus subject; (2) to identify the effectiveness of guided discovery learning in improving the students’ understanding toward integral calculus subject. this research was quasy experimental research with the students of even semester in mathematics education widya dharma university as the sample. cluster random sampling was conducted to determine control group that was taught using conventional model and experimental group that was taught using guided discovery learning model. the instruments of this research included pre-test, post-test, and student’s response questionnaire. the data of post-test was analyzed using t-test. the result was h0 was rejected for the level of significance 5%.  the result of this data analysis found out that guide discovery learning was more effective than conventional model. it was supported by the result questionnaire. the result of questionnaire that more than 75% questionnaire items got 67.65% positive response. it means guided discovery learning can increase students’ interest in joining integral calculus class. keywords: guided discovery learning model, understanding toward integral calculus concept abstrak penelitian ini bertujuan untuk (1) mengembangkan model pembelajaran penemuan terbimbing pada materi kuliah kalkulus integral dan (2) mengetahui efektivitas model pembelajaran penemuan terbimbing dalam meningkatkan kemampuan pemahaman mahasiswa. penelitian ini berbentuk eksperimen semu dengan sampel mahasiswa semester genap program studi pendidikan matematika universitas widya dharma klaten. sampel penelitian diambil secara cluster random sampling diperoleh satu kelas kontrol dengan perlakuan konvensional dan satu kelas eksperimen dengan perlakuan model penemuan terbimbing. instrumen pada penelitian ini meliputi tes kemampuan awal (pre-test), tes kemampuan akhir (post-test), dan angket respon mahasiswa. data post-test dianalisis menggunakan uji t dengan hasil menolak h0 untuk taraf signifikasi 5%.  hal ini memberikan kesimpulan bahwa rata-rata hasil post-test pada kelompok dengan model penemuan terbimbing lebih baik daripada kelompok dengan model konvensional sehingga dapat disimpulkan model penemuan terbimbing lebih efektif daripada model konvensional. disamping itu, butir angket sekurangkurangnya 75% mendapat respon positif dari mahasiswa sebesar 67.75%. hal ini menunjukkan pembelajaran penemuan terbimbing mendapatkan respon positif. kata kunci: model pembelajaran penemuan terbimbing, pemahaman konsep kalkulus integral how to cite: yuliana, tasari & wijayanti, s. (2017). the effectiveness of guided discovery learning to teach integral calculus for the mathematics students of mathematics education widya dharma university. infinity, 6 (1), 1-10. mailto:pakyulikids@unwidha.ac.id mailto:tasarilutfi@gmail.com mailto:septiana.wijaya@unwidha.ac.id yuliana, tasari & wijayanti, the effectiveness of guided discovery learning … 2 introduction the introduction presents the purpose of the studies reported and their relationship to earlier work in the field. it should not be an extensive review of the literature. use only those references required to provide the most salient background to allow the readers to understand and evaluate the purpose and results of the present study without referring to previous publications on the topic. integral is one of compulsory subject that must be learnt in even semester at widya dharma university in klaten. the material is included in integral calculus subject. the understanding of integral calculus concept is highly needed for other subjects in the next semester even integral calculus subject is the a prerequisite subject to join some other subjects, such as advanced calculus, mathematical statistics, ordinary differencial equations, and parcial differencial equations. besides, integral material is learnt for one semester at senior high school. therefore, it is very important to master this subject especially for future mathematics teachers that are learning in mathematics education. the facts make the material crucial to teach to students in mathematics education. integral material is partly taught in natural science program at senior high school. the students from this program have better understanding than those from social science program or vocational school. meanwhile, most of mathematics students at widya dharma university semester 1 in mathematics education program in the academic year of 2013/2014 come from social program and vocational school. they have less under standing of integral concept since the material was not deeply studied in the senior high school. only some of them come from natural science program. the data are supported with the students result test in calculus ii subject in the academic year of 2013/2014 that shows 65.7895% of the students haven’t had well understanding on calculus material.this condition encourages calculus integral teacher to do research on it. besides using students analysis, the reshearcher did interview to the students dealing with their problem in learning integral calculus. based on the result of interview, the identified problems are 1) students’ paradigma, the students think that integral calculus is difficult subject and it is not easy to learn since they have to memorize many equations that they don’t know how it can be like that or the meaning beyond the equations; 2) the students find difficulties on how to accomplishe the exercise.they are confused about the steps they should do to meet the answer; 3) the students have less strategy in accomplishing the task. they tend to follow how their teacher does the task without reading any references that may help them to face other type of exercise. therefore, they can’t solve other exercise variety; 4) learning process focuses on teacher, the students tend to be passive. they learn from what the teacher says. they don’t have enough chance to share idea in learning process. as the consequence, they can’t gain deep understanding toward the material because they don’t directly involve in the process. they tend to be listeners. therefore, a learning model that can enhance students’ motivation and activeness in learning integral subject is highly needed. active learning is a solution for attractive dan active students’ learning activity. active learning is a learning model that actively involves students in learning process (silberman, 1996). students actively participate in learning process by trying and practicing what they are studying. in this case, teacher’ role is creating an atmosphere that optimally develops students’ capability by giving them chance to identify and relate concepts based on their learning experience. volume 6, no. 1, february 2017 pp 1-10 3 there are many learning methods under active learning that can be implemented to solve the problems. one of them is guided discovery learning. prince and felder (2006) say that guided discovery learning is an inductive learning model that is relevant with constructivism theory. guided discovery learning independently lets the students do experiments and draw conclusion, opinion, intuition. this model lets them do trial and error. teacher’ role is facilitator in which he will help the students in learning process when he is needed. the students are encouraged to find out idea, concept, and skill by themselves. the teacher guides them so their learning process will lead them to intended understanding. later, it will be used to gain understanding about the next new concept. how far the teacher guides the students, it depends on the level of material complexity. in its learning process, guided discovery learning focuses on instruction between teacher, students and learning material. based markaban (2008), the interaction is shown as follows. figure 1. learning activity in guided discovery learning interaction happens between teacher to a certain student, some students, or all students in a classroom (s – g), students to students (s – s), students to learning material (s – b), students to learning material to students (s – b – s), and students to learning material to teacher (s – b – t). the interaction causes the influence each other. teacher stimulates the students to think by giving some questions that will lead them in understanding and contruction new concepts. method the research was conducted at mathematics education, teacher training and education faculty, widya dharma university, klaten. the research was quasy experimental research because only some of relevant variables that were controlled or manipulated.the sample was the students of mathematics education in even semester who took integral calculus subject with the use of certain integral in accomplishing a problem as the material. cluster random sampling was implemented to find out a representative sample of the population. based on the result of cluster random sampling, the control group that was taught using conventional model consisted of 22 students meanwhile the experimental group that was taught using guided discovery learning consisted of 34 students. the techniques of collecting data were documention, questionnaire, and test. documentation was implemented to collect data that supported for problem identification in analysis phase. meanwhile, questionnaire was used to find out students’ positive response after joining the learning process and to identify students’ perception on learning material; learning reference; and learning method as well. the results of perception questionnaire was used as the basic of hypothesis in the research. the questionnaire consisted of one statement or with two closed yuliana, tasari & wijayanti, the effectiveness of guided discovery learning … 4 alternative answers. they were “yes” if the statement was suitable with the reality and “no” if the statement was not suitable. test was used to measure students’ understandimg toward the material. figure 2. the instrument of student’s response questionnaire the test was given to experimental and control group. the test was done twice. those are pre test and post test. pre test was used to identify students’ capability before treatment in experimental group and control group as well, whether they have homogeneus capability or not. post test was implemented to find out the effectiveness of the learning method. it was done after treatment. the both test was in essay. figure 3. the instrument of post-test items volume 6, no. 1, february 2017 pp 1-10 5 there are two kinds of data in this research. those are qualitative and quantitative data. qualitative data covers validation sheet of learning instruments such as lesson plan, learning material, and student response questionnaire. meanwhile, quantitative data was the students’ score in pre test and post test as well. moleong (2006), says that data analyzing must be started by learning all data from any resources. data resources of this research were validation sheet, student response questionnaire, and test result. the quantitative data of the research are pre test and post test score. pre test score was used for preliminary research while post test was used to identify the effectiveness of learning model. pre test was used to determine whether the samples were homogeneus or not. this treatment was done to ensure that it was the research which influenced the students’ progress. quantitative data analysis was done twice using t-test for two independent samples : experimental group and control group. the first analysis was done for prerequisite research while the second was used to find out the effectiveness of guided discovery learning. t-test was conducted after normality and homogeneity test had been completed. normality test was carried out to identify whether the samples came from normal distribution population or not. normality test used liliefors method (budiyono, 2009). homogeneity test was done to find out whether the population has the same variance or not. homogeneity test implemented bartlett method with chi-square as the statistics test. balancing test was done to identify that the group coming from guided discovery learning and group coming from conventional model have balanced average. the pre test score was the result of mid term test of even semester. post test score was taken from the result of students’ test after having treatment. statistical test used for these two scores analysis was ttest. results and discussion results this research developed guided discovery learning model. the implementation of this model covers in lesson plan, learning material by the researcher, pre test instrument, post test instrument, and questionnaire. experimental and control group have the same duration in learning. each of them has 8 meetings. before conducting the research, the two groups had been ensured that they had similar condition. it was measured through pre test that was analyzed using t-test. after that, they got treatment. experimental group was taught through guided discovery learning while control group was taught using conventional model. post test was conducted after all groups had got treatment. the result of post test then was analyzed using t-test which then the effectiveness learning model could be identified. yuliana, tasari & wijayanti, the effectiveness of guided discovery learning … 6 figure 4. students’ learning activity in guided discovery learning. learning instrument that was developed by the researcher was learning material for integral calculus. the instrument was validated by using expert judgement method. there were 3 validators in this research. the validators were trusted and qualified validators. they were the masters on their field, mathematics education, and had experience in research. the validation covered learning material, lesson plan, pre test items, post items, and positive respond questionnaire. the learning material of the research was matched with the regular material that should be learnt by the students on that semester. this research was in line with rosita (2016) that says teacher should provide and develop learning material that is suitable with the student’s characteristic and social environment. the material that was developed in this research was adapted with the characteristic of integral calculus material and the learning model that was developed. validation test was applied toward learning material to ensure that it was in line with its objective. based on the result of validation the learning material has already had all indicators as a ready learning material to use for integral calculus in experimental group. the pre test items and post test items had already been validated by the validators. the pre test items were 3 essays about indefinite integral and integral technique. meanwhile, the post test items were 3 essays about the implementation of definite integral to answer a matter about the area and volume of spinning object. those both test items were regarded as valid item test by the validators. lesson plan consisted of eight meetings. lesson plan needed to be validated the learning objectives, standard competency, and learning step. the validators stated that the lesson plas was valid. therefore, the lesson plan was ready to use in experimental class. questionnaire that was used in this research was positive response questionnaire.the questionnaire consisted of 16 statement items, with yes or no response. before the questionnaire was used, it had been stated as a valid questionnaire by the validators. so, the questionnaire was ready to use in this research. the pre test was carried out on friday, may 5 th 2016. the pre test result can be seen as follows on table 1. volume 6, no. 1, february 2017 pp 1-10 7 table 1. the students’ pre test result experiment group control group the number of students 34 students 22 students the highest score 30 38 the lowest score 100 82 mean score 64.382 55.591 standard deviation 17.861 12.89 the data above on table 1 then was analyzed using normality test and homegeneity test as prerequisite test before t test. the result of normality test is shown on the table 2. table 2. the normality test result of pre test group lobs dk decision conclusion experiment 0.10271 0.151948 h0 is accepted normal control 0.167764 0.190000 h0 is accepted normal based on the result of normality test, samples in experimental and control group are in normal distribution. the second prerequisite test was homogenity test. bartlett method was used in variance homogenity test. the result was 2 obs = 2.5140 dk  with 2 (0.05;1) 3.84145915  . the result described that h0 is accepted, it means the both variances are homogeneus. having tested its normality and homogeneity, the next test was t test. t-test was conducted to ensure that both groups, experimental and control had similar condition. the result of t-test was t 1.9942562 dk,  with  dk t t -2.306 or t 2.306 .   so, h0 is accepted. it means the students in experimental and control group have similar capability. post test was carried out after both groups had treatment. the post test was done on monday, july 4 th 2016. the result of post test is described on the table 3 as follows. table 3. the result of post test experiment group control group the number of students 34 students 22 students the highest score 30 22 the lowest score 75 65 mean score 60.5 46.455 standard deviation 8.302 11.673 the data then was analyzed through normality and homogeneity test as the prerequisite test before t test. lilifoers test was done to know the normality of the data. the normality test can be seen on table 4 as follows. yuliana, tasari & wijayanti, the effectiveness of guided discovery learning … 8 table 4. the result of normality test in post test group lobs dk decision conclusion experiment 0.146792 0.151948 h0 is accepted normal control 0.12122 0.190000 h0 is accepted normal the data shows that the samples in both experimental and control group are in normal distribution. the second prerequsuite test is variance homogenity test. bartlett method was implemented in the test. the result was 2 obs = 3.01771123 dk  with 2 (0.05;1) 3.84145915.  based on the test, h0 is accepted. it means the variance of the samples in both groups are homogeneus. after all prerequisites test had been fullfiled, the next step is identifying the effectiveness of the learning model for each group. t test was done to find out the conclusion. t test result was t 5.26355152 dk  , with  dk t t 2.0049  . the decision could be drawn was that h0 was accepted. the average of post test in experimental group is better than that in control group. so, it can be concluded that guided discovery learning model is more effective than conventional model to teach integral calculus. questionnaire was distributed to the students after they had learning process through guided discovery learning. the students’ response toward learning process was identified through the result of questionnaire. the result of questionnaire that was given to 34 students was the item of the questionnaire at least got 52.94% positive response. more than 75% questionnaire items got 67.65% positive response. it means guided discovery learning can increase students’ interest in joining integral calculus class. the increase of students’ interest can improve the students’ understanding toward integral calculus material. discussion based on the result of data analysis, the conclusion was appropriate with the writer’s hypothesis. it is guided discovery learning model is more effective than conventional model to improve students’ understanding toward integral. guided discovery learning model was proven more effective because in its learning process students were involved in finding the concept. therefore, the students got unforgatable and meaningful learning experience that helped them to understand the material in which there were many equations to understand and remember. the role of teacher was as facilitator. he guided the students to find out the concept that they should understand. the result of this research was in line with dumitrascu (2009). it was stated that guided discovery learning model is a learning model that is able to improve students’ understanding of a concept. in guided discovery learning class, modification was done by grouping the students into some group discussions. this technique was the same with what the researcher had done in 2011, yuliana (2011). this learning step of guided discovery learning attracted the enthusiasm of the students in joining the class. the students’ interest can be seen in the result of positive response questionnaire. moreover, the finding of this research was supported by the result of cohen’s research (2008). the result was guided discovery learning was a more enjoyable and effective learning model than direct learning. he concluded that the students taught by using direct learning is faster in fulfilling the test, but they missed some important volume 6, no. 1, february 2017 pp 1-10 9 points of tests. therefore, guided discovery learning was more recommended. the results support the reseacher’s finding that states guided discovery learning is effective to improve students’ concept of integral calculus subject. conclusion based on the result of data analysis, the conclusions are: (1) learning process in experimental group was in line with the lesson plan. the result of validation is regarded as a very valid category. (2) the students’ achievement on integral calculus subject who are taught using guided discovery learning is better than those who are taught using convention model. it can be identified from the result of post test. the result of t test was t 5.26355152 dk  , with  dk t t 2.0049 .  it shows that the average score of the students coming from experimental group was better than those coming from control group. it means guided discovery learning is better than conventional model to teach integral calculus material. moreover, the suggestions that can be drawn are: (1) mathematics teacher is suggested to implement guided discovery learning to teach integral for the students. (2) researcher is suggested to apply other active learning model that is able to develop students’ ability and motivate them in learning mathematics. acknowledgments the researcher expresses his deep gratitude to widya dharma university that has given permission and support to conduct this research. besides, thank you very much for dikti (the ministry of high education) that has given financial support so the research can be well accomplished. references budiyono (2009). statistik untuk penelitian (edisi 2). surakarta: uns press. cohen, m.t. (2008). the effect of direct instructions versus discovery learning on the understanding of science lessons by second grade students. journal of northeastern educational research association (nera), 30, 1-28. dumitrascu, d. (2009). integration of guided discovery in the teaching of real analysis. journal of educational studies in mathematics. 19(4).370-380. markaban (2008). model penemuan terbimbing pada pembelajaran matematika smk. paket fasilitasi pemberdayaan kkg/mgmp matematika. yogyakarta: p4tk matematika. moleong, l. j. (2006). metodologi penelitian kualitatif. bandung: pt. remaja rosdakarya. prince, m.j & felder, r.m. (2006). inductive teaching and learning methods: definitions, comparisons, and research bases. journal of engineering education, 95 (2), 123138. rosita, c.d. (2016). the development of courseware based on mathematical representations and arguments in number theory courses. infinity, 5 (2), 131-140. silberman, m.l.(1996). active learning 101 strategies to teach any subject. usa: allyn and bacon. yuliana, tasari & wijayanti, the effectiveness of guided discovery learning … 10 yuliana.(2011). eksperimentasi pembelajaran matematika dengan model kooperatif tipe student team achievement divisions (stad) dan penemuan terbimbing ditinjau dari aktivitas belajar siswa pada pokok bahasan persamaan garis lurus kelas viii di smp negeri se-kabupaten klaten tahun ajaran 2011/2012. in budiyono, mardiyana, i. sujadi, & sutopo (eds). seminar nasional dan pendidikan matematika (153-162). surakarta : pelangi press. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 1, february 2020 e issn 2460-9285 https://doi.org/10.22460/infinity.v9i1.p111-132 111 the role of constructivism-based learning in improving mathematical high order thinking skills of indonesian students ani minarni*1, e. elvis napitupulu2 1,2universitas negeri medan article info abstract article history: received nov 4, 2019 revised feb 10, 2020 accepted feb 16, 2020 to make students actively involved in learning to grasp mathematical higherorder thinking skills (mhots) is not easy. meanwhile, the ability is so important for students to master for it takes place when students continue their studies to a higher level as well as work within a variety of professions, especially in the era of the industrial revolution such nowadays. many factors affect students' thinking abilities, including learning factors. this study, which implemented constructivism-based learning, aims to investigate the role and contribution of constructivism-based learning approaches as well as mathematical prior knowledge (mpk) to the achievement of mhots of middle secondary school students. the data tested through multivariate analysis at the 0.05 significance level. in general, this study found that: (1) in the experimental class, the learning approach plays an important role in the way it increased students' mhots significantly. (2) the average contribution of constructivism-based learning to mhots was at the range of 18% to 57%. (3) student activity in learning increased significantly. (4) in some cases, there is an effect of interaction between learning factors and mpk towards the achievement of mhots. the study recommended the teachers to have courageous in implementing constructivism-based teaching and learning to i e de mhots. keywords: constructivism-based learning, mhots, mpk copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: ani minarni, departement of mathematics education, universitas negeri medan jl. william iskandar ps. v, kenangan baru, deli serdang, north sumatera 20221, indonesia. email: animinarni10@gmail.com how to cite: minarni, a., & napitupulu, e. e. (2020). the role of constructivism-based learning in improving mathematical high order thinking skills of indonesian students. infinity, 9(1), 111-132. 1. introduction parents keep some will for their children, for example they want their children to be useful people for themselves, family & society, and to serve their parents, country and religion. god has the will of his creatures; god wants his creatures to be on a straight path in terms of ways that can make his creatures feel peace in living. the teachers want their students to have thinking competencies, life skills and useful characters. the teacher wants students to have the habit of lifelong learning and succeed in learning according to what nctm document emphasized (nctm, 2000). http://dx.doi.org/10.22460/infinity.v6i1.234 mailto:animinarni10@gmail.com minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 112 many factors influence student achievement, but the point is there are two, namely factors within themselves and external factors (brown, 1990). what can be considered as internal factors include intellectual level (intelligence level), learning ability, learning motivation, learning independence (self-regulated learning), attitudes, feelings, interests, psychological conditions, and due to socio-culture. meanwhile, external factors include the attitude of parents and teachers to students, learning factors applied in schools, curriculum, school discipline, teachers themselves, learning facilities, grouping students, social systems, student social status, teacher and student interaction, political economy conditions, circumstances time and place or climate. by paying attention to a series of influential factors, one of the things teachers can try in schools is to improve learning outcomes through the implementation of appropriate learning. learning should be empowering students to think and construct their knowledge, arise students' interest in learning and make students understand the topics they learn. learning that has characteristics like this is constructivism-based learning (resnick, 1987), such as problem-based learning, discovery learning, cooperative learning (arends, 2012; ronis, 2008), even contextual learning and open-ended approaches, and mathematical realistic approach. the learning achievement will be even better if parents help encouraging their children to understand what is learned in school through often asking what they learned, whether they understand the subject matter learned today, or trigger the student to do homework. if students understand they learned and can construct their knowledge for themselves, they will have the opportunity to gain understanding skills at the hots (high order thinking skills) level, the level that reaches the ability to apply knowledge to solve problems. in other words, the understanding ability will bring up the ability of problem solving. on the other side, problem-solving ability will build up hots as well. hots always plays an important role from time to time, especially in the present era that has entered the era of industrial revolution 4.0. hots, moreover mhots, has proven to be the basis/foundation in the development of that era (formaggia, 2017). to achieve mhots is not enough just to rely on learning factors, but the mathematical prior knowledge (mpk) also needs to consider since its also holds an important role in problem solving process. this is because according to the results of the research, the mpk factor contributes to the achievement of mathematical problem solving abilities (minarni, 2017). reasoning ability, connection ability & mathematical representation, even the interactions between the fac ca a affec de ea i g c e . the ef e, i i i e e ting to investigate how is the role, especially the contribution, of constructivism-based learning approaches to the achievement of mathematical thinking skills of middle secondary school students. the findings of this study could be triggered the teachers to grasp the courage in implementing constructivism-based learning approaches that they considered difficult to implement. mathematical high order thinking skills (mhots) high order thinking skills (hots) is the concept of education reform based on the taxonomy of learning objectives from bloom and its revisions in marzano & kendall (2007). the idea is that some types of learning not only require higher-level thinking skills but also require ways to teach it differently from other types of learning so there is the term hots. in bloom's taxonomy, for example, skills that involve analysis, evaluation and synthesis (the creation of new knowledge) are considered abilities at the highest level that require learning and teaching methods different from learning or teaching methods that require students to volume 9, no 1, february 2020, pp. 111-132 113 master facts and concepts (anderson et al., 2001). whereas in marzano & kendall (2007) the level of high-order thinking is self-system & metacognition. hots generally consist of include critical, logical, reflective, metacognitive thinking, problem solving, and creative thinking. the ability to think has the potential to develop and increase when a person faces a problem that is not familiar to him, uncertain, raises a dilemma or invites questions. hots that runs successfully produces explanations, decisions, a series of performance, and products that are valid in the context of existing experience and knowledge and it fosters the growth of other intellectual skills in a sustainable manner. hots is rooted in the skills of simply applying and analyzing knowledge and cognitive strategies that intertwined with prior knowledge. the appropriate learning strategy and learning environment is a facility for the growth of hots along with the growth of accuracy, self-supervision, openness, and flexibility in students. this explaination is in line i h he he e a ed h hots i ea ed a d de e ed i de c g i i e structure (kulm, 1990). one kind of the hots is critical thinking that has many different definitions. scriven & paul (1987) stated that critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action. in its exemplary form, it is based on universal intellectual values that transcend subject matter divisions: clarity, accuracy, precision, consistency, relevance, sound evidence, good reasons, depth, breadth, and fairness. therefore, a good critical thinker generally needs to be able to both analyze and synthesize information. another definition of critical thinking comes from cottrell (2005) that stated critical thinking is a cognitive activity that involves mental processes such as attention, categorization, selection, and judgment. thus, based on the definition of critical thinking, it can be said that mathematical creative thinking is the abilities that require elements to test, question, connect, evaluate, all aspects that exist in mathematical problems. all of the areas of mathematics requires critical thinking, including algebra and geometry. algebra work requires analysis, which is the ability to break apart the pieces of a problem to solve while doing geometry involves more synthesis than analysis, in that we take all the elements of geometry and combine them to solve problems (do geometric proofs). the term of mathematical problem solving skills (mpss) referred to the definition of problem solving defined by anderson et al. (2001), which is the process of applying mathematical knowledge in new and unfamiliar situations (problems). in the process of i g a he a ica b e e i g h gh p a f e (p a, 2004) which include understanding the problem, device a plan, carry out the plan and looking back. understand the problem is the ability to represent the problem in any other form that makes one easier to attain the solution. understanding skills also enable one to demonstrate mathematical connection skills (the ability to connect among mathematical knowledge/ideas/ procedure/concepts) (nctm, 2000). the stage of looking back or reflection can be interpreted as draw conclusions for the solutions. in solving mathematical problems, one should have creativity. the ability to think creatively in math problem solving is the ability to solve mathematical problems flexibly, involving convergent thinking and divergent thinking. mathematical creative thinking enable one to make connections between problems under consideration, mathematical knowledge, variety of strategies for possible solutions, variety solutions, models and related questions, evaluate the problem solving process and not just at the end, communicate with peers, teachers, and other interested adults while working on the problem as well as following its solution (jensen, 1976). minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 114 torrance (1960) had studied for long time to reveal an understanding of students mathematical creative thinking skills. just like other skills, mathematical creative thinking can be learned through the teaching and learning process that required students to construct their knowledge via problem solving. in addition, sheffield (2013) stated that encouraging and supporting the development of mathematical creativity have the added benefit to increasing students enjoyment of and engagement in mathematical reasoning, sense making, problem solving, and problem posing. mathematical creative thinking skills is consisted the aspects of to think flexibly, original (proposed new idea), fluency, detailed and depth explanation to the solution. constructivism-based learning for a long time, the learning approach teachers used in indonesian schools is dominated by direct learning or direct instructions and demonstrations to resolve routine problems with communication tend to be one-way from teacher to student (minarni, napitupulu, & husein, 2016). this learning approach can indeed foster mathematical problem solving abilities but the problem that can be solved is the problem that exists in the textbook which is often not related to real life problems (silver, 2013), does not have the characteristics of a problem that serves to increase mathematical high order thinking skills (hots) such as problem solving (ronis, 2008). the learning approach aimed at reaching hots requires clarity of communication to avoid ambiguity and confusion and to increase students' positive attitudes towards tasks that require them to think and as a way out to address the diverse needs of students. this kind of learning needs scaffolding techniques, i.e., the support and assistance as needed in students at the beginning of their problem solving. scaffolding should gradually reduce until finally the students are left to work independently. excessive or too little scaffolding can hinder a student's development or progress in reaching hots. at present, learning that is expected to improve hots and mhots is constructivism-based learning because this learning carries the principle that knowledge is the result of human construction (widodo, 2004), knowledge is the result of social construction (vygotsky, 1980). social interaction participates in giving an important role in the process of knowledge construction (phillips, 1997). knowledge is constructed in a particular social context and influenced by a variety of 'strengths', including ideology, religion, politics, economics, human interest, and group dynamics. therefore, individuals must construct their own knowledge due to knowledge cannot be simply transferred directly from the teacher to students or from the book to the readers. constructivism-based learning such as problem-based learning (pbl), discovery learning (dl), cooperative learning (arends, 2012), realistic mathematics education (rme) (gravemeijer & doorman, 1999), contextual teaching-learning (ctl), and the open-ended approach (becker & shimada, 1997) is designed by considering the factors that influence the learning outcomes. for example, pbl carries interdisciplinary learning, considers local culture and places emphasis on social interaction to foster students' problem solving skills and social skills (arends, 2012). meanwhile, social skills allow the improvement of academic achievement (minarni, 2013). instead of starting the learning process by presenting content for students to memorize and understand, pbl emphasizes the process of how humans learn naturally, that is, learning occurs when there are problems (hmelo-silver, 2004). to obtain a solution to the problem, people will be motivated to learn the skills and knowledge related to the problems they face, learn or recall the contextual knowledge related to the problem. pbl relies on problems that integrate useful knowledge for students in their personal lives or in volume 9, no 1, february 2020, pp. 111-132 115 facing their professional careers later on. problems are designed to be authentic, unstructured, and sufficiently challenging students to become active and reliable problem solvers. it can be inferred from savin-baden & major (2004) that the goal of pbl is to guide students to construct meaning rather than gathering facts and to become collaborative learner. those characteristics open the opportunities for the achievement of hots. another learning approach based on constructivism is realistic mathematics education (rme). rme is a teaching and learning theory in mathematics education that was first introduced and developed by the freudenthal institute in the netherlands (de lange, 1996). rme is an approach that insisted mathematics should be connected to reality and human activity, close to children and be relevant to everyday life situations (gravemeijer & doorman, 1999). h e e , he d ea i ic , efe j he c ec i i h he real-world, but also refers to problem situations which real in students' mind. the context of the problems presented to the students can be a real-world one but this is not always necessary. de lange (1996) stated that problem situations can also be seen as applications or modeling. there are two types of mathematization in rme formulated explicitly in an educational context (treffers, 1991). these are horizontal and vertical mathematization. in horizontal mathematization, the students come up with mathematical tools that can help to organize and solve a problem located in a real-life situation. examples of horizontal mathematization: identifying or describing the specific mathematics in a general context, schematizing, formulating and visualizing a problem in different ways, discovering relations, discovering regularities, recognizing isomorphic aspect in different problems, transferring a real world problem to a mathematical problem, and transferring a real world problem to a known mathematical problem. on the other hand, vertical mathematization is the process of reorganization within the mathematical system itself. examples of vertical mathematization: representing a relation in a formula, proving regularities, refining and adjusting models, using different models, combining and integrating models, formulating a mathematical model, and generalizing (gravemeijer, 1994). the learning process starts from contextual problems. using activities in the horizontal mathematization, for instance, the student gains an informal or a formal mathematical model. by implementing activities such as solving, comparing and discussing, the student deals with vertical mathematization and ends up with the mathematical solution. then, the student interprets the solution as well as the strategy used to another contextual problem. rme is closely related to socio-constructivism (de lange, 1996; gravenmeijer, 1994). in both approaches, students are offered opportunities to share their experiences with others. in addition, de lange (1996) stated that the compatibilities of socio-constructivist and rme are based on a large part or similar characterizations of mathematics and mathematics learning. those are: (1) both struggle with the idea that mathematics is a creative human activity; (2) that mathematical learning occurs as students develop effective ways to solve problems (streefland, 1991; treffers, 1991); and (3) both aim at mathematical actions that are transformed into mathematical objects (freudenthal, 2006). like wise rme, discovery learning (dl) is also based on constructivism. dl is also referred to problem-based learning, experiential learning and 21st century learning. discovery learning is the work of learning theorists and psychologists jean piaget, jerome bruner, and seymour papert (arends, 2012). jerome bruner is often credited to the origin of dl in the 1960s, but his ideas are very similar to those of earlier writers such as john dewey. bruner argues "practice in discovering for oneself teaches one to acquire information in a way that makes that information more readily viable in problem solving". minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 116 this philosophy later became the discovery learning movement of the 1960s. this philosophical movement suggests that people should "learn by doing". the label of dl can cover a variety of instructional techniques. a discovery-learning task can range from implicit pattern detection, to the elicitation of explanations and working through manuals to conducting simulations. dl can occur whenever they do not provide the student with an exact answer but rather the materials in order to find the answer. discovery learning takes place in problem solving situations where the learner draws on his own experience and prior knowledge and is a method of instruction through which students interact with their environment by exploring and manipulating objects, wrestling with questions and controversies, or performing experiments. it has been suggested that effective teaching using discovery techniques requires teachers to do one or more of the following: 1) provide guided tasks leveraging a variety of instructional techniques, 2) students should explain their own ideas and teachers should assess the accuracy of the idea and provide feedback, 3) teachers should provide examples of how to complete the tasks. a critical success factor to discovery learning is the teacher assistance. on the other hand, dl potentially make the students feel confused and frustrated. silver (2013) argued that pure unassisted discovery should be eliminated due to the lack of evidence that it improves learning outcomes. bruner (1961) who was one of the early pioneers of discovery learning cautioned that discovery could not happen without some basic knowledge (mathematical prior knowledge). i a , he eache e i di c e ea i g i c i ica he cce f learning outcomes. students must build foundational knowledge through examples, practice, and feedback. this can provide a foundation for students to integrate additional information and build upon problem solving and critical thinking skills. early research demonstrated that guided discovery had positive effects on retention of information at six weeks after instruction versus that of traditional direct instruction. it is believed that the outcome of discovery-based learning is the development of inquiring minds and the potential for lifelong learning. discovery learning promotes student exploration and collaboration with teachers and peers to solve problems. children are also able to direct their own inquiry and be actively involved in the learning process with the support of sufficient motivation (reid, 2007). the next one is contextual teaching and learning (ctl). ctl involves making learning meaningful to students by connecting to the real world (johnson, 2002). it draws de di e e ki , i e e , e e ie ce , a d c e a d i eg a e he e i what and how students learn and how they are assessed. in other words, contextual teaching situates learning and learning activities in real-life and vocational contexts to which students ca e a e, i c a i g c e , he ha , f ea i g b he ea h ha learning is important. some examples of contextual teaching and learning are interdisciplinary activities across content areas, classrooms, and grade levels; or among students, classrooms, and communities. problem-based learning strategies, for instance, can situate student learning in he c e f de c i ie . ma ki ea ed a a f c e a ea i g activities are transferable skills, can be used not only for successful completion of a current project but also in other content areas to prepare a student for success in later vocational endeavors. contextual learning, then, engages students in meaningful, interactive, and collaborative activities that support them in becoming self-regulated learners. additionally, these learning experiences foster interdependence among students and their learning groups. complementary outcomes assessments for contextual student learning are authentic assessment strategies, i.e., the assessment is not only limited to the results of the written test but also based on the students' performance in doing the assignments. volume 9, no 1, february 2020, pp. 111-132 117 another constructivism-based learning model or approach is cooperative learning, an educational approach aimed to organize classroom activities into academic and social learning experiences (arends, 2012). cooperative learning is actually not merely arranging students into groups; it is characterized as "structuring positive interdependence." students must work in groups to complete tasks collectively toward academic goals. unlike individual learning, which can be competitive in nature, students learning cooperatively can capitalize on one another's resources and skills (asking one another for information, evaluating one another's ideas, monitoring one another's work, etc.). furthermore, the teacher's role changes from giving information to facilitating students' learning. everyone succeeds when the group succeeds. ross & smyth (1995) describe successful cooperative learning tasks as intellectually demanding, creative, open-ended, and involve higher order thinking tasks. cooperative learning has also been linked to increase levels of student satisfaction. in cooperative and individualistic learning, student efforts are evaluated on a criteria-referenced base while in competitive learning teachers grade in a norm-referenced base. five essential elements are identified for the successful incorporation of cooperative learning in the classroom i.e: i. positive interdependence ii. individual and group accountability iii. promotes interaction (face to face) iv. teaching the students the required interpersonal and small group skills v. group processing. students in cooperative learning settings compared to those in individualistic or competitive learning settings, achieve more, reason better, and gain higher self-esteem. the next constructivism based learning approach is open-ended approach. open-ended approach provides students with experience in finding something new in the process of open problem solving (becker & shimada, 1997), while open problem solving is based on open-ended problems. it can be concluded that open-ended problems used in mathematics lessons from elementary through high school grades. these problems proposed have several or many correct answer, and several ways to get the correct answer. there are five advantages of open-ended approach: a. students participate more actively in lessons and express their ideas more frequently because open-ended approach provides free, responsive, and supportive learning environment. the problem has many different correct solutions, so each student has opportunities to get his own unique answer. hence, students are curious about other solutions and they can compare on and discuss their solutions. those activities bring a lot of interesting conversation to the classroom. b. students have more opportunities to make comprehensive use of their mathematical knowledge and skills. since there are many different solutions, students can choose their favorite ways toward the answer and create their unique solution. activities can be the opportunities to make comprehensive use of their mathematical knowledge and skills. c. every student can respond to the problem in some significant ways of his/her own. therefore, it is very important for every student to be involved into the classroom activities, and the lessons should be understandable for every student. the open-ended problems provide every student with the opportunities to find his/her own answer. d. the lesson can provide students with a reasoning experience. through comparing and discussing in the classroom, students are intrinsically motivated to give reasons of their minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 118 solutions to other students. it is a great opportunity for students to develop their mathematical thinking. there are rich experiences for students to have the pleasure of discovery and to receive approval from fellow students. since every student has each solution based on each i e hi ki g, e e de i i e e ed i fe de utions. there are also some disadvantages of the open-ended approach (sawada, 1997), such as the difficulty of posing problems successfully, the difficulty of developing meaningful problem situations, and the difficulty of summarizing the lesson. besides learning approach, ict also influences learning outcomes (agyei & voogt, 2011). moreover, affective aspects also affect learning achievement (minarni, napitupulu, lubis, & annajmi, 2018). therefore, this paper focuses more on the description of the contribution of learning approach as well as the influence of interactions between the learning approach and the mpk on the achievement of mhots. however, the mpk factor plays an important role as well since it is needed to be recalled previously learned or provide the results of a calculation, which were considered lower cognitive questions in previous studies, played key stages at introducing new mathematical content as well as in the stage of solving mathematical problems. the study first aimed to seek the answer on how the contribution degree of the constructivism-ba ed ea i g he achie e e i e e f de mhots i , and second to reveal if there exists an interaction between the learning approach and the mathematical prior knowledge on the achievement of mhots. 2. method the population of this study was junior high school (pjhs) students in medan, deli serdang, binjai, and padang sidempuan in the province of north sumatera, and banda aceh in the province of nanggroe aceh darussalam. because the school does not allow students to take randomly from each class, samples are taken per class. classes are taken through simple random sampling because the students at all classes assumed homogenous mathematical prior ability, two classes from each district. one class is used as the experimental class, the other one is the control class. the experimental class applied constructivism-based learning, while the control class applies conventional learning. this research runs in the odd semester in the 2017/2018 and 2018/2019 academic years. the instrument used in this study was an essay test of high-order mathematical thinking skills (mhots), which included tests of mathematical problem solving skills (mpss). the mpss indicators used in this study are modifications of polya (2004) and nctm (2000), including: 1. the ability of mathematical understanding shown by external representations and connections between ideas/facts/concepts/mathematical procedures. 2. the ability to propose problem solving strategies that are demonstrated by the existence of techniques/methods of problem solving in student worksheets, whether in the form of mathematical models, graphs, tables, diagrams, or others. 3. the ability to execute the proposed problem solving strategy, shown by calculations and mathematical manipulations to obtain a solution. 4. summing up the solution obtained by following the initial problem. problem 1 below is an example of a question for developing mathematical creative thinking skills (maharani, 2014) grade viii junior high school students. volume 9, no 1, february 2020, pp. 111-132 119 problem 1 look at the quadrilateral model below. 10 cm i. draw a quadrilateral that has same area with the image above. ii. create at least two different questions related to square and solve it. the following problem is an example of a problem that teacher proposed in the openended class (problem 2). problem 2 the average of mathematics score of students from a junior high school is 65. what is the additional score if the average score of the exam becomes 68. write down the steps you do to get a solution. implementation of constructivism-based learning after the learning tools are validated, the next stages of research are as follows: i. conduct the test of mathematical prior knowledge (mpk). ii. implement constructivism-based learning. iii. organizing post-tests iv. analyzing research data v. discuss the results of the research vi. conclude as long as the learning program took place, students are directed to solve mathematical questions contained in student worksheets (sws). each sws, which consists of three to four problems, is designed based on the aspects of mhots the students must achieve. the teacher directs students to work cooperatively and collaboratively in groups to solve them. in general, this is how the learning process takes place in the classroom, whatever type of constructivism-based learning approach used. of course, there are syntaxes differences between one learning approaches to the other, for example in the open-ended approach, the questions contained in the sws are open, that is, have various ways to get the solution and diverse solutions. in the rme approach, the questions are required to be c e a , a d i he di c e ea i g a ach, he e i e i e e gh eache guidance to enable the student in getting the solution. in general, the questions contained in the sws are designed based on the mpsa indicators modified from polya and nctm as mentioned in the introduction to this article. in the control classroom, the teacher implemented direct teaching to the whole class. in this case, the teacher is considered as an essential role model and is expected to be an expert or learned figure. a good grasp of the subject matter is more important and serves as a prerequisite for this kind of pedagogy. minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 120 data analysis univariate and multivariate analysis is used as a statistical tool to analyze the contribution of treatment towards mathematical high order thinking skills (mhots) achievement, while t-students is used as a statistical tool to determine the significant improvement of mhots (glass & hopkins, 1996). all analyses use a 0.05 level of significance. the role of the learning approach is elaborated through linking mhots achievements with the steps of the learning approach applied in the classroom based on the output of the regression analysis. 3. results and discussion the research took place in six different schools in the provinces of north sumatra and nanggroe aceh darussalam continuously from 2017 to 2019. the results of the study are presented in the following order. 3.1. mathematical communication achievement in pbl classroom the first study was conducted at public junior high school (pjhs) muara batu, aceh. the purpose of this study is to improve mathematical communication skills (mcs) as one of the mathematical high-order thinking skills (mhots). for the sake of this matter, we implemented instructional materials that integrated acehnese cultural context to problembased learning (pbl). instructional materials based on pbl is designed so that they meet valid, practical and effective criteria. table 1 show the data on mathematical communication skills at trial i and ii as a result of the research. table 1. students mcs achievement at pbl classroom category trial i trial ii highest 87.5 95.8 lowest 50.0 68.8 average 74.3 80.3 every aspect of average mcs scores in experiments i as well as experiment ii is presented in table 2. there was an increase in mathematical communication skills after pbl implementation. this supports the results of previous studies that pbl can improve the ability of mathematical high-order thinking skills (mhots), where mathematical communication is one of the mhots. the implementation of pbl also allows the development of social skills (arends, 2012) where one of the benefits of social skills is increased academic achievement (minarni, 2013). whether academic achievements in the field of mathematics or social fields, this requires separate research. table 2. average score of students mcs at each aspect aspect trial i trial ii explain the idea or situation of an image in his own words 10.3 11.2 describe a situation in image 13.1 13.9 describe the situation in mathematical equation 12.2 13.0 volume 9, no 1, february 2020, pp. 111-132 121 overall, instructional materials based on pbl that integrate aceh culture have fulfilled the criteria valid, practical and effective in accordance with the objectives of this study. the meaning of these criteria is: a. validity and practicality: 1) the average validity of rpp, student book, and students work sheets (sws) given by five validators is 4.60. 2) in trial i, this instructional material only requires a slight revision. in trial ii, the validators stated that this instructional material was valid. 3) based on the interview and questionnaire, the teachers and the students stated that there were no obstacles in using this instructional material. b. effectivity: 1) more than 75% of students involved in this study have achieved minimum learning completeness requirements, namely achieving test scores more than 65 (in accordance with what was agreed by the ministry of education) (table 1 and table 2). 2) time provided is sufficient for learning implementation. 3) both the teachers and the students respond positively to the instructional materials. 4) the mathematical communication skills of the students in the experimental classroom increased with average n-gain 0.61 (calculated based on table 1). these findings show that integrated pbl in the instructional materials affects significantly the achievement of mcs. the learning approach gives contribution to students learning outcomes.the study implies that if the teacher has the opportunity to design appropriate instructional materials based on the constructivism learning approach for developing mhots, then teacher's desire to improve students hots will be viable. furthermore, indonesia is a country with rich types of local culture; it should be easier to enrich the repertoire of cultural-based mathematical knowledge. another idea conveyed based on this research is that schools can ask the government to provoke the implementation of research results such as learning materials developed based on constructivism. in addition, based on an interview the students give positive responses to the implementation of problembased learning, so it makes sense that pbl gives contribution to the improvement of students' mcs. 3.2. mathematical understanding achievement in cooperative learning classroom the second study was carried out to investigate the effect of cooperative learning assisted mapping concept and microsoft visio (clmv) towards mathematical understanding concepts (muc). the sample consisted of 34 students of eighth-grade alulum islamic middle secondary school, medan, in the academic year 2017/2018. clmv was used in the experiment classroom, while hots to be developed was muc. previous research (arslan & altun, 2007) revealed mathematical prior knowledge (mpk) does not affect the achievement of mathematics learning outcomes, but many other researchers confirmed that it is influential. muc test scores from the experimental class and the control class are shown in table 3. minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 122 table 3. muc score test of the students mpk learning approach clmv conventional high 82 56 medium 62 42 low 63 44 table 3 shows that there is a difference in the muc test score between the experimental class and the control class. this suggests that there is a significant contribution of clmv to the achievements of the muc. thus, we used anova to test the contribution. the result of the test is presented in table 4. table 4. contribution of learning approach to muc source sum of squares df mean square f sig. corrected model 8065.8a 5 1613.2 8.5 0.000 intercept 108817.9 1 108817.9 571.4 0.000 mpk 960.3 2 480.2 2.5 0.088 learning approach 4303.3 1 4303.3 22.6 0.000 learning app.*mpk 416.2 2 208.1 1.1 0.342 error 11997.0 63 190.4 total 223107.0 69 corrected total 20062.8 68 a. r squared = 0.40 (adjusted r squared = 0.36) the test results in table 4 interpreted as follows: a. there is an effect of the learning model on the ability of muc. b. the contribution of learning factors to the achievement of muc is 40%. c. there is no effect of interaction between the learning approach and mpk factors on muc achievement. d. muc of the students taught through cooperative learning assisted mapping concepts and microsoft visio software is better than muc of the students taught through direct instruction. indeed, the involvement of software as part of ict undeniably gives a positive impact on student learning outcomes (agyei & voogt, 2011). furthermore, indonesia is a country that is responsive to the development of ict. almost all pjhs students have smartphones that can make it easier for them to download software or other applications needed in the learning process. therefore, the readiness of teachers is needed to integrate ict in mathematics learning. besides, the results of the observation indicate that learning in the experimental class is in line with the stages specified in the cooperative learning approach. activities to solve problems in the class that is done cooperatively give results in the form of increasing student muc achievements. this is the important role of cooperative learning in improving mathematical high order thinking of the students. this research is in line with the theory of cooperative learning, which states that learning through small groups enables increased volume 9, no 1, february 2020, pp. 111-132 123 learning achievement because in cooperative learning; tasks are designed so that they meet intellectually demanding, creative, open-ended, and involve higher-order thinking tasks (ross & smyth, 1995) which allow the growth of muc as one of mhots. the weakness found in this study is that the teacher is a little excessive in assisting because some students experience dead ends in solving problems. this needs to get the attention of policymakers so that teachers do not give up in applying this innovative learning. 3.3. the achievement of mpss in contextual and cooperative learning classroom subsequent research was carried out at medan budi agung middle secondary school. mathematical high order thinking skills (mhots) is investigated is mathematical problem solving skill (mpss). through the implementation of cooperative learning and contextual teaching learning (ctl) geogebra-assisted, this study aimed to investigate the diffe e ce be ee de mpss i c e a i e ea i g c a (e e i e a c a i) and contextual classroom (experimental class ii). in both experimental classes, learning was implemented with the help of geogebra software. student mpss test scores are presented in table 5. table 5. statistic of students mpss score learning approach statistic average sd cooperative 52.79 17.059 ctl 53.45 15.999 table 5 shows that there is a difference in the mpss score test between the two experimental classrooms. geogebra-assisted contextual learning (ctl-g) is superior in improving student mpss compared to geogebra-assisted cooperative learning. the difference in mpss achievements shows that there is an influence or contribution of the learning approach to the mpss. thus, we do the test of difference achievement of the mpss through two-way anova at 0.05 significance level. the test result is presented in table 6. table 6. test of the effect of learning approach to mpss source sum of squares df mean square f sig. corrected model 1206.4a 3 402.1 4.2 0.009 intercept 326446.6 1 326446.6 3.4e3 0.000 mpk 178.2 1 178.2 1.9 0.177 learning approach 571.4 1 571.4 5.9 0.018 learning app. * mpk 402.2 1 402.2 4.2 0.045 error 5340.2 56 95.4 total 332448.0 60 corrected total 6546.6 59 a. r squared = 0.18 (adjusted r squared = 0.14) minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 124 based on data in table 6, it can be concluded that the test result is significant, this means that: a. there is the mpss difference between students taught by geogebra-assisted cooperative learning and students taught geogebra-assisted ctl. this means, there is an effect of ea i g fac he de mpss achie e e . the effec i ea ed b he degree of contribution. the contribution of learning factors to mpss achievements is around 18%. both geogebra-assisted ctl and geogebra-assisted cooperative learning plays a substantial role in achieving mpss. b. there is an interaction effect between the learning approach and mpk factors on mpss achievement. it means the students from low and medium mpk get benefit from this kind of learning approach. the advantages of contextual learning that make it possible to play an important role in improving the mpss of the students are characteristics of problems that are designed to connect with the context of students' daily lives. based on interviews, students acknowledge that the problems given by the teacher are quite interesting and easier to understand because they are familiar with the theme of the problem. understanding the problem is the first and foremost thing in solving problems and according to the ctl theory, making learning meaningful to students by connecting to the real world is the core element in ctl (johnson, 2002). meanwhile, the weakness of this study mainly lies in the weakness of student mpk, in line with the results of other studies (minarni et al., 2016), such that the teacher is forced to remind students of mathematical knowledge that is not well stored in the cognitive structure of students. based on the results of this study, it is suggestions that: a. the teacher is advised to use cooperative learning and ctl to enhance student achievement in mathematical problem solving skills. b. in implementing constructivism-based learning such as cooperative learning and ctl, the teacher is advised to involve information and computer technology (ict) such as ge geb a f a e, e ecia f ge e a i g de i e e i d i g ge e . because through the help of the software, the display of geometric forms can be visualized more accurately and more 'eye-catching' which increases students' enthusiasm for learning and challenging them to explore other problems related to geometry problems. from this activity, it is hoped that the student's perseverance and life-long learning will be grown. c. the teacher is advised to strengthen students' comprehension of mathematical knowledge and mathematical concepts, as well as mpk. if mpk becomes an obstacle in achieving mpss then the implementation of innovative learning such as ctl and cooperative learning becomes increasingly important because the main advantage provided by these two constructivism-based learning is that learners will be able to store knowledge in long-term memory to guarantee the availability of mpk. this again shows that self-constructed knowledge can make a person firm in storing his knowledge. the following explanation is an example of the problem used in research and alternative solutions. volume 9, no 1, february 2020, pp. 111-132 125 problem: "a cube-shaped aquarium with a length of 85 cm is filled with water. if a decorative stone with a volume of 125 litres is put into the tank, determine the volume of water left in the tank." this problem is closely related to other disciplines, namely physics. moreover, this question is also related to the context of students' daily life where students are very familiar with aquariums as a container for keeping fish that require ornamental stones to mimic original fish habitat. solution: 𝑉 s × s × s cm 85 × 85 × 85 cm 614.125 cm 614.125 dm 614.125 liter 𝑉 𝑉 – 𝑉 614.125 – 125.000 489.125 liter. to solve the problem, students should execute four steps, i.e.: a. calculate the initial volume of water in the aquarium. b. convert water volume units (from cm3 to liters) to equal the volume units of ornamental stones. c. find the reduction of the initial volume because of the insistence of ornamental stones. d. conclude the volume of water left in the aquarium after ornamental stones press some water out. thus, this mathematical problem has characteristics as a good question, that is contextual, interesting, related to other disciplines, and requires multi-step to get a solution. all of these characteristics are in line with hots proposed by resnick (1987). 3.4. mpss achievement in realistic mathematics education classroom the fourth study was carried out at pjhs 2 beringin, district of deli serdang. this research is an effort to improve students' mathematical problem solving skills (mpss) through a realistic mathematical education (rme) approach assisted by autograph. the mpss score test of the students is presented in table 7. table 7. statistic of students mpss learning approach mpk statistic average sd rme high 19.02 2.452 medium 18.02 2,706 low 17.20 2,680 conventional high 17.03 1.150 medium 15.90 2.850 low 12.80 2.030 data from the research was analyzed through two-way anova. the result of the analysis is presented in table 8. it can be seen in table 8 that there are differences in mpss scores between students in the experimental class and students in the control class. the minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 126 difference in achievement is quite large, this means that there is an influence or contribution of the treatment to mpss achievements (glass & hopkins, 1996). the significance level test of the contributions is carried out through the analysis of variance. the output of the test is displayed in table 8. table 8. the contribution of rme towards mpss source sum of squares df mean square f sig. corrected model 175.9a) 5 35.2 3.3 0.012 intercept 12886.6 1 12886.6 1193.9 0.000 learning approach b) 59.5 1 59.5 5.5 0.022 mpk 98.5 2 49.3 4.6 0.014 learning app. *mpk 2.2 2 1.1 0.1 0.902 error 625.9 58 10.8 total 18292.0 64 corrected total 801.9 63 a) r squared = 0.22 (adjusted r squared = 0.15) b) learning approach: rme based on the result of the analysis in table 8, the research findings are: a. the enhancement of the student mpss in the experiment classroom is higher than the in the conventional classroom. b. the learning factor has a significant influence on the achievement of mpss. c. the adjusted r squared is 0.15. this means the contribution of the learning approach to mpss is 22%. d. there is no interaction effect between the learning approach and mpk factor on the mpss achievement. e. the process of solving mathematical problems shown by students in the experimental classroom is better (in terms of more mpss indicators that are met, systematic and directed). in this study, the teacher has implement rme properly, that is, the learning is conducted so that the students construct knowledge by themselves, in line with the socioconstructivism as one principle of rme (de lange, 1996; gravemeijer, 1994) and students are offered opportunities to share their experiences with others. besides, de lange (1996) stated that the compatibilities of socio-constructivist and rme are based on a large part or similar characterizations of mathematics and mathematics learning because they are struggling with the idea that mathematics is a creative human activity and mathematical learning occurs as students develop effective ways to solve problems (streefland, 1991; treffers, 1991). so, if this research is not successful enough in developing student mpss, it is shown by the low mpss score and the low contribution of rme to mpss achievements (only 15%), then that becomes a problem that we must think of a solution. the most striking obstacle in this study is the weakness of students in representing problems into various forms of representation such that they have trouble at the stage of de f a he a ica h i a age. a h gh de ha e b e i he h i a a he a ica age a d e ica a he a ica i he rme, he de f a d de f age ake de fee he ed i i g b e . i ee ha hi gives a role in increasing student mpss. volume 9, no 1, february 2020, pp. 111-132 127 one thing could be suggested from the study is time allocation. as in other studies, the time available is always insufficient to conduct the learning process that aims in enhancing high-order thinking skills. for this reason, it is recommended that the teacher prevent the debate or a prolonged argument among students. the teacher must immediately take over and decide firmly which correct solution is for a certain problem, and which solution still have shortcomings or mistakes. 3.5. mathematical creative thinking achievement in open-ended classroom the fifth study took place in the public middle secondary school number 2 at padangsidempuan. the constructivism-based learning applied here is the open-ended approach. the research objectives are: a. to investigate the influence of the open-ended approach integrated batak angkola culture (oebc) towards students' mathematical creative thinking skills (mcts). b. to investigate the effect of the interactions on mcts. the average score of mcts of the students is presented in table 9. table 9. statistic of students mcts learning approach mpk statistic average sd oebc low 61.99 8.39 medium 69,70 16.30 high 90.70 7.00 conventional low 32.30 5.40 medium 48.70 13.00 high 68.15 8.40 data from the research was analyzed through two-way anova. the anova output is presented in table 10. table 10. contribution of open-ended approach towards mcts source sum of squares df mean square f sig. corrected model 15436.534a 5 3087.307 15.350 0.000 intercept 195925.534 1 195925.534 974.151 0.000 mpk 9121.630 2 4560.815 22.677 0.019 learning app. 5829.018 1 5829.018 28.982 0.012 learning app.*mpk 499.494 2 249.747 1.242 0.296 error 11665.216 58 201.124 total 258944.000 64 a) r squared = 0.57 (adjusted r squared = 0.53) the results of the study indicate that: (a) there is an influence of the open-ended approach integrated batak angkola culture towards the achievement of students' mathematical creative thinking skills (mcts); (b) r squared = 0.57. this means the minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 128 contribution of learning factor to the achievement of mcts is 57%; (c) there is no common influence (interaction) between the open-ended approach integrated batak angkola culture and mpk towards the achievement of students' mathematical creative thinking skills. the mcts score test of the students at the experiment classroom is high enough, so does the contribution of learning approach towards mcts. it is possible since the openended approach implemented in the classroom provides students with experience in finding something new in the process of open problem solving (becker & shimada, 1997), while open problem solving is based on open-ended problems. these characteristics enable the students to participate more actively in lessons and express their ideas more frequently; have opportunities to get a unique answer; develop curiosity about other solutions and they can compare on and discuss their solutions; make comprehensive use of their mathematical knowledge and skills. since there are many different solutions, students can choose their favorite ways toward the answer and create their unique solution. the study also shows that the students involved in the classroom activities and enable the students to have reasoning experience and build intrinsic motivation to give reasons for their solutions to other students. it is a great opportunity for students to develop their mathematical thinking. all of these characteristics meet the demands of the open-ended approach. the weakness of this study is the students are less courageous in conveying ideas. this may be due to indonesian culture that children are generally educated not to argue with their parents or other family members. they are usually educated to obedient children. overall, based on the results of this study, the following suggestions are offered. a. the teacher is suggested to be creative in creating a learning atmosphere that allows students to express mathematical ideas in their language such that the students have selfconfidence, creativity, and courage to argue with their classmates. b. the teacher is suggested to provide a variety of mathematical problems that are in line with the context of the local culture and lure students to relate them to the subject matter or other mathematical problems. if this is done, it will build students' perception that mathematics is useful in their daily lives. c. the teacher should allocate a more accurate time so that this constructivist-based learning activity can run smoothly. preparation of discussion groups is only carried out once at the beginning of learning such that there is more time available for group discussion activities in the next session. 3.6. mpss achievement in discovery learning classroom the fi a e ea ch a c d c ed de e de achie e e i ma he a ica problem solving skills (mpss) through the implementation of instructional materials based on discovery learning approach. there are 40 students included in the experiment class and 40 students in the control class. all of the students are from public junior high school (pjhs) 17 medan. an essay test is used to c ec da a f he de mpss. s de ' mpss score test is presented in table 11. table 11. statistic of students mpss learning approach statistic average sd discovery 15.03 2.282 conventional 10.33 1.493 note: ideal score = 20 volume 9, no 1, february 2020, pp. 111-132 129 based on table 11, the mpss achievement of students in the experimental class (discovery learning class) is better than the conventional class. this shows that there is an influence/contribution of learning to student mpss achievements. the results of this study support the theory that through the discovery learning approach, students' trust in efforts to solve problems increases because they are accustomed to conducting investigations to find information needed to solve problems. according to dewey's opinion in finding such knowledge a person unknowingly stores information in ways that make information easier to use in solving new problems. (arends, 2012). thus, the achievement of problem solving skills become significant. the time limitation to implement discovery learning is a major obstacle in the completion of complete learning. the enhancement of mpss performance was presented in table 12. table 12. the enhancement of mpss f t df sig. mean diff. lower upper eq. var. 0.003 3.557 78 0.001 3.800 1.673 5.927 this significant increase in mpss encouraged researchers to statistically test the contribution of discovery learning to mpss achievements. test results are presented in table 13. table 13. test of learning approach effect towards mpss source type iii sum of squares df mean square f sig. 1. corrected model 308.112 1 308.112 19.852 0.000 2. intercept 11640.313 1 11640.313 750.011 0.000 3. app 308.112 1 308.112 19.852 0.000 4. error 1210.575 78 15.520 5. total 13159.000 80 r squared = 0.203 results of the research: (a) the e ha ce e f de mpss i ig ifica i h a mean difference of 3.80 (mpss ideal total score was 20) (table 12); (b) the contribution of the developed instructional materials based on discovery learning towards mpss is 20.3% (table 13). the results of this study indicate a high increase in mpss (3.8 points), but this research found that the increase is not due to the contribution of the learning approach because of its small contribution, which is only 20%. there may be other factors contribute more significantly to students' mathematical problem solving abilities. it is common in the world of education that there are indeed many factors that contribute to student learning outcomes in matthematics, including teacher factors, school environment, friends, and affective aspects such as mathematical disposition, social skills, self-confidence, selfregulated learning, and others (minarni et al., 2018), and others. finding out the dominant factors contribute to learning outcomes is the attention and interest of educational researchers. 4. conclusion based on the result of the research then the conclusion are, firstly, constructivismbased learning can improve mathematical high order thinking skills (mhots) such as mathematical connection, mathematical understanding, mathematical problem solving, and mathematical creative thinking skills/ability. in the experimental classroom, students' minarni, & napitupulu, the role of constructivism-based learning in improving mathematical … 130 mhots increased significantly. the contribution of constructivism-based learning to mhots is in the range of 18% to 57%. secondly, based on the results of observations made by the observer, the activity of students in the learning process increases significantly. third, in some cases, there is an influence of interaction between the learning approach and students' mathematical prior knowledge towards the achievement of mhots. fourth, based on observations and interview results, the integration of ict to the learning approach increases students' enthusiasm in solving mathematical problems. maybe there are other factors besides learning that contributes more to higher-order mathematical thinking skills (mhots), for example, affective factors such as mathematical disposition, social skills, and learning motivation. to approve this allegation, of course, requires special research. some suggestions can be drawn from the study are (1) we suggest that the teacher dares to implement constructivism-based teaching-learning approach to improve hots; (2) the teachers are advised to have the willingness to integrate local culture in the learning ce i e de i e e i olving mathematical problems; (3) the teachers are advised to integrate ict in explaining subject matter and describing the solution the student gets so that the explanation is easier for students to understand. references agyei, d. d., & voogt, j. 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accepted: august 28, 2016 abstract courseware have an important role in the achievement of the objectives of education. nevertheless, it does not mean any learning resources can be used for a type of learning. the teacher should provide and develop materials appropriate to the characteristics and the social environment of its student. number theory courses is one of the basic subjects that would be a prerequisite for courses at the next level, such as linear algebra, complex analysis, real analysis, transformation geometry, and algebra structure. thus, the student’s understanding about the essential concepts that exist in this course will determine their success in studying subjects that mentioned above. in trying to understand most of the topics in number theory required the abilities of mathematical argumentation and representation. the ability of argumentation is required in studying the topic of complex number system, special operations, mathematical induction, congruence and divisibility. ability representation especially verbal representations and symbols required by almost all the topics in this course. the purpose of this paper is to describe the development of teaching and learning number theory materials which facilitate students to develop the ability of mathematical argumentation and representation. the model used is a thiagarajan development model consisting phases of defining, planning, development, and deployment. this paper is restricted to the analysis of the results of the materials validation from number theory experts. keywords: courseware, numbers theory, argumentation, representation abstrak bahan ajar memiliki peranan penting dalam pencapaian tujuan pendidikan. meskipun demikian, tidak berarti sebarang sumber belajar dapat digunakan untuk suatu jenis pembelajaran. pengajar hendaknya menyediakan dan mengembangkan bahan ajar yang sesuai dengan karakteristik dan lingkungan sosial mahasiswanya. mata kuliah teori bilangan merupakan salah satu mata kuliah dasar yang akan menjadi prasyarat bagi mata kuliah-mata kuliah keprodian pada tingkat selanjutnya, seperti mata kuliah aljabar linear, analisis kompleks, analisis real, geometri transformasi, dan struktur aljabar. dengan demikian, pemahaman mahasiswa secara utuh terhadap konsep-konsep esensial yang ada pada mata kuliah ini akan sangat menentukan keberhasilan mereka dalam mempelajari mata kuliah-mata kuliah tersebut di atas. dalam memahami sebagian besar topik yang terdapat dalam mata kuliah teori bilangan diperlukan kemampuan argumentasi dan representasi matematis. kemampuan argumentasi diperlukan mahasiswa dalam mempelajari topik sistem bilangan kompleks, operasi khusus, induksi matematika, kekongruenan dan keterbagian. kemampuan representasi terutama representasi verbal dan simbol dibutuhkan oleh hampir semua topik pada mata kuliah ini. tujuan penulisan ini adalah untuk mendeskripsikan proses pengembangan bahan ajar teori bilangan yang memfasilitasi mahasiswa untuk dapat mengembangkan kemampuan argumentasi dan representasi matematisnya. model pengembangan yang digunakan adalah model pengembangan thiagarajan yang rosita, the development of courseware based on mathematical representations … 132 terdiri dari tahapan pendefinisian, perencanaan, pengembangan, dan penyebaran. tulisan ini dibatasi pada analisa penulis terhadap hasil validasi bahan ajar teori bilangan dari para ahli. kata kunci: bahan ajar, teori bilangan, argumentasi, representasi how to cite: rosita, c.d. (2016). the development of courseware based on mathematical representations and arguments in number theory courses. infinity, 5 (2), 131-140 introduction to train students as a problem solver can be done through the giving of the mathematical problems that are not routine to students. when students are faced with a problem which is not routine, indirectly students are practicing how to develop his thinking so hopefully able to utilize and apply mathematical concepts and mathematical thought processes also in solving real-world problems. this is in line with the opinion of piaget (suparno, 2001) which states that the exercise of thinking, formulate and solve problems and draw conclusions can help students develop thinking or intelligence. previous experience possessed by students in solving problems will affect their ability to solve similar problems. jonassen (2011) states that one of the factors that affect students' ability to solve problems is prior experience of students. role of prior experience for a problem solver is as a basis for interpreting the problem, giving signs of what is to be avoided and predict the consequences of a decision or action taken. this is reinforced by the results of research bereiter & miller (jonassen, 2011) which concluded that the problem solver base the identification of the problem on beliefs about the cause of the symptom ever found. based on the above opinion, the author tries to relate it to some concept of piaget's theory that experience is crucial in the development process of the formation of students' knowledge. the more experience on issues, environment or object faced by students will further develop ideas and knowledge. with more experience, the scheme of students will be a lot challenged and may be developed and modified by the process of assimilation and accommodation. the development of thought processes that experienced by students when trying to find the right solution is important in learning mathematics. according to jonassen (2011), in addition to finding an acceptable solution, a problem solver must also be able to recognize similar issues at different times. previously, silver & marshall (bergeson, 2000) states that, ‘‘while solving mathematical problems, students adapt and extend their existing understanding by both connecting new information to their current knowledge and constructing new relationship within their knowledge structure’’. my own interpretation on the statement is that when students solve mathematical problems, then it implies that students are adapting and expanding existing knowledge in a way to connect or associate the new information obtained by previous knowledge thus forming a new information that is interconnected in the structure of knowledge. fisher (suryadi, 2012) explains that the success in the thought process that will ultimately affect the development of intelligence a person, determined by the third operation of the acquisition of knowledge (input), the strategy of using the knowledge and problemsolving (output), as well as metacognition and decision making (control ). if the opinions of the foregoing linked to the quality of students prior knowledge, the authors interpret that in resolving a problem is not only based on the amount of knowledge that has been owned by the students but the quality of the knowledge itself is also a very important volume 5, no. 2, september 2016 pp 131-140 133 part. conceptual framework of the knowledge that has been acquired (prior knowledge) should be better and also integrated, in order to accommodate a variety of perspectives, methods, and solutions through the synthesis process and the conflict in cognitive structure. a rational from the above statement, there are some specific cognitive abilities that need to be owned by a problem solver, so that he can maximize the knowledge that has been gained and can use it optimally. novick & holyoak (english, 1994), kaur & har (2009), and polya (jonassen, 2011) expressed the opinion that is similar to that reasoning analogy have a significant role in solving the problem, because the ability to take advantage of the problems that are known to the new problems that have a structure identical will improve performance in solving the problem. meanwhile, according to gentner, holyoak, & kokinov (english, 2004) that, in essence, the analogy lies in the center of human cognition and appears closely linked to the development of the ability of representation. having regard to the opinions that have been described above, it can be concluded that the process of solving a mathematical problem is not a simple thought process, the activities require different types of cognitive abilities that are diverse and a complex cognitive activity. the ability to build schema problems, reasoning, arguing and represents knowledge, a cognitive process that allows a student to be able to solve the problem. students' ability to identify and represent the same concept through different representations regarded as a prerequisite for understanding certain concepts. the accuracy of the students in constructing various mathematical representations of a problem will make it a much simpler problem of making it easier for students to solve. conversely, if a representation which is constructed by the students mistakenly then the problem becomes difficult to solve. at the end of this study, the authors conclude that the ability of argumentation and mathematical representation is an important component in improving the performance in solving the problem. when these activities can be performed optimally and can be developed through the application of mathematics in various contexts, students will grow in a mathematical thinking habits that can help him realize about what they learned. therefore, the ability of argumentation and mathematical representation will assist students in gaining a deeper understanding. thus, the students will simultaneously acquire the subject matter of the lecture is are understood and it certainly will improve the effectiveness and efficiency of the learning process and a positive impact on the quality of the process and results of student learning. based on the experience of several years to build this course, lack of student learning outcomes is very likely caused by the teaching and learning materials that have not been in accordance with their needs, considering that every student will certainly experience the difference of learning trajectory. presentation of the material contained in the text book, still can not help the students to learn to understand it comprehensively. facts that occurred in number theory is learning the abstract concepts causes many examples of the concept cannot be recognized properly by the students. in general, students are not familiar with mathematics learning activities that involve their practice and communicate using various representations and mathematical argument. such a phenomenon even led to the low quality of student understanding of the number theory material. rosita, the development of courseware based on mathematical representations … 134 suryadi (2013) explains that the lack of didactic anticipation is reflected in learning planning can be less optimal impact on the learning process for each student. this is due in part to student responses on didactic situation that developed, beyond the reach of lecturers thought or unexplored so the diversity of learning difficulties that arise are not responded appropriately by the lecturers or not responded at all, so the learning processes cannot occur. therefore it is necessary the development of a teaching and learning materials in number theory which is expected to solve the problems mentioned above. method the research was conducted at the department mathematics of education swadaya gunung jati university in cirebon. this type of research is the research development (research and development). model development of teaching and learning materials used is a model developed by thiagarajan, semmel, and semmel (trianto, 2007) which consists of four phases of development namely definition, design, development, and deployment. the purpose of this phases is to produce a draft of learning material that have been revised based on input from the experts. validation is intended to get input in revising of learning materials that will be tested in the next phases. validation of learning materials is validation of content and the validation of construct that have been developed at this phases of design. based on the results of the evaluation by experts, hereafter used as input to improve the instrument before tested. assessment is done on the instrument and the instrument developed in the design phase (draft i), resulting a final instrument. based on the results of expert validation, revision of learning material (draft ii). validation of learning materials focused on the content, format, language and illustrations as well as compliance with the learning model that will be used in research. after doing validation, validator to write an assessment of the learning materials. assessment consists of five categories, ie not good (value 1), poor (2), is quite good (3), good (4) excellent (5). data were analyzed by expert judgments based on mean value of the scores. descriptions of the average scores are shown in table 1 below. table 1 instrument determining criteria interval criteria 1,00 ≤ x < 1,80 not good 1,80 ≤ x < 2,60 poor 2,60 ≤ x < 3,40 quite good 3,40 ≤ x < 4,20 good 4,20 ≤ x ≤ 5,00 excellent courseware can be used if the category is "quite good", "good" or "excellent". overall, phase / flow of learning material development that was done in this study are presented in figure 1 below. volume 5, no. 2, september 2016 pp 131-140 135 description: : sequence of events : result : type of event : decision : line cycles (if necessary) figure 1. flowchart modification phases of the development of learning materials with thiagarajan model results and discussion results definition phase aims to establish and define the terms of making teaching and learning materials developed by analyzing the goals and learning materials. activities undertaken in the definition phase includes an analysis of the beginning and the end, the students analysis, concept analysis, task analysis, and formulation of indicators. the front-end analysis of the study is to determine the basic problems that required in the development of learning instrument. the analysis is done by providing questions from number theory courses to students department of mathematics education, swadaya gunung jati university. the aims of the questions is to dig up information on learning obstacles experienced by the students when learning number theory. barriers to learning studied are epistemological. analysis of students concept analysis task analysis indicators formulation preparation of test media selection and props format selection early learning tool and research instruments design draf i validation draf i no ya draf ii valid ? revision no draf ii draf iii develop define ya effectiv front-end analysis design rosita, the development of courseware based on mathematical representations … 136 analysis of the student is to examine the characteristics of students accordance with the design and development of teaching and learning materials. these characteristics include cognitive development, academic background, knowledge background. analysis of the concept is an identification process that aims to identify, elaborate and systematically compile relevant concepts to be taught. there is a lot of number theory material that the concept can be built through previous concepts that have been received by the students. when learning most of the number theory topics, students must understand about real numbers systems and the operations properties of the real numbers set. the task analysis is done by identifying the major developed academic skills in the development of learning instrument. this analysis was compiled based on competency standards and indicators of achievement of learning outcomes of the number theory course. based on the analysis of the tasks students are expected to meet the specified mathematical argument indicators, namely: 1. be able to assess similarities and differences in the strategies used include identifying the data, warrants and claims; 2. be able to assess the validity of an argument which includes evaluating the claim based on the data and acceptable warrants; 3. can clarify the statement by providing the necessary and sufficient condition; 4. be able to compile an explanation based on the data that is relevant and irrelevant; and 5. can apply the principle to present sufficient data, the relevant warrants, and the acceptable claim in making decisions. specified mathematical representation abilities indicators are: 1. can present mathematical problems into a visual model; 2. can connect the procedures and processes at various representations of relevant concepts; and 3. can identify and use objects, processes, and procedures that are appropriate in different representations. arrangement of the test of mathematical argumentation and representations ability begins with making of test lattice. lattice (test blueprint or table of specification) is the description of the competence and the material to be tested. lattice test is based on learning objectives that includes a map of the spread of the questions that have been prepared in such a way that the item of the question can be determined with the appropriate level of mathematical arguments and representation completeness of a student. after arranging the lattice, then made instrument test questions. instrument test questions that has been prepared, then tested to count validity, reliability, level of difficulty and the distinguishing power. based on the lattice and analytical validity, reliability, level of difficulty and distinguishing power, then made final test questions instruments. media selection tailored to students analysis, task analysis, concept analysis, and facilities available on campus. in this case the media used include: (1) laptop and lcd, as a medium of learning; (2) student learning material, as the main learning medium for customized learning model that will be used. volume 5, no. 2, september 2016 pp 131-140 137 selection of media is an important part in the learning process because the media will greatly assist the smooth and successful learning process. therefore, the selection of these media will affect learning formats that will be developed. selection format of learning instrument developed in this research tailored to the learning formats that will be developed. selection of the format related to the learning objectives to be achieved, the learning model used, and specific indicators of the development circ learning model based ddr. the format of the selected learning instrument are used to design the content, the selection of learning strategies, and learning resources. the development phase is to produce a draft of the revised learning instrument based on the input of experts and data obtained from the test. validation of experts includes validation of content and construct validation of the courseware that have been developed at the planning phase. validation is done by five people who are competent to assess the feasibility of a learning instrument. revisions were made based on advice / instructions from the validator then generates draft ii. validator assessment of the courseware is based on the indicators contained in the validation sheet courseware. validator results of the assessment of the draft i courseware can be seen in table 2 below. table 2. score validator ratings draft i of courseware validator score v1 3,8 v2 3,8 v3 3,8 v4 3,8 v5 4,8 average 4,0 criteria: good, it can be used with little revision in general, validator stated that courseware can be used with minimal revision. aspects that need to be more emphasis are the substance of the material, kegrafisan, and special characters according to the circ learning model based on ddr. besides that, the tasks presented in courseware need to pay attention to relevant learning theory. based on the results of expert validation, some revisions were made to the teaching and learning material can be seen in table 3 below. table 3. revision of courseware based on the input validators aspects of the revised courseware before revisions courseware after revisions  relevance mathematical ability indicators as measured by the tasks presented  there a task that is not relevant of indicators measured  overall tasks are relevant to the indicators measured rosita, the development of courseware based on mathematical representations … 138 aspects of the revised courseware before revisions courseware after revisions  use of editorial or term  there is a term or editors who have multiple interpretations  change the terms or editors who have multiple interpretations  the use of verbal representation  verbal representation that is used is not appropriate  verbal representations being used matches these aspects are developed on courseware adjusted based on the aspects of the constructivist learning theory as presented in table 4 below. table 4. aspects of courseware development number courseware general components characteristics of constructivism 1 the indicators of achievement of competence standard and basic competence presentation of the prerequisites material 2 contains learning objectives in accordance with the basic competence the presentation of the subject matter 3 the suitability of the content with the aim transforming the knowledge 4 the truth of concept presentation of task 5 description of concept emphasis social nature and the zone of proximal development on learning materials 6 exercises of concept support knowledge is constructed through cognitive apprenticeship 7 utilization of languages effectively and efficiently presentation of the material with the principle scaffolding 8 suitability language and level of development 9 completeness of student books as learning material volume 5, no. 2, september 2016 pp 131-140 139 discussion instrument development process begins with preparing the initial draft (draft i). the draft of the first device was further validated by experts (validator) and made revisions in accordance with the input validators to obtain draft ii. revision of the special character of courseware meant that the product produced in accordance with the learning model that will be developed and have special characteristics as the author wish. explained in free writing subjects that a book will typically contain about something that is a fruit of the mind of an author. if a teacher is preparing a book which is used as teaching and learning material, the mind must be derived from the basic competencies set out in the curriculum, so that the book will give a meaning as courseware for students to learn. the availability of courseware for students expected to provide facilities in learning each competency to be mastered in order to provide the opportunity for students to learn independently and reduce dependence on the presence of a teacher. through courseware capable of actively involving intellectual students will make students learn independently trained, guided in constructing new understandings associated with the concept of understanding already exists in students so that their knowledge will be perceived as a science related to each other. thus the effective learning of communication between teachers and students can be built. conclusion overall the above discussion shows that the development of courseware in this study have been through the validation phase and the revision until the product is obtained in the form of a final draft of instructional materials that meet the content validity and construct validity. this means that this study has produced a valid courseware. references english, l. (1994). reasoning by analogy in constructing mathematical ideas. centre for mathematics and sciences education. queensland university of technology. english, l. (2004). mathematical and analogical reasoning of young learners. london: lawrence erlbaum associaties. jonassen, d. h. (2011). learning to solve problems. a handbook for designing problemsolving learning environment. newyork: routledge. kaur, b. & har, y. b. (2009). mathematical problem solving in singapore schools. in b kaur at al (2009). yearbook. mathematical problem solving. singapore: world scientific. suryadi, d. (2012). membangun budaya baru dalam berpikir matematika. bandung: rizqi press. rosita, the development of courseware based on mathematical representations … 140 suparno, p. (2001). teori perkembangan kognitif jean piaget. yogyakarta: kanisius. infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.216 99 analysis of students mathematical representation and connection on analytical geometry subject muchamad subali noto 1 , wahyu hartono 2 , mohammad dadan sundawan 3 1,2,3 department of mathematics education, swadaya gunung djati university, cirebon indonesia 1 balimath61@gmail.com received: may 20, 2016; accepted: august 10, 2016 abstract the importance ability of mathematical representation and connection to owned by the student really help students in understanding the mathematical concepts in the form of pictures, symbols, and the written word. the use of mathematical representation and the correct connection by students will help students make mathematical ideas more concrete and can connect a concept to another concept, so that students can develop a view of mathematics as a whole integration. this research aims to describe and analyze the ability of representation and mathematical connection on the topics of analytical geometry. the research method was descriptive with the subject as much as 22 mathematics students. data collected through tests and interviews. the results show that the average ability of representation is 46.00; the average mathematical connection ability is 36.77. this means both the abilities still belongs to low, particularly for the ability of mathematical connection. keywords: mathematical representation, mathematical connection, and analytical geometry abstrak pentingnya kemampuan representasi dan koneksi matematis untuk dimiliki oleh mahasiswa sangat membantu mahasiswa dalam memahami konsep matematis berupa gambar, simbol, dan kata-kata tertulis. penggunaan representasi dan koneksi matematis yang benar oleh mahasiswa akan membantu mahasiswa menjadikan gagasan-gagasan matematis lebih konkrit dan dapat menghubungkan suatu konsep ke konsep yang lain, sehingga mahasiswa dapat mengembangkan pandangan matematika sebagai integrasi yang utuh. penelitian ini bertujuan untuk mendeskripsikan dan menganalisis kemampuan representasi dan koneksi matematis mahasiswa calon guru matematika pada matei geometri analitik. metode penelitian adalah penelitian deskriptif dengan subjek mahasiswa calon guru matematika sebanyak 22 mahasiswa. teknik pengumpulan data menggunakan tes dan wawancara. hasil menunjukkan bahwa rata-rata kemampuan representasi sebesar 46,00; rata-rata kemampuan koneksi matematis sebesar 36,77. ini berarti kedua kemampuan tersebut masih tergolong rendah, terutama untuk kemampuan koneksi matematis. kata kunci: representasi matematis, koneksi matematis, dan geometri analitik how to cite: noto, m.s., hartono, w. & sundawan, m.d. (2016). analysis of students mathematical representation and connection on analytical geometry subject. infinity, 5 (2), 99-108 noto, hartono & sundawan, analysis of students mathematical representation … 100 introduction a committee on undergraduate mathematics or cupm recommends that each program should include activities that help students in developing analytical thinking, critical reasoning, problem solving, communication skills and acquire the habit of thinking mathematically. those activities should be designed to promote and measure progress of students in learning mathematics, one of them with communicating mathematical ideas clearly and coherence through writing and speaking. student understanding of the mathematical concepts, and the ability of students in using mathematical ideas can be seen from how students choose the right way in representing its mathematical ideas. nctm (2000) states that when students have access to mathematical representation and ideas that they show, then they have a bunch of tools that will significantly expand their capacities in thinking mathematically. as expressed by jones (2000), there is a need for some reason the ability of representation, namely: is the ability to build basic concepts and mathematical thinking, and to have a good understanding of the concepts that can be used in problem solving. wahyudin (2008) also added that representation could help the students to organize his thoughts. learning by emphasizing mathematical representation is a demanding mental activity learning students optimally in understanding a concept. bruner (hasanah, 2004) said that to understand the most important mathematical concepts is not storage of past experience but how to get back the knowledge that has been stored in memory and are relevant to the needs and can be used when needed. the importance of the ability of mathematical representation for the owned by the student really help students in understanding the mathematical concepts in the form of pictures, symbols, and the written word. the use of the right of representation by students will help students make mathematical ideas more concrete. a complex problem would be much simpler if you use a representation that corresponds to a given problem, otherwise false representation of construction make the problem difficult to be solved. with regard to the theory of bruner, according to ruseffendi (2006), in mathematical learning needs to pay attention to the four propositions, namely; drafting (construction), the notation, contrast and diversity (variation), and connectivity. the evidence for the preparation explaining that in studying mathematics, it will be more embedded if student conduct themselves the order of representation. the evidence for notation explained that in the learning needs to consider the use of a notation which corresponds to the mental development of the child. the evidence for contrast and diversity explaining that in order to make the concept becomes more meaningful, it should be a contrasting concept dishes and diverse. while the evidence for connectivity explains that the learning process needs to consider the opportunity of studying the relation between concepts, topics, and between branches of mathematics. the ability to connect between the concepts, topics and between branches of mathematics called mathematical connection abilities. according to fisher, daniels, & anghileri (suhendar, 2007) making connection is a way to create understanding and instead understand something means making a connection. to understand an object in depth one must know: (1) the object itself; (2) his relations with other similar objects; (3) the relationship with other volume 5, no. 2, september 2016 pp 99-108 101 similar objects; (4) the dual relation with other objects of its kind; and (5) relationship with other objects in the theory (suhendar, 2007). the mathematical connection means the activities connect between mathematical concepts; connect mathematical concepts with the concept of other lessons; apply thought and modeling mathematics to solve the problems that appear in other disciplines such as art, music, psychology, science and business; even also is the activities connect the concept of mathematics with the daily life. nctm popularising the mathematical connection in his native language of english called mathematical connection and make it as one of the six curriculum standards. in addition, students who have the ability of a good connection will be easier to learn the multitude of learning materials with how to connect the material to one another. nctm (2000) justify the statement and proposed that without the mathematical connection ability, students must learn and remember many concepts. continued nctm (2000), when the students are able to connect a concept to other concepts, then they have developed views of mathematics as a whole integration. this means that the purpose of the mathematical connection is intended to broaden the students, see mathematics as a unified whole not stand alone and know the relationship and the benefits of mathematics both in school and outside the school. the geometry is a mathematics lesson materials that need a good mathematical ability to understand it. according to nctm (siregar, 2009: 5) abilities that must be owned by the student in learning geometry is: 1) the ability to analyze the characters and the nature of the geometry either two dimensional or three dimensional, and able to build these arguments regarding the relationship mathematics geometry with the other; 2) the ability to determine the position of a point with more specific and image spatial relationship with using the coordinates geometry and connect it with the other system; 3) application abilities transformation and its use in the symmetric to analyze mathematical situation; 4) is able to use visualization, spatial reasoning, and model geometry to solve problems. by mastering the abilities, students are expected to master the material in the geometry with better. sunardi (2007) stated that in comparison with other math materials, the geometry of the position of the most concern. the difficulty of college students in learning geometry occurs starting from elementary school to college (pt). if studied more about the link between geometry objects that abstract with the difficulty students in learning geometry, then it would appear that there are indeed problems in learning geometry in school are associated with the formation of abstract concepts. learn the abstract cannot be done only with the transfer of information, but it takes a process of formation of concepts through a series of activities that is experienced directly by the student. the series of activities of formation of abstract concepts that hereafter the process of abstraction. nurhasanah (2010) states that fit the characteristics of geometry, abstraction processes must be integrated with the process of learning that goes so should pay attention to some aspects like, learning methods, models of learning, learning materials, availability and use of props or teacher skills in managing learning activities. one of the branches of the geometry is analytic geometry. this courses are intended to give an understanding to the students about the basic concept in the analytically geometry so that the students can solve the problems related to the concepts in analytically geometry. the noto, hartono & sundawan, analysis of students mathematical representation … 102 courses of analytic geometry also has an important role in providing a strong foundation for students to learn more advanced courses such as transformation geometry, calculus ii, advance calculus i and ii. this courses presents many visual representation such as a picture or graph, representation symbols as the common denominators of mathematics and demanding the student to explain it verbally/orally. based on these problems, this article will discuss the related to the ability of mathematical representations and connection students of analytical geometry with the formulation of the problem as follows. 1. how the mathematical representation ability of mathematics students? 2. how the mathematical connection ability of mathematics students? method this research is a descriptive research, with the aim to describe and analyze the ability of mathematical representation and connection students based on the data obtained. description of the method is selected for researchers attempting to uncover the factual situation regarding the ability of mathematical representation and representation students. research subject as much as 22 students of mathematics from unswagati that contracted courses analytical geometry. data collection method in the form of a test and an interview. research instruments include test of mathematical representation and connection, and interview sheet. results and discussion based on the mathematical representation of ability, obtained the following results. table 1 shows the results of a mathematical representation ability (mra). table 1. statistics krm n valid 22 missing 0 mean 46,0000 median 46,0000 mode 10,00 based on table 1 of the output above, seen that the results of the average (mean) and median are the same value. average mra reach 46.00. this shows the average mra are not optimal, because it is still under 50. it means to mra remains to be improved again. seen from the mode value of 10.00. these values are still below average, it means the mra unswagati student tend to the left. median value = 46.00, meaning that there is a 50% (100 students) get value under 46.00. this indicates that the mra is still low or less than optimal, it also reinforced with a value of mode which is still below average. volume 5, no. 2, september 2016 pp 99-108 103 table 2. max and min mra value n valid 22 missing 0 std. deviation 24,82510 variance 616,286 minimum 9,00 maximum 95,00 percentiles 15 11,3500 25 27,2500 50 46,0000 75 65,0000 based on the table 2, obtained that the standard deviation 24,82. this means that the spread of the mra data is 24,82 from the average. minimum value = 9 and maximum value 95.00, this means that there is a student with low and the highest mra. there are 25% (22 students) that mra is under 27.25 and 15% is under 33.33. in the percentiles 75, the value of mra is 65, it means that 25 percent students get mra value above the average. based on its achievements of each mathematical representation indicator, presented in table 3 below. table 3. achievements of mra no. measured indicator average score max. scor e achieveme nts (%) r1. presents the mathematical problems in the visual model. 1. drawing straight lines forming a specific angle with another straight line. 5 10 50 5. drawing a circle with the center of the known and alluded to a straight line. 4,82 10 48,2 r2. identify and use the object, process and procedures that are appropriate in various representation. 3. determine the equation of a tangent on circles if known gradient. 2,59 10 25,9 4. identify the center, radius and the equation of a circle through three points. 19,59 40 48,98 noto, hartono & sundawan, analysis of students mathematical representation … 104 no. measured indicator average score max. scor e achieveme nts (%) r3. linking the procedures and processes in various representations in relevant concept. 2. determine the area of the image of the triangle through the long sides. 8,64 20 43,2 table 3 above shows that the indicator r1-r3 measured with questions no 1-5 obtained its achievements by 50%; 43.2%; 25.9%; 48,98%; 48,2%. this shows that overall the indicator the average has not yet reached 75%. the first indicator is measured with questions no. 1 and 5, presented the problems related to the straight line and the circle, students complete the problem with are modeling visually to describe the straight line and the circle. there are 25 percent of the total amount of students are modeling problems visually correctly, the remaining 75% mistaken in applying the concept to resolve the issue so the model that made it wrong. the second indicator is measured with questions no. 3 and 4, identify and use the object, process and procedures that are appropriate in a representation of the circle equation. 10% of students can identify and use the three coordinates of the point that passed by the circle, to apply the procedure of elimination or substitution the linier equation to get the circle equation, 80% mistaken in the concept of the count and one procedure of elimination or substitution. the third indicator is measured with questions no. 2, linking the procedures and processes in various representations relevant concept. 36% of students can linking procedures and the process of searching for broad and provide an explanation in writing related to the two straight lines that are perpendicular to each other on a visual representation in the form of a picture triangle. 36% of students can linking with appropriate procedures and processes, 64% mistaken in the concept and count. based on the mathematical connection ability data obtained the following result. table 4 shows the results of the students mathematical connection ability (mca). table 4. statistics mca n valid 22 missing 0 mean 36,7727 median 30,5000 mode 25,00 based on the table 4 in the above output, seen that the results of the average (mean) and median almost the same value. the average mca reach 36,00. this shows the average mra is not optimal, because the value is below 50. that means the mca also still must be improved again. the mode is 25.00, below the average, that means mra unswagati students tend to the left. the median value = 30.00, that means 50% (100 students) get under 36.00 volume 5, no. 2, september 2016 pp 99-108 105 value. this shows that the mca still low or less than optimal, is also strengthened by mode value that is still below the average. table 5. max and min value of mca n valid 22 missing 0 std. deviation 25,4893 7 variance 649,708 minimum 6,00 maximum 81,00 percentiles 15 7,4500 25 10,5000 50 30,5000 75 62,5000 based on the table 5, obtained that the standard deviation 25,48. this means that the spread of mca data is approximately 25,48 from the average. minimum value = 6 and maximum value 81,00, this means that there is a student with low mca and the highest mr a of 81. there are around 25% (22 students) that mra under 10,50 and 15% under 7.45. there is also 25 percent students get mca value above the average. based on its achievements of each mathematical connection indicator, presented in table 6 below. table 6. achievements of mca no. measured indicator average score max. score achieveme nts (%) k1. looking for relations of various representations of the concept. 6. explain the relationship between the two straight line based on the image. 5,62 10 56,2 k2. apply math in other fields or in everyday life. 7. apply the concept of an elliptical surface in the field of health 1,72 10 17,20 8. apply the concept of an elliptical surface in the the bridge construction 1,90 10 19,00 noto, hartono & sundawan, analysis of students mathematical representation … 106 no. measured indicator average score max. score achieveme nts (%) 9. apply the concept of a circle related to the problem of radar 7,45 15 49,70 table 6 above shows that the indicator k1 and k2 measured with questions no.6-9 obtained its achievements of 56.2%; 17.2%; 19%; 49,70%. this shows that overall the indicator the average has not yet reached 75%. the indicator k1 measured with questions no.6, presented the problems related to the straight line, students explain the relationship between the representation visually, that explain the relationship between the two straight line based on the image of the given triangle. there are 40 percent of the total amount of students can give an explanation about the relationship between the two straight lines is analytically correctly, the remaining 60% mistaken in applying the concept and create a symbol. the indicator k2 measured with questions no. 7-9, apply mathematics in other areas or in everyday life. 22,7% students can apply the concept of an elliptical surface in the field of health or related to the bridge construction, 77.3% mistaken in the concept and count. 36,36% students can apply the concept of a circle related to the problem of radar, 63,64,3% mistaken in the concept and count. conclusion the ability of mathematical representation students on the material of analytical geometry especially related to the system of the coordinates kartesius, straight line, circle and ellipse still low. they are not able to apply the concepts, error in performing operations, cannot visuallize (drawing a straight line or circle) and cannot use the procedures related to the specific representation. mathematical connection ability of the mathematics students also still low. they are not able to understand the problem, cannot apply the concepts in everyday life, error in performing operations and cannot create a symbol properly. after knowing the mathematical abilities, the authors give suggestions for the next researcher: 1) implementing a learning geometry software-assisted model related in mathematical representation and connection ability, the purpose using this software is to assist students in visualizing geometric objects, so that it can build an internal representation of the students. 2) designing teaching materials with the activities of mathematical representations and connection on the analytical geometry. the development of teaching materials must be based to the description of the abilities of the attention to the difficulty/confusion created by the students. volume 5, no. 2, september 2016 pp 99-108 107 references committee on the undergraduate program in mathematics. (2004). undergraduate programs and courses in the mathematical sciences: cupm curriculum guide 2004. http://www.maa.org/cupm/summary.pdf. coxford, a.f. (1995). the case for representation.dalam p.a. house dan a.f coxford (eds). yearbook connecting mathematics across the curriculum. reston, va: the national council of teachers of mathematics. hasanah,a. (2004). mengembangkan kemampuan pemahaman matematika siswa smp melalui pembelajaran berbasis masalah yang menekankan pada representasi matematika.tesis pps upi. bandung: tidak diterbitkan. hudojo, h. (2002). representasi belajar berbasis masalah.journalmatematika atau pembelajarannya. issn:085-7792. tahun viii, edisi khusus. hwang,w. y., chen, n. s., dung, j. j., & yang, y. l. (2007). multiple representation skill and creativity effects on mathematical problem solving using a multimedia whiteboard system.educational technology and society.vol 10 no. 2: 191-212. jones, b.f., & knuth, r.a. (1991).what does research about mathematics?[online]. tersedia: http://www. ncrl.org/sdrs/areas/stw_esys/2math.html. jones, a.d. (2000). the fifth process standard: an argument to include representation in standar 2000. [online]. available: http://www.math.umd.edu/~dac/650/jonespaper.html. kaput, j. j dan goldin, g. a. (2004).a join perspective on the idea of representation in learning and doing mathematics.[online]. tersedia: http://www.simmalac.usmassad.edu. nctm (2000).principles and standards for school mathematics.reston: virginia. nurhasanah, f. (2010).abstraksi siswa smp dalam belajar geometri melalui penerapan model van hiele dan geometer’s sketchpad (junior high school students’ abstraction in learning geometry through van hiele’s model and geometer’s sketchpad). tesis sps upi bandung: tidak diterbitkan. ruseffendi, e.t. (2006). pengantar kepada membantu guru mengembangkan kompetensinya dalam pengajaran matematika untuk meningkatkan cbsa. bandung: tarsito. siregar, n. (2009). studi perbandingan kemampuan penalaran matematik siswa madrasah tsanawiyah kelas yangbelajar geometri berbantuan geometer’s sketchpad dengan siswa yang belajar tanpa geometer’s sketchpad. tesis sps upi bandung: tidak diterbitkan. http://www.maa.org/cupm/summary.pdf http://www/ http://www.math.umd.edu/~dac/650/jonespaper.html. http://www.simmalac.usmassad.edu/ noto, hartono & sundawan, analysis of students mathematical representation … 108 suhendar (2007). meningkatkan kemampuan komunikasi dan koneksi matematika siswa smp yang berkemampuan rendah melalui pendekatan konstektual dengan pemberian tugas tambahan. tesis pada sps upi: tidak diterbitkan. sumarmo, u. (2010). berfikir dan disposisi: apa, mengapa dan bagaimana dikembangkan pada peserta didik. fpmipa upi.: tidak diterbitkan. sunardi.(2007). hubungan tingkat penalaran formal dan tingkat perkembangan konsep geometri siswa. jurnal ilmu pendidikan. jakarta: lptk dan ispi. wahyudin. (2008). pembelajaran dan model-model pembelajaran. bandung: upi. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.223 69 the effect of problem posing approach towards students’ mathematical disposition, critical & creative thinking ability based on school level adi nurjaman 1 , indah puspita sari 2 1,2 mathematics education stkip siliwangi, cimahi, indonesia 1 hendrialfianto@gmail.com, 2 chiva.aulia@gmail.com received: october 27, 2016 ; accepted: january 16, 2017 abstract the background of this study is the school of the new students of mathematics education courses came from grade high, medium and low. here the writer wants to see how much influence of the school level on new students’ critical thinking skills and creative mathematical. the purpose of this study was to examine differences in new students’ mathematical disposition, critical & creative thinking ability through the mathematical problem posing approach based on school level (high, medium, low). the method used in this research is the experimental method, with only posttest design. the population of this study is all the students of mathematics education department in cimahi; while the sample is selected randomly from one college. then from this chosen college is taken two samples from random class. the instrument of essay test is used to measure students’ critical and mathematical creative thinking ability; while non-test instrument is questionnaire of attitude scale. the results show that: 1) based on the school level (high, medium, and low); there is difference in students’ mathematical critical thinking ability through problem posing approach. 2) based on the school level (high, medium, and low); there is difference in the students’ mathematical critical thinking ability through problem posing approach. 3) based on the school level (high, medium, and low); there is difference in students’ mathematical disposition. keywords: critical, creative, problem posing, disposition abstrak latar belakang dari penelitian ini yaitu sekolah dari mahasiswa baru program studi pendidikan matematika berasal dari grade atas, menegah dan rendah disini penulis ingin mengatahui seberapa besar pengaruh level sekolah terhadap kemampuan berpikir kritis dan kreatif matematik mahasiswa baru. tujuan penelitian ini adalah menelaah perbedaan kemampuan berpikir kritis, kreatif dan disposisi matematik mahasiswa melalui pendekatan problem posing berdasarkan level sekolah (tinggi, sedang, rendah). metode yang akan digunakan dalam penelitian ini adalah metode eksperimen, dengan desain postes only. populasi dalam penelitian ini adalah seluruh mahasiswa program studi pendidikan matematika di kota cimahi, sedangkan sampel akan dipilih secara acak satu perguruan tinggi. hasil penelitian menunjukkan bahwa 1) terdapat perbedaan kemampuan berpikir kritis matematik mahasiswa melalui pendekatan problem posing berdasarkan level sekolah (tinggi, sedang, rendah). 2) terdapat perbedaan kemampuan berpikir kreatif matematik mahasiswa melalui pedekatan problem posing berdasarkan level sekolah (tinggi, sedang, rendah). 3) terdapat perbedaan disposisi matematik mahasiswa berdasarkan level sekolah (tinggi, sedang, rendah). kata kunci: kritis, kreatif, problem posing, disposisi mailto:hendrialfianto@gmail.com nurjaman & sari, the effect of problem posing approach towards students’ … 70 how to cite: nurjaman, a. & sari, i. p. (2017). the effect of problem posing approach towards students’ mathematical disposition, critical & creative thinking ability based on school level. infinity, 6 (1), 69-76. introduction criteria of secondary schools according to peraturan bersama antara menteri pendidikan nasional dan menteri agama nomor 04/vi/pb/2011 nomor ma/111/2011 tentang penerimaan siswa baru (joint regulation of the minister of education and minister of religious affairs number 04 / vi / nt / 2011 number ma / 111/2011 concerning admission) is having been graduated from smp (junior high school)/mts (islamic junior high school)/ smplb (junior high school for the disabled)/ program paket b (package b program), having a diploma and highest age of 21 years old. in accordance with the theory of piaget (budiningsih, 2004), there are four stages of cognitive development, namely: 1) stage of sensory-motor (0-2 years old) 2) stage of pre-operational (2-7 years old) 3) stage of concrete operational stage (7-11 years old) 4) stage of formal operational (11 – adult years old) based on the theory, it can be said that students’ developmental stage at 18 years old and beyond is formal operational stage in which students can work and think effectively and systematically, analyze combination, think proportionally and generalize fundamentally on the kinds of content. when entering college courses, the students are already in the range of 18 – adult years old where the stage of thinking is different from students who are in concrete operational stage. stkip siliwangi is a higher education institution that embodies the community who want to gain knowledge and continue education to a higher one. of course, students who enrolled in stkip siliwangi come from high school educational background and different areas. school background of new students study math education comes from grade high, medium and low. here the writers want to see how much influence of school level towards mathematical critical and creative thinking of new students. sumarmo (sugandi, 2010) says, "it is important to train students high level mathematical thinking skills (kbmtt) trained the students, supported by the educational goals of mathematics that has two directions of development that meets the needs of the present and future" the ability to think critically and creatively of students from diverse secondary school background also gives impact on the mindset of the students themselves, but it is possible if there are students coming from high schools with lower school levels in cognitive ability can be equal to those of schools with high or moderate school level. in addition to cognitive domain, new students’ affective ability, such as disposition, will also be studied. one way to find out the influence of students’ cognitive and affective ability is by using problem posing approach. there are several terms related to mathematical thinking (sumarmo in hidayat & hamidah, 2014), among others are mathematical thinking, mathematical abilities, doing mathematic, and mathematical task. students thinking ability is not the same. there are differences in mindset; students from schools with high grade are probably better than those from schools with medium or low grade. here the writers want to see how much influence of school level towards mathematical critical and creative thinking of new students. critical thinking according to johnson (zetriuslita, ariawan & nufus, 2016) is a focused and clear process volume 6, no. 1, february 2017 pp 69-76 71 used in mental activities such as solving problems, making decisions, persuading, analyzing assumptions, and conducting scientific research. in line with it, lipman (zetriuslita, ariawan & nufus, 2016) argues, critical thinking is the focus, reasons, inferences, situation, clarity and reviewing. creative thinking can be defined as a mental activity that is used to build new ideas. according sumarmo (choridah, 2013), creative thinking deals with the characteristics as follows: the characteristics of fluency include: 1) sparking many ideas, many answers, a lot of problem solving, and many questions smoothly. 2) provide lots of ways or suggestions to do various things. 3) always think about more than one answer. the characteristics of flexibility are: 1) generating ideas, answers, or questions varied 2) an issue from diverse viewpoint. 3) finding many alternatives or different directions. 4) being able to change the approach or way of thinking. skills of sharing within the whole class can be done by pointing couples who volunteer or take turn to report on the work of their group, so about a quarter of couples already have the opportunity to report. in addition to seeing an increase in mathematical critical and creative thinking skills, we can also analyze students’ mathematical disposition. sumarmo (hidayat & hamidah, 2014) argues, "through students’ mathematical disposition we can see their confidence, expectations and meta-cognition, passion and serious attention in learning mathematics, persistence in facing and solving problems, high curiosity, and the ability to share opinions with other people". in line with it, mahmudi (sugilar, 2013) argues that attitudes and habits of thought would essentially establish and grow a mathematical disposition. problem posing approach emphasizes students to form or ask questions based on the information or the given situation so that students can discover and construct their own knowledge. problem posing approach provides the opportunity for students to be more active in learning activities in the classroom. in addition, students are free to expend their ideas at the time of submitting the matters. there are three stages of problem posing as proposed by zakaria (afgani, saputro & darmayasa, 2016), namely; 1) identifying whether or not the problem can be solved, 2) identifying the category of content matter, and 3) providing score based on the students’ creativity. problem posing as proposed by hamzah (2003) are: 1) formulating simple math problem or reformulation of the problem that has been given through some means in order to solve complex problems. 2) formulating of mathematical problems related to the terms of the problem to be solved in order to find alternative solutions that are relevant. 3) formulating or asking a question of mathematics of a given situation, whether filed before, during or after troubleshooting. silver and cai (1996) classify three cognitive activities in manufacturing questions as follows. 1) pre-posing solution, which is making items based on circumstances or information provided 2) within-posing solution, i.e. manufacturing or formulating items that are being resolved. making items is intended as a simplification of the problem being solved nurjaman & sari, the effect of problem posing approach towards students’ … 72 3) post-solution posing. this strategy is also called the strategy "find a more challenging problem." students modify or revise objectives or conditions of items that has been completed to generate more challenging new problems. making such problems refers to a strategy of "what-if-not ...?" or "what happen if ...". based on the above description, the authors want to investigate how much influence of students’ previous school level towards their mathematical critical and creative thinking ability. therefore, the authors take the title the effect of problem posing approach towards students’ mathematical disposition, critical & creative thinking ability based on school level. based on the background above, the question for this research is whether or not there are differences in students’ ability to think critically, creatively, and disposition through mathematical problem posing approach based on school level (high, medium, low)? the purpose of this study was to examine differences in students’ the ability to think critically, creatively and disposition through the mathematical problem posing approach based on school level (high, medium, low). method the method used in this research is the experimental method, with only posttest design. the population in this study is all students of mathematics education courses in cimahi, while samples are selected randomly at one college. then from this chosen college is taken two samples from random class. the instrument of essay test is used to measure students’ critical and mathematical creative thinking ability; while non-test instrument is questionnaire of attitude scale and observation to see the students’ confidence, expectations and metacognition, passion and serious attention in learning mathematics, persistence in dealing with and solving problems, high curiosity, and the ability to share their thoughts with others. the method used in this research is the experimental method, with only posttest design. the design of this research is: a x o a x o in this study will also be given scale post to examine the learning with problem posing approach to the students’ position with the following design: notes: a: the research subjects selected randomly. o: posttest (test of mathematical disposition, critical & creative thinking ability). x: treatment of learning with posing problem approach. volume 6, no. 1, february 2017 pp 69-76 73 results and discussion results table 1. recapitulation of results of research ability experimental class control class pretest % posttest % pretest % posttest % matemathical critical thingking ̅ 6.93 34.65 16.64 83.20 6.48 32.40 15.57 77.85 s 1.47 2.02 1.27 1.73 matemathical creative ̅ 5.69 28.45 15.52 77.60 5.71 28.55 14.57 72.85 s 2.00 1.77 1.70 1.71 disposition ̅ 72.28 60.23 69.45 57.87 s 10.97 7.31 notes: smi test of matemathical critical = 20 smi test of matemathical creative = 20 smi scale of mathematical disposition= 120 table 1 above shows that in experimental group, the students’ pretests mean for the category of their mathematical critical thinking ability is 6.93 and the control group’s mean is 6:48. it is seen that the deviation of mean for the category of mathematical critical thinking ability of both classes is 0.45. so, it can be said that the mathematical critical thinking ability of both classes is not much different. this means that before the treatment, both classes have the same mathematical critical thinking ability. experimental class’ standard deviation of pretest for their mathematical critical thinking ability is 1.47, while control class’ is 1.27. the difference between the two groups is 0.20, which means the experimental group or the control group had a relatively equal distribution of data. furthermore, in experimental group, the students’ posttests mean for the category of their mathematical critical thinking ability is 16.64 and control class’ is 15.57 which shows significant difference of 1.07; meaning that there is big difference between mathematical critical thinking ability in both groups. if the mean of the two groups is changed in terms of percentage, the percentage of experimental class’ pretest mean score for their mathematical critical ability is 34.65% and the control group’s mean is 32.40%, which means that the percentage of mathematical critical thinking abilities for both groups is almost the same. percentage of score is obtained from the mean score division of the ideal score multiplied by 100%. but after being treated, the percentage of students’ posttest mean for their mathematical thinking ability in experimental class and control class becomes 83.20% and 77.85%, which means the percentage of mathematical critical thinking abilities in experimental group is higher than the percentage of the control group posttest data analysis of students’ mathematical critical thinking ability table 2. results of test of two-way anova for students’ mathematical critical ability source type iii sum of squares df mean square f sig. class .836 1 .836 1.099 .298 school level * class 4.877 2 2.438 3.204 .046 nurjaman & sari, the effect of problem posing approach towards students’ … 74 based on table 2, the probability is 0.000 for 0.000 < 0.05 then h0 is rejected. thus, by using a significance level of 0.05 then we can conclude that there are differences in students’ mathematical critical thinking ability based on school level. the probability based on experimental and control class is 0298, for 0.298 > 0.05 then h0 is accepted. therefore, by using significant level 0.05 it can be concluded that there is no difference for both experimental and control group in the category of their mathematical critical thinking ability. the interaction between the classroom and school level generates probability of 0.046 > 0.05. so, by using significance level 0.05, it can be inferred that there is interaction between the experimental class and control class with the school level. results of analysis of students’ mathematical creative ability table 3. results of test of two-way anova for students’ mathematical creative thinking ability source type iii sum of squares df mean square f sig. class .241 1 .241 .422 .518 school level * class 2.073 2 1.036 1.813 .170 based on table 3, the probability is 0.000 for 0.000 < 0.05 then h0 is rejected. thus, by using a significance level of 0.05 then we can conclude that there are differences in students’ mathematical creative thinking ability based on school level. the probability based on experimental and control class is 0518, for 0518 > 0.05 then h0 is accepted. therefore, by using significant level 0.05 it can be concluded that there is no difference for both experimental and control group in the category of their mathematical creative thinking ability. the interaction between the classroom and school level generates probability of 0.170 > 0.05. so, by using significance level 0.05, it can be inferred that there is no interaction between the experimental class and control class with the school level. results of analysis of students’ mathematical disposition ability table 4. results of test of two-way anova for students’ mathematical disposition ability source type iii sum of squares df mean square f sig. class .200 1 .200 .005 .946 school level * 2434.861 2 1217.431 28.374 .000 based on table 4, the probability is 0.000 for 0.000 < 0.05 then h0 is rejected. thus, by using a significance level of 0.05 then we can conclude that there are differences in students’ mathematical disposition ability based on school level. the probability based on experimental and control class is 0946, for 0946 > 0.05 then h0 is accepted. therefore, by using significant level 0.05 it can be concluded that there is no difference for both experimental and control group in the category of their mathematical disposition ability. the interaction between the classroom and school level generates probability of 0.000 > 0.05. so, by using significance level 0.05, it can be inferred that there is interaction between the experimental class and control class with the school level. volume 6, no. 1, february 2017 pp 69-76 75 discussion the purpose of this study was to examine differences in new students’ mathematical disposition, critical & creative thinking ability through the mathematical problem posing approach based on school level (high, medium, low). in general, the implementation of learning by problem posing approach goes as expected. some of the things that researchers have found in the implementation of research on learning by problem posing approach include: 1) firstly, researchers give directions to the students about learning to be carried out in accordance with the schedule of events organized. on this occasion, researchers also convey the subject to be examined with a question and answer, recalling previous relevant materials. 2) at the second meeting, the researchers inform learning objectives in accordance with the basic competencies and approaches that will be used in learning. 3) at its next meeting, the researchers present the learning material with appropriate strategies and try to always engage the students in activities 4) at the first and second meeting, the students are still not used to follow each step of the preliminary activities. 5) at this meeting, the researchers provide opportunities for the students to ask things that are still not clear 6) engaging the students in problem posing approach by allowing them to create questions of a given situation. the activities can be done in groups or individually. 7) at this stage, the researchers allow the students to solve problems made by their own. 8) in the final stage, the researchers direct the students to make inferences from the material already learned. there is no difference for initial mathematical critical thinking ability in both classes. after being given the treatment of learning through problem posing approach, the mean score of experimental group for their mathematical critical thinking ability is classified as high category while control group’s is middle category. the differences in mathematical critical thinking ability based on the school level use twoway anova. probability of 0.000 for 0.000 < 0.05 then h0 is rejected. thus, by using significance level of 0.05 then we can conclude there are differences in the ability of mathematical critical thinking based school level. based on the probability of class experimental and control, namely 0298, for 0.298 > 0.05 then h0 is accepted. therefore, by using significance level of 0.05 then we can conclude there are no differences in the ability of mathematical critical thinking for the experimental class and control class. the interaction between the classroom and school level generated probability of 0.046 > 0.05. so, by using significance level 0.05, it can be inferred that there is interaction between the experimental class and control class with school level. equivalent initial mathematical creative thinking ability. after the experimental group was given problem posing learning and the control group was given conventional learning, the mean score for mathematical creative ability in each group increased. there is no difference for initial mathematical creative thinking ability in both classes. after being given the treatment of learning through problem posing approach, the mean score of experimental group for their mathematical creative thinking ability is classified as high category while control group’s mean score is in middle category. nurjaman & sari, the effect of problem posing approach towards students’ … 76 the differences in the ability to think creatively based on school level by use two-way anova. probability of 0.000 for 0.000 < 0.05 then h0 is rejected. thus, using a significance level of 0.05 then we can conclude that there are differences in mathematical creative abilities based on the level of the school. based on the probability of class experimental and control, namely 0.518, for 0.518 > 0.05 then h0 is accepted. therefore, by using significant level 0.05, it can be inferred that there is no different mean score of mathematical creative thinking ability for the experimental class and control class. the interaction between the classroom and school level generated probability of 0.170> 0.05, so by using 0.05 significance level, it can be concluded that there is no interaction between the experimental class and control class with school level. conclusion it can be concluded that there is different ability to think critically, creatively and disposition possessed by the students through the mathematical problem posing approach based on school level (high, medium, low). references afgani, m. w., saputro, b. a., & darmayasa, j. b. (2016). pembelajaran matematika menggunakan pendekatan problem posing berbasis komputer pada siswa sma kelas x. infinity, 5(1), 32-41. budiningsih, a. (2004). belajar dan pembelajaran. yogyakarta: rinika cipta. choridah, d. t. (2013). peran pembelajaran berbasis masalah untuk meningkatkan kemampuan komunikasi dan berpikir kreatif serta disposisi matematis siswa sma. infinity, 2(2), 194-202. hamzah. (2003). meningkatkan kemampuan memecahkan masalah matematika siswa sekolah lanjutan tingkat pertama negeri di bandung melalui pendekatan pengajuan masalah. disertasi upi. bandung: not published. hidayat, w., & hamidah. (2014). retensi daya matematik siswa sma melalui pembelajaran meas (model-eliciting activities). jurnal penelitian dan pembelajaran matematika, 7(1), 15-24. silver, e. a., & cai, j. (1996). an analysis of arithmetic problem posing by middle school students. journal for research in mathematics education, 521-539. sugandi, a. i. (2010). mengembangkan kemampuan berpikir matematis tingkat tinggi dan kemandirian belajar melalui pendekatan berbasis masalah dengan setting kooperatif tipe jigsaw pada siswa sma. disertasi upi. bandung: not published. sugilar, h. (2013). meningkatkan kemampuan berpikir kreatif dan disposisi matematik siswa madrasah tsanawiyah melalui pembelajaran generatif. infinity, 2(2), 156-168. zetriuslita, ariawan, r., & nufus, h. (2016). analisis kemampuan berpikir kritis matematis mahasiswa dalam menyelesaikan soal uraian kalkulus integral berdasarkan level kemampuan mahasiswa. infinity, 5(1), 56-65. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p191-202 191 the relationship between musical intelligence and the enhancement of mathematical connection ability using ethnomathematics and the mozart effect dianne amor kusuma*1, estiyan dwipriyoko2,3 1universitas padjadjaran, indonesia 2universiti tun hussein-onn malaysia, malaysia 3universitas langlangbuana, indonesia article info abstract article history: received feb 26, 2021 revised apr 16, 2021 accepted apr 18, 2021 the background of this study is mathematics learning outcomes of junior high school students in agricultural areas are still low because they are less motivated to learn mathematics, so that is has an impact on their low learning outcomes. this study aims to find the relationship between musical intelligence and the enhancement of mathematical connection ability by applying ethnomathematics and the mozart effect for increasing students’ motivation to learn mathematics. this study used a quasi-experimental nonequivalent control group design in grade 7 students at smpn bojongsoang 1, kabupaten bandung. the instruments used were the test of mathematical connection ability, musical intelligence questionnaire, and observation sheets. the results showed that: (1) there were differences in mathematical connection ability of students who received ethnomathematics and the mozart effect learning with students who received direct learning; (2) musical intelligence has a positive impact on the enhancement of students' mathematical connection ability; and (3) students have a positive attitude towards learning with application of ethnomathematics and the mozart effect, and more motivated to learn mathematics. the conclusion of this study is that there is a relationship between musical intelligence and the enhancement of mathematical connection ability, and students are more motivated to learn mathematics. implication of this research for future research and learning practice is that students' mathematical connection ability can be explored and improved in various ways, one of which is by applying ethnomathematics and the mozart effect in mathematics learning, and can be influenced by many things, one of them is musical intelligence. therefore, in future research, it would be recommended to study the relationship between musical intelligence and the enhancement of other mathematical abilities using ethnomathematics and the mozart effect. keywords: ethnomathematics, mathematical connection, mozart effect, musical intelligence copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: dianne amor kusuma, department of mathematics, faculty mathematics and natural sciences, universitas padjadjaran jl. raya bandung sumedang km. 21, jatinangor, west java 45361, indonesia. email: amor@unpad.ac.id how to cite: kusuma, d. a., & dwipriyoko, e. (2021). the relationship between musical intelligence and the enhancement of mathematical connection ability using ethnomathematics and the mozart effect. infinity, 10(2), 191-202. https://doi.org/10.22460/infinity.v10i2.p191-202 kusuma & dwipriyoko, the relationship between musical intelligence and the enhancement … 192 1. introduction most of students in any levels of education feel that mathematics is a difficult subject. the impact on their low motivation to learn. the low motivation of students can impact to their low learning outcomes (kusuma, 2019a). the motivation to learn will emerge if students feel that what they are learning has benefits in real life. in other words, students will be motivated to learn if what they learn is related to their daily lives, so they feel the need and are interested for learning it. students' interest in learning mathematics can be influenced by two things, namely: the relation of mathematics to their real life and culture, and their comfort when learning mathematics in class. a preliminary study conducted through interviews with several mathematics teachers in several public junior high schools in the agricultural area of bandung district shows that mathematics learning outcomes are still low because students do not understand the internal and external relationships of mathematics. in a sense, students do not understand the relationship between mathematical concepts and their procedures, mathematics and other fields, and the relationship between mathematics and everyday life, so they feel that mathematics is not useful in their lives. the purpose of the mathematics learning process, besides changing student behavior, is to explore and improve mathematical abilities, one of which is mathematical connection. mathematical connection ability needs to be improved so that students understand the following: (a) there is a relationship between mathematical concepts and its procedures; (b) mathematics has links with other fields; and (c) mathematics is useful in their daily life and culture (rohendi & dulpaja, 2013). therefore, it is necessary to apply a learning approach that can improve students' mathematical abilities and can make them feel comfortable when learning mathematics. in this study, the measured mathematical connection ability includes the following indicators: (1) linking mathematical concepts; (2) using one mathematical idea to understand another mathematical ideas; (3) using mathematics to solve problems in everyday life that contain local culture; and (4) exploring mathematical problems and describing the results with graphs, surrounding objects, and mathematical modeling. ethnomathematics is knowledge that links mathematics with cultural elements (d'ambrosio, 1985). the form of the relationship can be seen from the aspect of applying mathematical concepts in a culture and how to teach mathematical concepts that are adapted to local culture and the uniqueness of the character of students (kusuma, 2019b). ethnomathematics is also a study that focuses on how to teach mathematical concepts in a cultural context (orey & rosa, 2004). thus, it is expected that students will feel that mathematics is part of their culture and are more motivated to learn mathematics. implementation of ethnomathematics in this research are: a) conducting a survey in tegalluar village, bojongsoang, bandung regency, and conducting interviews with residents and community leaders in the village; b) compiling learning tools containing the culture of tegalluar village; c) carry out a learning process that includes the culture of tegalluar village; d) compile a research instrument containing the culture of tegalluar village. in order for students to understand the material being taught, it is necessary to create a learning atmosphere that makes them feel comfortable and do not feel tense or pressured when learning mathematics. one way is to implement classical music into learning, because music can strengthen the brain's nervous system, can improve memory, and make students more focused. music composed by mozart (known as the mozart effect) is a classical music that can increase a person's intelligence (stough et al., 1994; verrusio et al., 2015). the composition of notes in mozart's music can stimulate the performance of the forebrain in humans, which has a positive impact on reasoning abilities, so it is hoped that it can explore volume 10, no 2, september 2021, pp. 191-202 193 and improve students' mathematical connection skills and make them feel comfortable when studying mathematics. in this study, application of the mozart effect is combined with ethnomathematics learning. the technical implementation, mozart k. 448's music is played with the dynamics of mezzo piano (rather soft), frequency of 8000-32.000 hertz, and tempo of allegro con spirito (120-168 beat/minute) during learning activities, with amplitude settings that has been adjusted to capacity of the classroom and number of students. application of the mozart effect is combined with ethnomathematics learning because it have many advantages as follows: (a) mozart k. 448’s music that is played during mathematics learning activities can stimulate students' brains and facilitate the complex nervous system involved when students learn mathematics (campbell, 2009), so that they can better understand the relationship between the mathematical concepts learned and their culture; and (b) mozart k. 448's music can manipulate students' moods and improve their cognitive performance (rauscher et al., 1993), thus making them more creative in finding relationships between mathematical concepts taught with their culture, and in providing examples of situations which is usually done in their culture which contains mathematical concepts. gardner (2011) suggests that everyone has different type of intelligence, some of which are mathematical logic intelligence and musical intelligence. mathematical logic intelligence is ability to reason, recognize patterns, and logically analyze problems (gardner, 2011). musical intelligence is ability to recognize patterns, rhythms and sounds, and express musical forms (widhianawati, 2011). the students' mathematics skills with high logicalmathematical and musical intelligence could meet all indicators of problem interpretation problems, solution applications, solution evaluation, and concluding results using facts (rifqi et al., 2021). mathematical logic intelligence and musical intelligence have one tendency in common, namely being able to think conceptually about patterns. therefore, it is expected that musical intelligence can have an impact on increasing mathematical connection ability. in this study, indicators of musical intelligence that were measured as follows: (1) like playing a musical instrument; (2) can hum and sing; (3) easily recognize songs and memorize song lyrics; and (4) sensitive to tone and rhythm. studies related to students' mathematical connection ability have been conducted by several researchers, including: hendriana et al. (2014), noto et al. (2016), siregar and surya (2017), haji et al. (2017), and yaniawati et al. (2019). these studies showed that efforts have been made to improve students' mathematical connection abilities by applying various learning approaches. however, none of these studies has examined the impact of musical intelligence on students' mathematical connection ability. so far, study on the implementation of ethnomathematics in learning has been carried out by several researchers, including: rubio (2016), muhtadi et al. (2017), kusuma, et al. (2017), supiyati et al. (2019), hartinah et al. (2019), peni and baba (2019), and vitoria and monawati (2020). in these studies, it shows that ethnomathematics has been implemented in learning as an effort to improve student learning outcomes in mathematics. however, so far there has been no study that combines ethnomathematics with the mozart effect. because this study involves the mozart effect, researcher is interested to find the relationship between musical intelligence on enhancing certain mathematical abilities, namely mathematical connection ability. studies related to the mozart effect have been conducted by roth and smith (2008), taylor et al. (2012), verusio et al. (2015), and kusuma (2020). based on description above, problems of this study are: (1) is mathematical connection ability of junior high school students who get ethnomathematics and the mozart effect learning better than students who get direct learning? (2) how is the impact of musical intelligence on mathematical connection ability of junior high school students who learn kusuma & dwipriyoko, the relationship between musical intelligence and the enhancement … 194 mathematics by applying ethnomathematics and the mozart effect? (3) how do students’ attitude and motivation towards learning mathematics with application of ethnomathematics and the mozart effect? 2. method the method research strategy used in this study was quasi experiment non equivalent control group. research philosophy used in this study was positivism, because this is an objective research (dwipriyoko, 2020). research approaches used in this study was deductive, because scientific research based on theories. research timeline used in this study was longitudinal, because historical time orientation is forward (dwipriyoko & sari, 2021). data collection was conducted on two classes of the 7th grade students of smpn bojongsoang 1 bandung district, each class containing 30 students. experimental class was treated with ethnomathematics and the mozart effect (em) learning, and control class was given direct learning (dl) treatment. figure 1 shows the class condition of the 7th grade students of smpn bojongsoang 1 bandung district. figure 1. students with mozart's musical background research instruments used were mathematical connection ability test, musical intelligence questionnaire, and questionnaire on students' attitude towards learning ethnomathematics and the mozart effect. the research instrument was tested before use, then it was analyzed. results of testing validity of mathematical connection ability test items volume 10, no 2, september 2021, pp. 191-202 195 using pearson product moment correlation formula show that all the items used are valid. this research is descriptive and inferential. data of mathematical connection ability test results were analyzed in the following steps: 1) normality test using one-sample kolmogorov smirnov; 2) homogeneity test of variance using levene statistical test; 3) hypothesis test using one way anova; 4) normalized gain analysis, to determine the enhancement of mathematical connection ability of students who get em and those who get dl; and 5) regression analysis, to determine the relationship between musical intelligence and the enhancement of mathematical connection ability. data of musical intelligence questionnaire and students' attitude questionnaire on application of ethnomathematics and the mozart effect in mathematics learning were analyzed descriptively. 3. results and discussion 3.1. results of mathematical connection ability test analysis mathematical connection ability test was given at the first meeting (pretest) and the last meeting (post test) to experimental and control class students. pretest questions are the same as post test, there are seven essay questions, with subject of social arithmetic. pretest was given with the aim of determining students' initial ability of both classes in mathematical connections. results of pretest showed that experimental class students got the highest score of 46 and the lowest was 21. control class students got the highest score of 43 and the lowest was 25. mean of experimental class students was 29.17 and control class was 31.23. pretest descriptive statistics are presented in table 1. table 1. pretest descriptive statistics n minimum maximum mean std. deviation pretest_experiment 30 21 46 29.17 5.509 pretest_control 30 25 43 31.23 4.853 valid n (listwise) 30 after normality and homogeneity test were carried out and it was stated that two sample groups were normally distributed and homogeneous, then mean difference test of initial mathematical connection ability was carried out using one-way anova. results of initial ability mean difference test are presented in table 2. table 2. result of mean difference test of initial mathematical connection ability sum of square df mean square f sig. between groups 64.067 1 64.067 2.335 0.132 within groups 1591.533 58 27.440 toal 1655.600 59 table 2 show the p-value of 0.132 > α, so there is no mean difference between initial mathematical connection ability of experimental class and control class students. after the treatment was given to students of experimental class (em) and students of control class (dl) for five lectures, students of both classes were given post test of their mathematical kusuma & dwipriyoko, the relationship between musical intelligence and the enhancement … 196 connection ability. the test is given with aim of seeing the impact of em and dl treatment on students' mathematical connection abilities. results of post test showed that the highest score achieved by experimental class students was 95 and the lowest score was 35. the highest score achieved by control class students was 67 and the lowest was 33. mean achieved by experimental class students was 61.57 and control class was 49.50. descriptive statistics in detail can be seen in table 3. table 3. post test descriptive statistics n minimum maximum mean std. deviation post test_experiment 30 35 95 61.57 18.822 post test_control 30 33 67 49.50 10.312 valid n (listwise) 30 after normality and homogeneity test were carried out and it was stated that two sample groups were normally distributed and homogeneous, mean difference test of mathematical connection ability was carried out using one-way anova. results of post test mean difference test are presented in table 4. table 4. result of mean difference test of mathematical connection ability sum of square df mean square f sig. between groups 2184.067 1 2184.067 9.484 0.003 within groups 13356.867 58 230.291 toal 15540.933 59 table 4 show the p-value is 0.003 < α, so there is mean difference between mathematical connection ability of experimental class students and control class students. that is, mathematical connection ability of students who get ethnomathematics and the mozart effect with students who receive direct learning is significantly different. 3.2. results of normalized gain analysis normalized gain analysis was carried out to determine the enhancement of mathematical connection ability of experimental class students (em) and control class students (dl). table 5 presents descriptive statistics of normalized gain. table 5. descriptive statistics of normalized gain n minimum maximum mean std. deviation gain_experiment 30 0.16 0.91 0.4738 0.23098 gain_control 30 0.11 0.42 0.2719 0.10213 valid n (listwise) 30 table 5 showed that mean gain of experimental class is 0.4738 and control class is 0.2719. it means, there is a tendency that implementation of ethnomathematics and the mozart effect in mathematics learning increases the variability of students' mathematical connection ability. volume 10, no 2, september 2021, pp. 191-202 197 after normality and homogeneity test were carried out, and it was stated that two sample groups were normally distributed and homogeneous, then mean difference test of normalized gain was carried out using one-way anova. results of mean difference test of normalized gain are presented in table 6. table 6. result of mean difference test of normalized gain sum of square df mean square f sig. between groups 0.611 1 0.611 19.167 0.000 within groups 1.850 58 0.032 toal 2.461 59 table 6 showed that the p-value (sig.) is 0.000 < α, there is mean difference between gain of mathematical connection ability of experimental class students and control class students. therefore, it can be said that gain (enhancement) of students’mathematical connection ability who get learning ethnomathematics and the mozart effect, with students who get direct learning, is significantly different. the enhancement of mathematical connection ability achieved by students who get ethnomathematics learning and the mozart effect (em) is due to em can stimulate the brain to compose patterns that produce meaning by linking the concepts learned with real life experienced by students (johnson, 2002). in addition, application of the mozart effect in mathematics learning strengthen the brain nervous system, so it can improve memory and make students more focus on material taught by the teacher. 3.3. results of musical intelligence questionnaire analysis experimental class students are asked to fill out a likert scale questionnaire after all learning activities on the subject of social arithmetic have been completed and post test have been carried out, to find out the impact of musical intelligence on mathematical connection ability of students who get ethnomathematics and the mozart effect learning. the questionnaire contains 30 statements. results of students’ musical intelligence questionnaire analysis are presented in table 7. table 7. results of musical intelligence questionnaire analysis unstandardized coeff. unstandardized coeff. beta t sig. model b std. error 1 (constant) -33.656 4.571 -7.363 0.000 musical int 1.438 0.068 0.970 21.199 0.000 table 7 showed that regression coefficient is 1.438, so it can be said that musical intelligence has positive impact on students' mathematical connection ability. because the p-value (sig.) is 0.000 < α, there is an effect of musical intelligence on students' mathematical connection ability. this results match with rifqi et al. (2021) article about mathematics skills with high logical-mathematical and musical intelligence could be related. kusuma & dwipriyoko, the relationship between musical intelligence and the enhancement … 198 3.4. results of students’ attitude questionnaire analysis for obtaining information about students’ attitude towards learning mathematics using ethnomathematics and the mozart effect (em), experimental class students were asked to fill out a likert scale questionnaire. the questionnaire contains 30 statements and is given after all learning activities have been completed. recapitulation of results of students’ attitude questionnaire analysis is presented in table 8. table 8. recapitulation of results of students’ attitude questionnaire analysis score frequency 80.00 – 86.00 5 87.00 – 92.00 8 93.00 – 98.00 14 99.00 – 104.00 3 sum 30 mean score 92.42 table 8 show that those who get scores between 93 and 98 (from maximum score of 120) are almost half of experimental class students. mean score achieved by students of experimental class was 92.42. it means, students have positive attitude towards learning mathematics by applying ethnomathematics and the mozart effect. these results are supported by the findings of ponte et al. (1994) that students' attitudes in learning mathematics are one of the factors that underlie student achievement in mathematics, so student achievement is also influenced by student attitude towards learning. 4. conclusion mathematical connection is one of mathematical abilities that exist in every students, which must be explored and developed. this ability can be explored and developed by applying appropriate learning approaches and in accordance with academic ability and cognitive development of students. based on the results of research and discussion, it was found that mathematical connection ability of junior high school students who received ethnomathematics and the mozart effect learning was better than students who received direct learning. it can be seen from post test mean score of experimental class which is greater than control class, and gain average of experimental class is greater than control class. because mathematical logic intelligence and musical intelligence have one tendency in common, namely being able to think conceptually about patterns, it is necessary to study more deeply the impact of musical intelligence on students’ mathematical connection ability when they learn mathematics by applying ethnomathematics and the mozart effect. results showed that musical intelligence has a positive impact on mathematical connection ability of junior high school students. it means, there is a relationship between musical intelligence and the enhancement of mathematical connection ability. students’ achievement in mathematics is influenced by several factors, one of which is students’ attitude and motivation towards learning. students' attitude and motivation regarding emotions related to mathematics and their beliefs in mathematics. the results of this study indicate that students have a positive attitude and more motivated towards learning with application of ethnomathematics and the mozart effect. volume 10, no 2, september 2021, pp. 191-202 199 acknowledgments the author would like to thank the center of science and technology studies of fmipa unpad for providing support in this research, also to mr. cucu dermawan, s.pd., m.m.pd. as the principal of smpn 1 bojongsoang, bandung district, and mrs. dra. siti sa'adah, m.m.pd. as mathematics teacher, who has helped and facilitated the researcher. references campbell, d. 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(2019). core model on improving mathematical communication and connection, analysis of students’ mathematical disposition. international journal of instruction, 12(4), 639-654. https://doi.org/10.29333/iji.2019.12441a https://doi.org/10.1080/10790195.2012.10850354 https://doi.org/10.9734/bjmmr/2015/17192 https://doi.org/10.1088/1742-6596/1460/1/012021 https://doi.org/10.29333/iji.2019.12441a kusuma & dwipriyoko, the relationship between musical intelligence and the enhancement … 202 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.238 37 the enhancement of mathematical problem solving ability of senior high school students through quantum learning julita graduate student of indonesian education university, indonesia julita33@yahoo.co.id received: september 21, 2016 ; accepted: january 4, 2017 abstract this study is aimed to examine the quality of quantum learning imfluence toward the enhancement of mathematical problem solving ability of senior high school students, both viewed entirely and based on mathematical initial ability (mia) category. in particular, this study is aimed to examine enhancement difference of students’ mathematical problem solving ability in a whole and in each level of mathematical initial ability (high, medium and low) between students who receive quantum learning and students who receive conventional learning. this study use experimental quasi with pretests-posttest control group design. population of this study are senior high school students in bogor city. data is obtained through problem solving ability test and mathematical initial ability data. the result of study showed that students who receive quantum learning have enhancement of mathematical problem solving ability which is higher than students who receive conventional learning. there is no difference enhancement of mathematical problem solving ability both entirely and in each level of mathematical initial ability, except for students with high level of initial mathematical ability. keywords: mathematical problem solving ability, quantum learning abstrak penelitian ini bertujuan untuk mengkaji kualitas pengaruh pembelajaran quantum terhadap peningkatan kemampuan pemecahan masalah matematis siswa sma, baik ditinjau secara keseluruhan maupun berdasarkan kategori kemampuan awal matematis (kam). secara khusus, penelitian ini bertujuan untuk mengkaji perbedaan peningkatan kemampuan pemecahan masalah matematis siswa secara keseluruhan dan setiap tingkat kemampuan awal matematis (tinggi, sedang, dan rendah) antara siswa yang mendapatkan pembelajaran quantum dengan siswa yang mendapatkan pembelajaran secara konvensional. penelitian ini menggunakan metode quasi eksperimen dengan desain kelompok kontrol pretes-postes. populasi penelitian siswa sma di kota bogor. data diperoleh melalui tes kemampuan pemecahan masalah dan data kemampuan awal matematis. hasil penelitian menunjukkan bahwa siswa yang mendapatkan pembelajaran quantum memiliki peningkatan kemampuan pemecahan masalah matematis lebih tinggi daripada siswa yang mendapatkan pembelajaran secara konvensional. baik secara keseluruhan maupun setiap tingkat kemampuan awal matematis, kecuali siswa dengan tingkat kemampuan awal tinggi tidak terdapat perbedaan peningkatan kemampuan pemecahan masalah matematis. kata kunci: kemampuan pemecahan masalah matematis, pembelajaran quantum how to cite: julita (2017). the enhancement of mathematical problem solving ability of senior high school students through quantum learning. infinity, 6 (1), 37-50. julita, the enhancement of mathematical problem solving ability … 38 introduction problem solving is integral part of mathematics learning process which require students to think. according to sabandar (2008), thinking process can be triggered and developed through challenging and non routine mathematical problems. in non routine problem, its solution problem need further thinking because its solution procedure is not the same with those taught in class. sumiati and asra (2009) argued that problem solving process give opportunity to students to actively involved in studying, searching, finding by themselves the information to be processed into concept, principle, theory or conclusion. besides, problem solving is ability to process the information to make decision in problem solving. student ability in processing information to solve the problem is varied depended on background of student ability in using reasoning, that is ability to see causal effect relation to draw conclusion. problem solving ability is ability which shows directed thinking process to generate ideas or develop the possibility to solve problems solved to achieve desired goal (sumiati and asra, 2009). according to santrock (2009), problem solving is finding a right way to achieve a goal. based on some opinions which had been explained, it can be synthesized that problem solving ability is ability to process information and arrange various alternative of solutions to achieve desired goal. besides, problem solving is solution of non routine problem and higher level thinking process, and really needed in mathematics learning. according to polya (1973), the steps in mathematical problem solving are: understanding the problem, arranging the plan of problem solving, implementing the plan which had been arranged, and rechecking the correctness of problem solving result which had been done. in first step, student should understand clearly the problem faced and it will easier by drawing a picture, diagram, or table of known things. in next step, student find the relation between given information and unknown information which will enable student to arrange the plan of problem solving. student can decide the way of problem solving which is suitable and use given information or unknown information to arrange new information. in third step, students implement the plan which had been arranged in second step, that is implement the problem solving. in implementing the plan, students should check each stage of plan and write the detail which prove that each stage is correct. students can solve the problem in accord with steps of problem solving they use with correct result. the last step is recheck the steps of problem solving which had been done. according to national council of teachers of mathematics (nctm, 2012) that each student has mathematical problem solving ability, if that student is able to apply and adjust various strategies which are appropriate to solve the problem, able to solve the problem occurred in mathematics and everything which involve mathematics in another context, able to build ne w mathematical knowledge through problem solving, and able to observe and reflect mathematical problem solving. the process to determine solution of a problem require thinking ability. the ability to collect information and data, express the argument, determine the supporting theory, determine the plot of problem solving is a process which enable students to be able to solve the problem volume 6, no. 1, february 2017 pp 37-50 39 (soekisno, 2015). this is in accord with the goal of problem solving which expect students to have problem solving ability which comprise ability to understand the problem, arrange mathematical model, solve the problem and interpret solution obtained. this presuppose that students should have problem solving ability to master mathematics. in fact, the result of pisa (program for international student assessment) which measure students mathematical ability in various countries found that the level of mathematical problem solving ability of indonesian students is very less satisfying (still low). according to indonesia pisa center, in 2012 the rank of indonesia in mathematics field down to 64 th of 65 participating countries from 61 st rank in 2009. one factor which result in low of indonesian students’ achievement in pisa is lack of problem solving ability in non routine or high level problem. the study conducted by ibrahim (2008) also found that mathematical problem solving ability of secondary school and higher education students in indonesia is still low. besides, the ability of senior high school students in mathematical problem solving in bogor city also had not yet showed satisfying achievement. in senior high school mathematics olympic in bogor city level, students who occupied top five rank are senior high school students with high category in bogor city, this condition shows that mathematical problem solving ability of senior high school students with medium and low category in bogor city is still low. mathematical problem solving ability which had not meet the expectation shows that students had not been able to develop their thinking ability optimally so mathematical learning process is needed to be improved. students will not able to solve the problem if they don’t posses many concept, theorem or rule from various aspects. another ability which should be possessed by students in problem solving is ability to identify the problem, namely: what the problem is, where the problem come from, what of type and nature of problem, why the problem is solved, how to solve the problem, and for what aim the problem is solved (thoifuri, 2008). the effort to enhance students’ mathematical problem solving ability depend on teacher’s ability to implement learning process which is effective in school. it is expected that teacher implement learning process which is inspired, enjoyable, challenging, and motivate students to become autonomous learner and capable to solve the problem in their span of life. teacher needs to do change toward learning process he/she implement. teacher habit to implement mathematical learning process which only require students to memorize ways or formulation which had been taught in solving the problem need to be changed. students do not need to solve the problem with only one way exampled by teacher because it make students’ thinking ability not developed and effected on their mathematical problem solving ability. besides, mathematics learning process had not involved students to participate actively, that is still using lecture method for all learning materials. this makes mathematics become a boring and unpleasant subject. learning which can create comfortable and enjoyable atmosphere and which optimize students’ problem solving ability is through quantum learning. according to deporter & hernacki (1999), quantum learning is learning which try to create conducive learning atmosphere which is comfortable and enjoyable by combining self confidence, study skill, and communication skill. quantum learning arise students’ interest toward learning by ambak (apa manfaat bagiku) or what benefit for me, that is give learning motivation to julita, the enhancement of mathematical problem solving ability … 40 students by choosing mentally between benefit and consequence of decision, and create effective learning environment. figure 1. the scheme framework of quantum learning quantum learning process is process of active student learning which balance left brain and right brain which enable students to combine logical thinking and creative thinking. according to deporter & hernacki (1999), the combination of logical and creative thinking is ability needed in mathematical problem solving and ability to process information and arrange various alternative of solutions to achieve the desired goal. the problem solving done is solving the non routine or unfamiliar problem and higher order thinking process, and it is very important in mathematics learning. one’s ability in solving the problem depend on potential ability (intelligence) he/she posses (skinner in sumiati & asra, 2009). besides, quantum learning process also maximize the potency of student brain in teaching learning process which is active and contextual by increasing togetherness in enjoyable atmosphere. learning atmosphere is said enjoyable if it creates communicative and relax learning (yosodipuro, 2013). the technique which can be done in quantum learning to support this condition can be done by: 1) creating study room which is conducive to build positive suggestive, for example by arranging classroom with good lighting, set background music in class, class wall which is decorated by slogan posters to trigger the spirit, temperature in room which is comfortable, plants placed in classroom, 2) increasing students participation in learning process, 3) teacher not only master teaching material, but also the art which give positive suggestion. one characteristic of quantum learning is humanistic, that is learning which drive students to learn humanly. according to hendriana (2012), the characteristic of students who learn humanly is students who learn by building the meaning of mathematics by themselves by using information or knowledge they just acquire. building the meaning from what is learned by using new information to change, complement or make perfect the understanding which had been inculcated before. herbat (in sumiati & asra, 2009) suggested that, before teacher implement learning process, teacher should first know the level of knowledge which had been possessed by students before, because learning as cognitive process is influenced by their initial knowledge. this is in accord with opinion of ausubel (in cahyo, 2013) about meaningful learning, that is a quantum learning problem solving mia : high medium low left brain right brain balance process creative thinking logical thinking volume 6, no. 1, february 2017 pp 37-50 41 process to relate new information to existing knowledge in students’ cognitive structure and the most important factor influencing learning is students’ initial knowledge. based on that background, it is needed to conduct the study which aims to examine the enhancement of mathematical problem solving ability of students who are taught by quantum learning and students who are taught by conventional learning viewed from a whole students and based on category of students’ mathematical initial ability level (ial), namely high, medium and low level. method this study is conducted by using experimental quasi with pretest-posttest control group design which involve two groups selected in random, namely experiment group and control group. pretest is given to two groups before first learning is started, which aimed to enhance level of students’ initial ability in mathematical problem solving. next, posttest is given in final learning (study) which aimed to find out the enhancement of mathematical problem solving ability (psa) after two groups received learning. experiment group receive quantum learning, whereas control group receive conventional learning. to see more deeply the quality of quantum learning influence toward mathematical psa, this study consider students’ mathematical ial namely high, medium and low level which is taken from the average of daily math test both in experiment class and control class. this study involve three variables, namely independent, dependent, and control. independent variable consist of quantum learning and conventional learning, whereas students’ mathematical psa is dependent variable. students’ mathematical ial included in control variable. population in this study are all students of class x senior high school in bogor city and sman 10 is school which is selected as sample of this study with school qualification is non rsbi in bogor city. the selection of school sample is done in random with lottery method to select one school from seven non rsbi schools. from school which is selected, two classes are taken in simple random as sample of study. this is done because based on information from school staff, the grouping of students in that school not based on ranking. thus, students’ ability in each class is varied. those two classes which had been selected are selected again to decide experiment class and control class. the class which is selected as experiment class is class x-7 with sample size is 39 students, whereas control class is x-8 with sample size is 39 students. in this case, class which is selected is class x based on material tested, namely three dimension is learning material which is taught in class x. instrument of study is set of test items and observation sheet whose level of validity, reliability and distinguishing ability and difficulty index had been measured. instrument used had fulfill validity. data obtained from this study is quantitative study as analysis toward students’ answer for test item of mathematical problem solving ability and is processed by aid of microsoft excel and software spss version 16.0 for windows program. data analysis of study result is done descriptively and inferentially, that is by displaying descriptive data of students’ mathematical psa and its inferential statistic analysis use independent sample t-test (mann whitney test) in confidence level of 5%. julita, the enhancement of mathematical problem solving ability … 42 results and discussion results the distribution of students in experiment class and control class based on mia level is presented in table 1 as follow. table 1. the number of students in class of study based on mia level class level of mia total high medium low experiment 5 27 5 37 control 5 30 3 38 total 10 57 8 75 analysis of mathematical initial ability (mia) data 1. descriptive analysis of mathematical initial ability (mia) data mia data is obtained from the average score of daily test in experiment class and control class. this mia data is taken to find out the equality of students’ mathematical ability average in experiment class and control class, and to group students based on their mia. the description of mia in this study is presented in data descriptive in table 2 as follow. table 2. data descriptive of students’ mia based on learning approach level of ability combination high medium low experiment class sample size 5 27 5 37 average 87,50 74,24 52,00 73,03 deviation standard 3,06 3,87 7,58 10,48 control class sample size 5 30 3 38 average 89,00 72,75 48,33 72,96 deviation standard 3,35 5,55 5,77 10,54 data in table 2 shows that the average and deviation standard of students’ mia in each ial (high, medium and low) for experiment class and control class is relatively the same. as for average and deviation standard of all students for experiment class and control class is relative the same. therefore, it can be concluded that the quality of students’ mia in experiment class and control class in each ial, or all students combined is relatively the same. 2. inferential analysis of mathematical initial ability (mia) data inferential analysis of mia is done to find out the equality of mia average of all students and each ial (high, medium and low) between experiment class and control class. the first step before doing equality test of students’ mia average is doing normality test of mia data in both classes of study based on psa and its combination. volume 6, no. 1, february 2017 pp 37-50 43 the calculation result of data normality test of medium mia psa and its combination is showed in table 3 as follow. table 3. result of normality test of mia data based on medium ial and combination of all samples result of normality test in table 3 shows that all pairs of group of students’ mia with medium ial and its combination have sig. < 0.05, so h0 is rejected. this shows that two groups of learning not all normal distributed, so to test the average equality of students’ mia with medium ial and its combination use mann-whitney test. the summary of average equality test result of students’ mia from two classes of study based on ial and its combination is presented in table 4 as follow. table 4. summary of average equality test result of students’ mia from two classes of study based on ial and its combination group of sample n z asymp. sig. (2-tailled) decision ec cc between ec dan cc with high ial 5 5 -.775 .439 accept h0 between ec and cc with dengan medium ial 27 30 -1.146 .252 accept h0 between ec and cc with low ial 5 3 -.769 .442 accept h0 between ec and cc (combination) 37 38 -.400 .689 accept h0 annotation: ec = experiment class, cc = control class in table 4, it can be seen that asymp. sig. (2-tailleds) of two classes of study (combination) is bigger than 0.05 so ho is accepted. this means that median of students’ mia pretest in experiment class is not different significantly with median of students’ mia pretest in control class. in other word, students’ mia in experiment class (who receive quantum learning) is not different with students’ mia in control class (who receive conventional learning). besides, in table 4 also it can be seen that asymp. sig. (2-tailleds) of each ial is bigger than 0.05, so h0 is accepted. this means that median of students’ mia pretest of each ial in group of sample kolmogorov-smirnov decision db sig. experiment medium 27 0,000 reject h0 combination 37 0,000 reject h0 control medium 30 0,074 accept h0 combination 38 0,040 reject h0 julita, the enhancement of mathematical problem solving ability … 44 experiment class is not significantly different with median of students’ mia pretest of each ial in control class. in other word, students’ mia of each ial in experiment class (who receive quantum learning) is not significantly different with students’ mia of each ial in control class (who receive conventional learning). because students’ mia of two classes and students’ mia of each ial of two classes are not different, then the requirement is fulfilled to give different treatment in each class of study. if there is difference of mathematical psa in the end of learning, then it is as influence from different treatment in each class and not caused by mathematical ability difference before learning. analysis of mathematical problem solving ability (psa) data 1. descriptive analysis of psa data based on learning approach students’ mathematical psa data is obtained from pretest and posttest, then n-gain is calculated. this data is analyzed based on factor of quantum model learning and conventional learning, and mathematical ial of students with high, medium and low category. students’ mathematical psa data which is based on learning approach is presented in table 5 as follow. table 5. data descriptive of psa based on learning approach class descriptive statistic pretest posttest n-gain experiment sampel size 37 37 37 average 18,00 27,41 0,29 deviation standard 6,41 5,01 0,11 control sample size 38 38 38 average 19,89 23,89 0,14 deviation standard 6,24 6,80 0,10 annotation : maximum ideal score of psa test is 50 descriptive statistic data shows that enhancement of students’ psa who receive quantum learning is higher compared to students who receive conventional learning. in table 5 it can be seen that psa pretest average of students who receive quantum learning is 18.00 which is relatively the same with students who receive conventional learning, that is 19.89. after learning process, students’ psa is enhanced. this can be seen from posttest average in students who receive quantum learning which is increased to become 27.41, that is enhanced of 0.29, whereas students who receive conventional learning is increased to become 23.89, that is enhanced of 0.14. according to hake (1998), the enhancement of 0.29 and 0.14 is fall in low category. 2. descriptive analysis of psa data based on learning approach and mathematical ial psa data is based on learning approach and students’ mathematical ial is presented in table 6. that descriptive statistic data shows that psa enhancement in all mathematical ial of students who receive quantum learning is higher than students who receive conventional learning volume 6, no. 1, february 2017 pp 37-50 45 . table 6. descriptive data of psa based on learning approach and ial high medium low pretest posttes n-gain pretest posttes n-gain pretest postes n-gain experiment class sample size 5 5 5 27 27 27 5 5 5 average 26,40 36,00 0,40 18,15 26,89 0,27 9,60 21,20 0,29 deviation standard 4,34 4,00 0,15 5,29 3,00 0,09 0,89 2,28 0,06 control class sampel size 5 5 5 30 30 30 3 3 3 average 27,60 33,60 0,27 19,93 23,80 0,12 6,67 8,67 0,05 deviation standard 2,19 0,89 0,04 4,47 4,25 0,10 3,06 3,06 0,00 annotation: maximum ideal score of psa test is 50 in table 6, it can be seen that before learning is implemented (pretest data), mathematical psa for students with high ial from two classes of study is relatively the same. after learning is implemented, there is enhancement of mathematical psa in each level of students’ ability. this is happened in experiment class and control class. learning with quantum model in students with high mathematical ial has enhancement of 0.40, whereas students with medium and low iai also has enhancement of 0.27 and 0.29. mathematical psa with conventional learning for students with high ability has enhancement of 0.27, whereas for students with medium and low ial has enhancement of 0.12 and 0.05. psa enhancement of each ability level of students who receive quantum learning and conventional learning is categorized low, except for students with high ial who receive quantum learning is categorized medium (hake, 1998). 3. inferential analysis of mathematical problem solving ability (psa) data data analysis of mathematical psa is continued by statistic test toward difference of psa enhancement of two groups of learning which is done based on all samples combined and students’ mathematical ial (high, medium and low). before doing that statistic test, normality test is done first toward data of pretest, posttest and n-gain of mathematical psa of two group of learning based on mathematical ial and combination of all samples as requisite to choose appropriate statistic test. normality test only done on mathematical psa of all samples combined and ial only. this test cannot be dome for data of mathematical psa with high and low ial because data available is very little, that is less than 10. statistic test toward difference enhancement of mathematical psa with high and low ial use mannwhitney test. hypothesis of normality test for mathematical psa of all sample and medium ial are as follow: h0 : data is normal distributed. h1: data is not normal distributed. julita, the enhancement of mathematical problem solving ability … 46 criteria of hypothesis test based on p-value (sign.), ho is rejected if sig. < α, for α = 0.05 and h0 is accepted in another thing. the result of normality test of pretest, posttest and n -gain data which use kolmogorov-smirnov is presented in table 7 as follow. table 7. result of data normality test of pretest, posttest and n-gain of mathematical psa based on medium ial and combination of all sample group of sample kolmogorov-smirnov db pretest posttest n-gain sig. decision sig. decision sig. decision experiment medium 27 0,093 accept h0 0,069 accept h0 0,200 accept h0 combination 37 0,068 accept h0 0,128 accept h0 0,200 accept h0 control medium 30 0,179 accept h0 0,037 reject h0 0,000 reject h0 combination 38 0,200 accept h0 0,169 accept h0 0,000 reject h0 in table 7, it can be seen that all data have sig. > 0.05 which means that h0 is accepted, except for posttest data and n-gain of all samples combined and medium ial in control class. this shows that mathematical psa data for two classes of study for all samples combined and medium ial is normal distributed, except for posttest data and n-gain of all samples combined and medium ial in control class is not normal distributed. in the next step, statistic test is done toward psa pretest data of all samples combined and each students’ ial for two classes of study to find out the equality of its average. for pretest data of all samples combined and medium ial use independent sample t-test) because two groups of data compared are independent. in independent-sample t-test, there are two value of significance (sig.), that is sig. with assumption that variance of two groups of data compared are homogenous and sig. with assumption that variance of two groups of data are not homogenous, so homogeneity test needs to done toward each pair of psa mathematical data from class of study for all samples combined and medium ial. homogeneity test toward variance of two groups of data use levene test (levene statistic) with hypothesis formulation as follow: h0 : variance of two groups of homogenous mathematical psa data h1: variance of two groups of non homogenous mathematical psa data. the criteria of testing is based on probability value (sig.). h0 is rejected if sig. < α, for α = 0.05 and h0 is accepted in another thing. the calculation result of homogeneity test of mathematical psa data variance for two classes of study based on medium ial and combination of all samples is presented in table 8 as follow. volume 6, no. 1, february 2017 pp 37-50 47 table 8. result of homogeneity test of mathematical psa pretest data variance for two classes of study based on medium ial and combination of all samples initial ability level n f db1 db2 sig. decision medium 57 1,958 1 55 0,167 accept h0 combination 75 0,490 1 73 0,486 accept h0 in table 8, it can be seen that probability value (sig) > 0.05 for medium ial and combination of all samples, so h0 is accepted. this means that mathematical psa data between experiment class and control class in medium ial and combination of all samples have homogeneous variance. after homogeneity test is done, then statistic test is done toward equality of mathematical psa average in two groups of learning based on ial and combination of all samples. result of equality test of mathematical psa from two classes of study based on ial and combination of all samples is presented in table 9 as follow. table 9. result of equality test of mathematical psa average from two classes of study based on ial and combination of all samples ial n statistics decision mann-whitney test independen sample t-test z asymp. sig.(2tailled) t db sig. (2-tailled) high 10 0,110 0,913 accept h0 medium 57 -1,380 55 0,173 accept h0 low 8 -1,537 0,124 accept h0 combined 75 -1,297 73 0,199 accept h0 in table 9, it can be seen that probability values (asymp, sig and sig.) > 0.05 for each ial and combination of all samples, so h0 is accepted. this means that there is significant difference between mathematical psa average of students in experiment class and control class for each ial and also combination of all samples. if there is enhancement difference of mathematical psa in the end of learning, then it is influence of different treatment in each class and not caused by difference of mathematical ability before learning. based on information which had been obtained, statistic test is done toward enhancement difference of mathematical psa toward students of two groups of learning and enhancement difference of mathematical psa in each students’ ial of two groups. this statistic test use ngain data of mathematical psa of students in two classes of study. based on earlier explanation about data for high and low ial which is too small and n-gain normality data of medium ial and combination of all samples, then statistic test is done by using mann whitney test. the summary of test result of mathematical psa average difference between students from two classes of study based on ial and its combination is presented in table 10 as follow. julita, the enhancement of mathematical problem solving ability … 48 table 10. the summary of test result of mathematical psa average difference of two classes of study based on ial and its combination no. hypothesis group of sampel n z asymp. sig. (1-tailled) decision ec cc 2 between ec and cc with high ial 5 5 -1,293 0,098 accept h0 3 between ec and cc with medium ial 27 30 -4,597 0,000 reject h0 4 between ec and cc with low ial 5 3 -2,236 0,013 reject h0 1 between ec and cc (combination) 37 38 -5,158 0,000 reject h0 annotation: ec = experiment class, cc = control class in table 10 it can be seen that value of asymp. sig (1-tailled) of two classes of study (combination) is smaller than 0.05, so h0 is rejected. this means that median of mathematical psa n-gain of students with medium and low ial in experiment class is higher significantly than median of mathematical psa n-gain of students with medium and low ial in control class (who receive conventional learning). besides, in table 10 also it can be seen that value of asymp. sig. (1-tailled) of high ial is bigger than 0.05, so h0 is accepted. this means that median of mathematical psa n-gain of students with high ial in experiment class is not significantly higher than median of mathematical psa n-gain of students with high ial in control class. in other word, it can be concluded that students with high ial in experiment class (who receive quantum learning) have mathematical psa enhancement which is higher than students with high ial in control class (who receive conventional learning). discussion based on analysis result of data descriptive, it can be known that the average of mathematical psa of all students and in each ial is enhanced, both for experiment class and control class. this shows that implementation of learning in those two classes had been able to stimulate development of students’ mathematical psa. this condition is normal because it is an effect of learning process. result of statistic test in table 10 shows that mathematical psa of students in all ial who receive quantum learning is enhanced significantly higher than students who receive conventional learning, except for students with high ial. mathematical psa of students with high ial is not enhanced significantly compared to students who receive conventional learning. this is because of students with high ial will ready to receive learning with whatever methods (sumiati & asra, 2009), so there is no significant difference of mathematical psa enhancement between students with high ial who receive quantum learning and students who receive conventional learning. besides, result of statistic test in table 10 also shows that mathematical psa in group of students who receive quantum learning is enhanced significantly higher than group of students who receive conventional learning. this means that in a whole, quantum learning is better in enhancing students’ mathematical psa compared to conventional learning. volume 6, no. 1, february 2017 pp 37-50 49 the success of teaching learning process is very influenced by potency of all people involved and interaction created in class. the higher of potency of all people involved and the more optimal of interaction activity in learning process with conducive and enjoyable atmosphere, then the higher will be the effectiveness of teaching learning process occurred. according to reigeluth (uno, 2007), the effectiveness of teaching usually measured by level of students’ achievement in teaching goal which had been determined. conclusion based on data analysis and discussion of study result in earlier chapter, the conclusions are obtained as follow: 1. the enhancement of mathematical psa in students who receive quantum learning is higher than students who receive conventional learning. 2. the enhancement of mathematical psa in students with medium and low ial who receive quantum learning is higher than students who receive conventional learning. whereas, there is no significant difference of mathematical psa enhancement between students with high ial who receive quantum learning and students who receive conventional learning. references cahyo, a. n. (2013). panduan aplikasi teori-teori belajar mengajar teraktual dan terpopuler. yogyakarta: diva press. deporter, b., & hernacki, m. (1999). quantum learning: membiasakan belajar nyaman dan menyenangkan. bandung: penerbit kaifa. hake, r. r. (1998). interactive engagement versus traditional methods: a six-thousandstudent survey of mechanics test data for introductory physics courses. american journal physics, 66, 64-74. hendriana, h. (2012). pembelajaran matematika humanis dengan metaphorical thinking untuk meningkatkan kepercayaan diri siswa. infinity, 1(1), 90-103. ibrahim. (2008). pembelajaran matematika untuk meningkatkan kemampuan pemecahan masalah matematis siswa sekolah menengah atas. jurnal pendidikan matematika, 2, 90-99. nctm. (2012). 2011-2012 nctm program report of spa assessments. reston, va: the national council of teachers of mathematics inc. polya, g. (1973). how to solve it: a new aspect of mathematical method. new jersey, usa: princeton university press. sabandar, j. (2008). thinking classroom dalam pembelajaran matematika di sekolah. simposium internasional. bandung: universitas pendidikan indonesia. santrock, j. w. (2009). psikologi pendidikan. jakarta: penerbit salemba humanika. soekisno, r. b. (2015). pembelajaran berbasis masalah untuk meningkatkan kemampuan argumentasi matematis mahasiswa. infinity, 4(2), 120-139. sumiati, & asra. (2009). metode pembelajaran. bandung: cv wacana prima. thoifuri. (2008). menjadi guru inisiator. semarang: rasail media group. julita, the enhancement of mathematical problem solving ability … 50 uno, h. b. (2007). model pembelajaran menciptakan proses belajar mengajar yang kreatif dan efektif. jakarta: pt. bumi aksara. yosodipuro. (2013). siswa senang guru gemilang. strategi mengajar yang menyenangkan dan mendidik dengan cerdik. jakarta: pt. gramedia pustaka utama. infinity journal of mathematics education p–issn 2089-6867 volume 5, no. 2, september 2016 e–issn 2460-9285 doi 10.22460/infinity.v5i2.215 83 profile of cognitive structure of students in understanding the concept of real analysis wahyu widada the graduate study program mathematics education at the university of bengkulu, indonesia wahyu.unib@gmail.com received: may 24, 2016; accepted: july 15, 2016 abstract the purpose of this research is to describe proil cognitive structure of students in understanding the concept of real analysis. this research is part of the research development of the theory of cognitive structure of students mathematics education program at the university of bengkulu. the results of this research are: 1)there are seven models decompositions of genetic students mathematics education reviewed based on the srp model about the concepts of real analysis namely pra-intra level, level intra, level semi-inter, level inter, level semi-trans, trans level, level and extended-trans (only theoretic level while empirically not found); 2) there are six models decompositions of genetic students mathematics education reviewed based on ka about the concepts of real analysis namely level 0: objects of concrete steps; level 1: models semi-concrete steps; level 2: models theoretic; level 3: language in domain example; level 4: mathematical language; level 5: inferensi model. profile of cognitive structure of mathematics education student at the university of bengkulu is 6.25% students located on the basic level (pra-intra level with concrete objects), there is 8.75% students located at level 0 (intra level with concrete objects), there are 15,00% students located at level 1 (semi-level inter with semi-concrete model), there are 33.75 percent students located on level 2 (level inter with theoretical model), there are 22.50 percent students located at level 3 (semi-trans level with the bible in domain example), there are located on the student percent during the level 4 (trans level with the language of mathematics), and there are 0 percent students located at level 5 (level extended-trans with inferensi model). students education mathematics at the university of bengkulu pembangunnya element is functional can achieve trans level, students will be able to set up activities and make the algorithm that formed the concept/principles with the right. functional students can also perform the process of abstraction using the rules in a system of mathematics. keywords: cognitive structure of the understanding of the concepts, profile abstrak tujuan penelitian ini adalah mendeskripsikan profil struktur kognitif siswa dalam memahami konsep analisis real. penelitian ini merupakan bagian dari penelitian pengembangan teori struktur kognitif mahasiswa program pendidikan matematika fkip universitas bengkulu. hasil penelitian ini adalah: 1) terdapat tujuh model dekomposisi genetic mahasiswa pendidikan matematika ditinjau berdasarkan model srp tentang konsep-konsep analisis real yaitu level pra-intra, level intra, level semi-inter, level inter, level semi-trans, level trans, dan level extended-trans (hanya level teoretik sedangkan secara empiric tidak ditemukan); 2) terdapat enam model dekomposisi genetic mahasiswa pendidikan matematika ditinjau berdasarkan ka tentang konsep-konsep analisis real yaitu level 0: objek-objek konkret; level 2: model-model semi-konkret; level 2: model-model teoretik; level 3: bahasa dalam domain contoh; level 4: bahasa matematik; level 5: model inferensi. profil struktur kognitif mahasiswa pendidikan matematika fkip universitas bengkulu adalah 6,25% mahasiswa berada pada level dasar (level pra-intra dengan objek konkret, terdapat 8,75% mahasiswa berada pada level 0 (level intra dengan objek konkret), terdapat 15,00% mahasiswa berada pada level 1 (level semi-inter dengan model semi-konkret), terdapat 33,75% mahasiswa berada pada level 2 (level inter dengan model widada, profile of cognitive structure of students in understanding … 84 teoritis), terdapat 13,75% mahasiswa berada pada level 4 (level trans dengan bahasa matematika), dan terdapat 0% mahasiswa berada pada level 5 (level extended-trans dengan model inferensi. mahasiswa pendidikan matematika fkip universitas bengkulu yang unsur pembangunnya adalah fungsional dapat mencapai level trans, mahasiswa tersebut mampu mengatur, menyusun kegiatan dan membuat algoritma sehingga terbentuk konsep/prinsip dengan tepat. mahasiswa fungsional juga dapat melakukan proses abstraksi dengan menggunakan aturan—aturan dalam suatu system matematika. kata kunci: struktur kognitif, pemahaman konsep, profil how to cite: widada, w. (2016). profile of cognitive structure of students in understanding the concept of real analysis. infinity, 5 (2), 83-98. introduction the understanding of the concepts and mathematical strategic competence still considered optimal yet owned students (afrilianto, 2012), and needs to be repaired through the implementation of appropriate learning methods (darusman, 2014). to overcome this, the berbagai mathematics educators efforts in improving the ability learners in understanding the concepts of mathematics continues to be done. dimaksdu efforts done start from the elementary education up to higher education. one of the efforts is the development of students mindset to be able to do the process of abstraction, generalisation and idealisation in the achievement of the concept of mathematics. through the development of this mindset, students can define with both the concepts of mathematics with the pattern defining "name" concept is "genus proksimum" that "special distinguished". with the pattern defining, makes it easier for students to understand the concept of mathematics. the understanding of the concept of mathematics done system processing information students through the understanding of abstract ideas using media/objects of concrete steps to classified into the example/non-examples from the concept. according to the hendriana (2012), konsep abstract concepts organized through think metaforik, stated in concrete things based on the structure and ways of reasoned based system of melting texture of the motor that is called with the conceptual metaphors. widada & herawaty (2016) stated that pemahaman about the concept of mathematics is the result of the construction or reconstruction of the objects of mathematics. construction or reconstruction was done through the activities in the form of action mathematics, process, objects organized in a scheme to solve a problem. to know the understanding of the concept of students, can be done by analysis of the process of kongnitif students through an analysis of the genetic decompositions. analysis of the genetic decompositions is a be from the theory apos (action, patterns, object, and schema) (dubinsky, 2000). he said the apos theory is a theory about how the possibility of continuous konstruktivis achievement/learning a concept or principle of mathematics which can be used as a elaborasi about mental construction of action, process objects and the scheme. listen to the explanation above, in mathematics teaching requires creativity educators in optimizing cognitive process students to take advantage of the ability to start/basis which belongs to the students. in line with this hendriana (2012) stated that mathematics teaching must pay attention to the aspects of basic skills students. basic skills students this can be seen from the perception of the students about an object of mathematics. on the basic level of students, usually the perception of students on a mathematics objects (subject) is the initial capital of understanding he on advanced mathematical skills. volume 5, no. 2, september 2016 pp 83-98 85 hits widada (2006), there are students of misconceptions about the concept of fractions. they have the misconception that every number that can be written in a/b with a,b numbers round and b≠0 is fractions. misconceptions are found in a scientific meeting prodi even written in student bachelor theses, like 6/3 he call with fractions. but when asked what 2 fractions?", so too complex not fractions. this shows the extent of the inconsistency the conception is in the process of kognitifnya, when mention 6/3 fractions once asked about 2 that equivalent with 6/3, he answered not fractions. the above conception, indicated that they dim between the concept of fractions with the concept of rational numbers. should there are still constraints that must be met in order for the number of rational called with fractions, namely (a,b) = 1 or simply a not increments of b. this constraint as diferensia spesifika (special distinguished) so that the association of rational numbers can be partitioned into two aggregate namely the association of round numbers and aggregate fractions (often called with the association bipartisi). this means that every fractions is the number of rational, but no rational numbers not fractions and numbers would round numbers. the same thing was also found when the students met with the concept of the limit function. on the concept of the limit of f(x) for x near a, there are antecedents that 0<|x -a|<∂. they argue that more than 0 is not necessary because the absolute value of x-a sufficient (must have been more than 0). even though the sign more than not only suggests that x of a more than 0, but more than the x≠0. but not only was found also students who have no conception of the limit function with -∂, moreover if we take on the concept of the neighborhood, then the students more trouble. this may indicate that the level of analysis of the student to interkoneksitas between absolute value scheme and the core of the concept of the limit itself has not been an integrated sequence. so to reach the mature scheme needed cognitive process that right. when this process is achieved, then the system processing information (short-term memory or working memory) will perform the task in the form of encoding to mature scheme about the concept of the limit function that is stored in a neurons in long-term memory. mature schema which is stored is the main basic competency that can selectively-triggers the other competency components to be able to operate a performance that means. the process of abstraction is a mathematics activity about the reorganization of the vertically (vertical mathematizing) from the object of mathematics that previous dikonstruk (previous schema) on a new structure. mereorganisasi on the new structure of the objects of mathematics includes creating new hypothesis (conjecture), find or find back (reinvention) the object of mathematics is more complex and new strategies for problem resoursces'. vertical matematisasi is an activity placing objects of mathematics together, structured organized and developed on the objects of the other more abstract or more formal than originally. idealisation occurs when we are dealing with an object that was not perfect (unperfect), and considered perfect. as the line, we image is not too straight, field of drawn not too flat, then the image of the line we assume that the straight path and image of our field flat. while generalisations is a process to find the object of mathematics (concept/principles) general, not just the conception that has not been accepted the truth in the structure of mathematics. based on the explanation above, cognitive process students in mathematics teaching focuses on the ability of abstraction, idealisation and generalisations to reach the level of widada, profile of cognitive structure of students in understanding … 86 understanding of the concept of/principles cautiously. therefore, in this article discussed about how the cognitive structure of students/students in understanding the concept in real analysis courses. the structure of the knowledge representation is a development network/scheme about the fact, konsi cognitive things or situations that arise from an experience. the structure is the scheme, frame, or scripts. each network can be mapped into a level in level-level triad (piaget & garcia, 1989). based on the results of research grants researchers (widada, 2010) a characteristic of the structure of the epresentasi knowledge (srp) based on the level of a new level of triad (i.e. extended triad level++). competency grants researchers previous year (widada, 2009) about the theory and the model of teaching mathematics based on the level of triad++ obtained an additional new levels namely pre-level intra. the results of the earlier research as fundamental research (widada & herawaty, 2005; widada, 2006), and the results of research widada (2001, 2002, 2003, 2004), in the theoretic and empiric found that on pelevelan scheme structure (triad) students in learning mathematics, there is a new level in between the level intra and inter level i.e. semiinter level and there are new levels in between the level inter and trans level, namely semitrans level. this means that there are five development level student scheme in learn mathematics namely level 0 (intra level), level 1 (semi-level inter), level 2 (inter level), level 3 (semi-trans level), and level 4 (trans level), and then pelevelan are named level triad+ (widada, 2007). but in the real analysis learning is to be interdicted by widada. (2015), when reviewed based on the ability of abstraction students mathematics education study program at the university of bengkulu, found the model of the structure of representation outside of the extended characters level triad++, one level student intra able to do abstraction until on the symbolic level. research on, emphasized the structure of cognitive learners in understanding mathematics, which refers to the various reference about cognitive theory. cognitive theory sees the individual as an active information processing, so that the individual is able to represent any information in accordance with the level of knowledge and make it as a representation of the structure of knowledge in the form of frame, or in the form of the scheme, or a script to be stored in the memory (beddely, 1998; hunt & ellis, 1999; solso, 1995; dubinsky, 1986, 1987, 1989, 1995, 2000; dubinsky & mcdonald, 2000; dubinsky & yiparaki, 2001; devries, 2000). the results of research widada (2001) about the development of student scheme in learn calculus (especially the problem graph functions and rounded up takhingga) located on three levels that herarkis and functional. the sequence of the level-level is low level of intra as (level 0), inter level as middle level (level 1), and trans level as the highest level (level 2). this means that the knowledge of the students to the level of i contained in the knowledge level students to-(i+1), for i=0, 1. figure 1. triad scheme development triad trans level level inter level intra volume 5, no. 2, september 2016 pp 83-98 87 based on the results of research on the widada (2002) develop the same research about the model of the interaction during the scheme. interaction model the scheme referred to in the research is the model that is used to describe the data process ( completed) problems with level pairs from the triad (double triad). double triad in the problems of the sketch of the graph of the function is the triad in pairs that bases the two different scheme, but both used in the scheme mensketsa graphs. two schemes is a condition of the scheme function nature (property schema) given (the first instance, the second instance, limit, and kekontinuannya), and the interval scheme (interval scheme) on the domain (interval the interval near each other or overlap). any scheme described separately, and then demonstrated in a pair of scheme interaction model. data shows that in the mensketsa graph a function there are nine level scheme interaction double triad. nine level interaction scheme are not clearly unordered sequence, only the lowest level (i.e. intra-property+intra-level interval) and the highest level (namely trans-property+trans-level interval) that can be seen clearly. while the level of the other level that is not sorted explicitly. now the nine interaction level double triad intended scheme is as follows. figure 2. interscheme action double triad but from the research above seen that results from the interaction of the scheme double triad is not herarkis and functional, for that widada (2003) examine about the interaction of the new model scheme. the interaction of the new model scheme is defined as follows. e.g. to resolve a specific problem must involve interkoneksitas two different scheme, just mentioned the scheme a and scheme b, then there are five levels in the interaction of the new model scheme. five levels to be ordered by remain, intra level as level 0, semiinter level as level 1, inter level as level 2, semitrans level as level 3, and trans level as level 4, pelevelan this can be seen in the picture 1.3 follows. the interaction scheme of double triad intraproperty +intralevel interval intraproperty +interinterval intralevel property +trans interval interlevel property +intralevel interval interproperty +interintervel interlevel property +transinterval translevel property +intra interval translevel property +inter translevel interval property +translevel widada, profile of cognitive structure of students in understanding … 88 picture 3 the interaction of the new model scheme (ismb) based on widada (2004), the structure of the representation of the knowledge student scheme based mathematics triad level, so that obtained the network development of the scheme as follows. figure 4. scheme development network based on the results of research on the widada & herawaty perform fundamental research (2005), about decompositions of genetic student in learn real analysis (based on the development of the scheme using the theory apos). the fundamental research results 2005 is found two new levels in pelevelan triad (namely semiinter level and semitrans level) as an initial step to improve the level triad. and that importantly pelevelan is herarkis and functional, so that there are five level triad (which consists of three levels triad and plus two new levels) is herarkis and functional, although this is still in the substance of the real analysis. level 0 (intra) level 0 (intra) level 2 (trans) level 2 (trans) a p o s o s jaringanjaringan perkembanganperkembangan skemaskema triadtriad level 1 (inter) level 1 (inter) a p o a p o the interaction of the new model scheme level 1 level 1 level 0 level intra-a+intra-b level intra-a+inter-b level inter-a+intra-b level intra-a+trans-b level inter-a+inter-b level trans-a+intra-b level 2 trans-level a+inter-b level inter-a+trans-b level 3 level trans-a+trans-b volume 5, no. 2, september 2016 pp 83-98 89 to develop the theory, then widada perform fundamental research (2006) about the development of the theory of the development of the scheme (triad level+) about calculus in mathematics students. the results of the research is to be a network of development of student scheme as follows. figure 5. image of the development of the appropriate network scheme level triad+ if reviewed based on the ability of abstraction students mathematics education study program at the university of bengkulu, and based on the explanation above literature study and observation during the lesson real analysis is to be interdicted by widada t.a. 2014, then found the model of the structure of representation outside of the extended characters level triad++, one level student intra able to do abstraction until on the symbolic level. this suspicion toward the formation of a model of the structure of the new representation, i.e. the level above the trans with symbolic abstraction level. method to answer the problem of research about "how profiles of cognitive structure of students/students in understanding the concept in real analysis courses?", research development of the theory of cognitive structure of students mathematics education program at the university of bengkulu. this model development research with two approaches. the first approach through research theoretic form of library research, through reflection, review and validation experts. the second approach through qualitative research and quantitative descriptive. in qualitative research done by the method of analysis of the comparison remains, with how to apply the theory of the austrian glaser proved & strauss (1967), a teoretisasi process through the four stages. the main instrument in this research interviewer (in this case the researcher himself) and guided by other instruments in the form of the task sheet about the issues of the attributes of the real analysis courses, and guidelines interview. each stage of the election of the subject directly conducted data collection process through the interview based on the task and continued with the analysis of the data (either genetic decompositions analysis). based on the analysis of the data known to level 0 (intra) level 0 (intra) level 3 (semitrans) level 3 (semitrans) level 4 (trans) level 4 (trans) a p o s o s level 2 (inter) level 2 (inter) a p o level 1 (semiinter) level 1 (semiinter) a p o o a p a p o s o widada, profile of cognitive structure of students in understanding … 90 the level of the extended level triad++ is fully charged and the level of the extended level triad++ that have not yet been fully charged. results and discussion based on the analysis of the research data obtained overview of cognitive structure profile mathematics education student at the university of bengkulu reviewed based on the ability of abstraction in the formation of the concepts of real analysis courses. research results of revelation and the widada & herawaty (2016) concluded that there are seven models decompositions of genetic students mathematics education reviewed based on the srp model about the concepts of real analysis namely pra-intra level, level 0 (intra level), level 1 (semi-level inter), level 2 (level inter, level 3 (semi-trans level), level 4 (trans level), and level 5 (level extended-trans, only theoretic level while empirically not found). the conclusions of the two (widada & herawaty, 2016), that there are six models decompositions of genetic students mathematics education reviewed based on ka about the concepts of real analysis namely level 0: objects of concrete steps; level: models semi-concrete steps; level 2: models theoretic; level 3: language in domain example; level 4: mathematical language; level 5: inferensi model. the conclusions of the three, there are seven models decompositions of genetic students mathematics education reviewed based on the model of the srp and the ka about the concepts of real analysis namely basic level (pra-intra level with concrete objects), level 0 (intra level with concrete objects), level (semi-level inter with semi-concrete model), level 2 (level inter with theoretical model), level 3 (semi-trans level with the bible in the example domain), level 4 (trans level with the language of mathematics), and level 5 (level extended-trans with inferensi model, but found students with the combination of the srp levels and ka namely trans level with inferensi model. the following could be served by one profile of cognitive structure of mathematics education student at the university of bengkulu in chart form as has been disseminated in semirata mipa ((bks west 2016 (widada, 2016)). the explanation of the data analysis results are presented as follows. cognitive structure of students reviewed from the ability of abstraction in the formation of the concepts of real analysis figure 6. frequency of students based on cognitive structure description: 1) pra-intra level with concrete objects 2) level intra with concrete objects 0 5 10 15 20 25 30 1 2 3 4 5 6 7 5 7 12 27 18 11 0 volume 5, no. 2, september 2016 pp 83-98 91 3) level semi-inter with semi-concrete model 4) level inter with theoretical model 5) level semi-trans with the bible in domain example 6) level trans with the language of mathematics 7) the extended level-trans with inferensi model figure 6 shows that the cognitive structure of 80 students mathematics education at the university of bengkulu reviewed based on the ability of abstraction in the formation of the concepts of real analysis courses can be mapped in the level of the following level. there are 5 students who are at the level of pra-intra with concrete objects, 7 students located on the intra-level with concrete objects, 12 students located at the level of semi-inter with semiconcrete model, 27 students located at the level of inter with theoretical model, 18 students located at the level of semi-trans with the bible in domain example, 11 students located at the level of trans with the language of mathematics and no students located at the level of extended-trans with inferensi model. based on the figure 6 frequency of students based on the structure of cognition, so it can be served the diagram circle cognitive structure of mathematics education student at bengkulu universotas as finances on figure 7 following. figure 7. percentage students based on cognitive structure description: 1) pra-intra level with concrete objects 2) level intra with concrete objects 3) level semi-inter with semi-concrete model 4) level inter with theoretical model 5) level semi-trans with the bible in domain example 6) level trans with the language of mathematics 7) the extended level-trans with inferensi model based on the figure 7, obtained map cognitive structure of mathematics education student at bengkulu university in the form of a percentage as follows: there is 6.25% students located on the basic level (pra-intra level with concrete objects), there is 8.75% students located at level 0 (intra level with concrete objects), there are 15,00% students located at level 1 6.25 8.75 15 33.75 22.5 13.75 0 the precentage based on student cognitive structure 1 2 3 4 5 6 7 widada, profile of cognitive structure of students in understanding … 92 (semi-level inter with semi-concrete model), there are 33.75 percent students located on level 2 (level inter with theoretical model), there are 22.50 percent students located at level 3 (semi-trans level with the bible in domain example), there are located on the student percent during the level 4 (trans level with the language of mathematics), and there are 0 percent students located at level 5 (level extended-trans with inferensi model). cognitive structure of students reviewed from the ability of abstraction in the formation of the concepts of real analysis and gender according to the data from the cognitive structure of mathematics education student at the university of bengkulu, can be reviewed based on sex (women). the results of the analysis can be presented as follows: figure 8. frequency of students based on cognitive structure and gender description: 1) pra-intra level with concrete objects 2) level intra with concrete objects 3) level semi-inter with semi-concrete model 4) level inter with theoretical model 5) level semi-trans with the bible in domain example 6) level trans with the language of mathematics 7) the extended level-trans with inferensi model based on the figure 8, cognitive structure of mathematics education student at the university of bengkulu reviewed based on the ability of abstraction in the formation of the concepts of real analysis courses and gender (women) can be mapped in the level of the following level. there are 3 male students and 2 students of women who are at the level of pra-intra with concrete objects, 5 male students and 2 students women located on the intra-level with concrete objects, 2 students of male and female students 10 students located at the level of semi-inter with semi-concrete model, 10 students of male and female students 17 located at the level of inter with theoretical model, 5 students of male and female students 13 located on the levels of semi-trans with the bible in domain example, 4 male students and 7 students women located in the trans level with the language of mathematics and no students located at the level of extended-trans with inferensi model. in accordance with the image of cognitive structure frequency 1.4 mathematics education student at the university of bengkulu reviewed based on the ability of abstraction in the 0 5 10 15 20 1 2 3 4 5 6 7 3 5 2 10 5 4 0 2 2 10 17 13 7 0 t h e f re q u e n cy cognitive structure the frequency of students based on sex male women volume 5, no. 2, september 2016 pp 83-98 93 formation of the concepts of real analysis courses and gender (women), can be arranged the diagram circle each level as follows. 1) pra-intra level with concrete objects figure 9. structure of student cognitive pra-intra level with concrete objects figure 9 structure of student cognitive pra-intra level with concrete objects shows that from the 5 students that is located at the level of pra-intra with concrete objects, there are 40 percent women and 60% man. this may indicate that the cognitive structure of students at level pra-intra with concrete objects occupied by the majority of men with the difference in 20%, which means that there are differences in the structure of the student cognitive men and women on pra-intra level with concrete objects. 2) intra level with concrete objects figure 10. cognitive structure level student intra with concrete objects figure 10 cognitive structure level student intra with concrete objects shows that from 7 students who are located on the intra-level with concrete objects, there are 29 percent women and 71% man. this may indicate that the cognitive structure of students at level intra with concrete objects occupied by the majority of men with the difference that very far namely 42%. with this data obtained the conclusions that there is a difference between the very means student cognitive structure of men and women on the levels of intra with concrete objects. 3) semi-level inter with semi-concrete model figure 11. cognitive structure level student semi-inter with semi-concrete model male 60% [cate gory name] [perc male 71% women 29% male 17% wome n 83% widada, profile of cognitive structure of students in understanding … 94 based on the figure 11 cognitive structure level student semi-inter with model semiconcrete finances that from 12 students located on the semi-level inter with semi-concrete model, there are 83 percent women and 17% man. this may indicate that the cognitive structure of students at level semi-inter with model semi-concrete occupied by the majority of women with the difference that very far namely 66%. based on precentage can disimpulan that there is a difference between the very means student cognitive structure of men and women on the levels of semi-inter with semi-concrete model, i.e. 83% is a student of women. 4) the inter-level with theoretical model figure 12. cognitive structure inter level students with theoretical model figure 12 cognitive structure inter level students with theoretical model shows that from 27 students who are at the level of inter with theoretical model, there are 63 percent women and 37% man. this may indicate that the cognitive structure of students at the level of inter with theoretical model occupied by the majority of women with the difference that very far namely 36%. based on precentage can disimpulan that there is a difference between the very means student cognitive structure of men and women at the level of inter with theoretical model, i.e . 63% is a student of women. 5) semi-trans level with the bible in domain example figure 13. cognitive structure level student semi-trans with the bible in domain example based on the figure 13 structure of student cognitive level semi-trans with the bible in domain example finances that from 18 students located at the level of semi-trans with the bible in domain example, there are 72 percent women and 28% man. this may indicate that the cognitive structure of students at level semi-trans with the bible in the example domain inhabited by the majority of women with the difference that very far namely 44%. based on precentage can disimpulan that there is a difference between the very means student cognitive male 37% wome n 63% male 28% women 72% volume 5, no. 2, september 2016 pp 83-98 95 structure of men and women on the levels of semi-trans with the bible in domain example, i.e. 72% is a student of women. 6) trans level with the language of mathematics figure 14. cognitive structure trans level students with the language of mathematics based on the figure 14 cognitive structure trans level students with the language of mathematics finances that from 11 students located at the level of trans with the language of mathematics, there are 64 percent women and 36% man. this may indicate that the cognitive structure of students on the trans level with the language of mathematics occupied by the majority of women with the difference that very far namely 28%. based on precentage can disimpulan that there is a difference between the very means student cognitive structure of men and women on the trans level with the language of mathematics, i.e. 64% is a student of women. cognitive structure of students reviewed from the ability of abstraction in the formation of the concepts of real analysis and predikatif/functional based on the analysis of the research data, cognitive structure of mathematics education student at the university of bengkulu reviewed based on the ability of abstraction in the formation of the concepts of real analysis courses and predikatif/functional can be served in the circle diagram as follows: figure 15. cognitive structure predikatif/ functional mathematics education student at the university of bengkulu based on the figure 15, cognitive structure of mathematics education student at the university of bengkulu reviewed based on the ability of abstraction in the formation of the concepts of real analysis courses and predikatif/functional there are as many as 74% male 36% women 64% predika tif 26% functio nal 74% widada, profile of cognitive structure of students in understanding … 96 functional pembangunnya elements and 26 percent students pembangunnya predikatif elements. it can thus be disimpukan that the majority of the cognitive structure of mathematics education student at the university of bengkulu is functional. in accordance with this indicator in accordance with the results of research schwank (1993) and widada (2015), mathematics education student at the university of bengkulu pembangunnya element is functional can build connectedness between the action, process, objects and other scheme (do retrieval of the previous schema) so formed a mature scheme (mature scheme). the students are able to set up activities and make the algorithm that formed the concept/principles with the right. students can perform the process of abstraction using the rules in a system of mathematics. conclusion based on explanation of research results above can be summarized as follows: 1) there are seven models decompositions of genetic students mathematics education reviewed based on the srp model about the concepts of real analysis namely pra-intra level, level intra, level semi-inter, level inter, level semi-trans, trans level, level and extended-trans (only theoretic level while empirically not found); 2) there are six models decompositions of genetic students mathematics education reviewed based on ka about the concepts of real analysis namely level 0: objects of concrete steps; level 1: models semi-concrete steps; level 2: models theoretic; level 3: language in domain example; level 4: mathematical language; level 5: inferensi model. profile of cognitive structure of mathematics education student at the university of bengkulu is 6.25% students located on the basic level (pra-intra level with concrete objects), there is 8.75% students located at level 0 (intra level with concrete objects), there are 15,00% students located at level 1 (semi-level inter with semi-concrete model), there are 33.75 percent students located on level 2 (level inter with theoretical model), there are 22.50 percent students located at level 3 (semi-trans level with the bible in domain example), there are located on the student percent during the level 4 (trans level with the language of mathematics), and there are 0 percent students located at level 5 (level extended-trans with inferensi model). students mathematics education at the university of bengkulu pembangunnya element is functional can achieve trans level, students will be able to set up activities and make the algorithm that formed the concept/principles with the right. functional students can also perform the process of abstraction using the rules in a system of mathematics. references afrilianto, m. 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(2016). dekomposisi genetik mahasiswa pendidikan matematika ditinjau berdasarkan model struktur representasi pengetahuan (srp) dan kemampuan abstraksi (ka) tentang konsep-konsep analisis real. artikel dimuat dalam prosiding jambi international seminar on education. 3-5 april 2016. widada, w. (2016) kemampuan abstraksi mahasiswa pendidikan matematika dalam memahami konsep-konsep analisis real ditinjau berdasarkan struktur kognitif. artikel dimuat dalam prosiding semirata mipa bks barat di unsri 22-24 mei 2016. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.234 11 developing students’ ability of mathematical connection through using outdoor mathematics learning saleh haji 1 , m. ilham abdullah 2 , syafdi maizora 3 , yumiati 4 1,2,3 department of mathematics education bengkulu university, bengkulu, indonesia 4 department of mathematics education open university, indonesia 1 salehhaji25@gmail.com, 2 ilhamabdullah418@gmail.com, 3 syafdiiemaizora@yahoo.com, 4 yumiatis@gmail.com received: october 20, 2016; accepted: november 12, 2016 abstract the purpose of this study is to determine the achievement and improvement of students’ mathematical connectionability through using outdoor mathematics learning. 64 students from the fifth grade of primary school at sdn 65 and sdn 67 bengkulu city were taken as the sample of this study. while the method of the research used in this research is experiment with quasi-experimental designs nonequivalent control group. the results of the study are as follows: (1) there is an increasing ability found in mathematical connection of students whom taught by using outdoors mathematics learning is 0,53; (2) based on statical computation that achievement of students’ ability of mathematical connection is taught by using outdoor mathematics learning score is 71,25. it is higher than the students score 66,25 which were taught by using the conventional learning. so as to improve students’ mathematical connection, teachers are suggested to use the outdoors mathematics learning. keywords: mathematical connection ability, outdoor mathematics learning abstrak tujuan penelitian ini adalah untuk menentukan pencapaian dan peningkatan kemampuan koneksi matematika siswa melalui pembelajaran matematika di luar ruangan. sampel dalam penelitian ini sebanyak 64 orang siswa yang berasal dari kelas lima sekolah dasar pada sdn 65 dan sdn 67 kota bengkulu. metode yang digunakan adalah eksperimen dengan desain kuasi-eksperimental nonquivalen kelompok kontrol. hasil dari penelitian ini adalah sebagai berikut: (1) terdapat peningkatan kemampuan koneksi matematis siswa yang belajar di luar ruangan, dengan peningkatan sebesar 0.53. (2) pencapaian kemampuan koneksi matematis siswa yang belajar di luar ruangan (71.25) lebih tinggi daripada siswa yang belajar dengan cara convensional (66.25). sehingga untuk meningkatkan kemampuan koneksi matematis siswa, hendaknya guru dapat melakukan pembelajaran matematika di luar ruangan. kata kunci: model pembelajaran penemuan terbimbing, pemahaman konsep kalkulus integral how to cite: haji, s., abdullah, m. i., maizora, s. & yumiati (2017). developing students’ ability of mathematical connection through using outdoor mathematics learning. infinity, 6 (1), 11-20. introduction generally, primary schools in indonesia organized learning in the classroom. learning in the classroom has many weaknesses in instilling mathematical concepts to students. the weaknesses among other things, teachers are not flexible in linking the mathematical concepts mailto:salehhaji25@gmail.com mailto:2ilhamabdullah418@gmail.com mailto:syafdiiemaizora@yahoo.com mailto:yumiatis@gmail.com haji, abdullah, maizora & yumiati, developing students’ ability of mathematical … 12 to everyday life. similarly, students are limited in observing the real objects of everyday life are related to mathematics. this led to the ability of understanding mathematical concepts, especially the ability of mathematical connection students do not develop properly. mathematical connection capability is one capability that suggested by the nctm. according to the nctm (1989), mathematics skills need to be developed through the study of mathematics as follows: 1. problem solving, 2. reasoning and proof, 3. communication, 4. connection, and 5. representation. mathematical connection capability is the ability of students in linking the various issues related to mathematics. the connectionis included in mathematics and between mathematics with things outside mathematics. as link the concepts included in algebra with the concepts included in geometry.the linking of mathematics to other disciplines and to everyday life. connection between mathematics and outside mathematics should be developed through the study of mathematics. because mathematics is a science that includes a lot of the linking between concept.as the link between the concept of relationship with the concept of function.the linking of the addition operation with multiplication operations on numbers.the linking of the concept of the derivative function with the concept of profit and loss in the economic field.as well as the linking of the concept of the exponential growth of bacteria. to improve students' mathematical connection required learning related to everyday life. learning that includes such properties are learning outside the classroom (outdoor education). kennard (2007) link the recreational ride trains children with their understanding of numbers. moffett (2010) describes the things are interlinked in a museum. like other forms of geometry contained in ancient objects. results of research daher and bayaa (2011) found that cell phone use can increase students' understanding of mathematical objects. while burriss and burris (2011) found that learning outside the classroom through game activities can improve students' ability to solve a problem. the problems of this study as follows: 1. is there an increase in the ability of mathematical connection students taught through outdoor learning mathematics? 2. is the achievement ofstudents mathematical connectiontaught through outdoor learning mathematics is higher than those taught by conventional teaching? mathematical connections ability mathematical connection ability is the ability of linking between components in mathematics, the mathematics to other disciplines, and between mathematics to everyday life. kutz in yusepa (2002) states that the mathematical connection includes internal and external relationships mathematically.mathematical knowledge about organized structure (ruseffendi, 1991). organizations in mathematics linking the various elements contained therein. the elements in mathematics may consist of: algebra, geometry, arithmetic, probability, and calculus. in addition, the mathematics may also consist of facts, concepts, principles, and skills. as the link between the concept of a square with a parallelogram. the square is a parallelogram that all sides are equal in length and all the right-angled corners. kusuma (2003), classifies the mathematical connections consist of: 1. the connections between topics volume 6, no. 1, february 2017 pp 11-20 13 and mathematical processes, 2. connections between mathematics with other sciences, and 3. connections between mathematical concepts to everyday life. the linkage between concepts in mathematics and between mathematics to everyday life can help students understand mathematical concepts. as to understand the concept of integer multiplication through its association with the concept of the sum of the integers.because the integer multiplication is repeated addition of integers. fisher in ruspiani (2000) mentions that the mathematical connection was an attempt to foster students' understanding. sumarmo (2000) describes the importance of mathematics connection ability in mathematics, namely: 1. expanding horizons, 2. clarify mathematics as a whole, and 3. clarify the benefits of mathematics. the outdoor learning in mathematics the outdoor learning in mathematics is mathematics that relies on learning activities outside the classroom. it focuses on the learning activities of students and teachers outside the classroom. the places of classroomoutside, such as: the school yard, garden, market, health hospital, police offices and others. husamah (2013) explains an outdoor learning outside of school activities that contain activities outside the classroom and the other in the wild. amin (2008) explains that the method of outdoorlearning process is a method of learning science by doing adventure in the neighborhood, accompanied by thorough observation that the results recorded in the observation worksheets. according to bartlet in husamah (2013), an outdoor education learning model is a learning outside the room or outside the classroom. activities outside the classroom as an environment for children to learn mathematics.real and dynamic environment that can facilitate the child understand the mathematics material. bratton (2005) explains that learning mathematics outdoors provides an environment for children to learn. maizora & haji (2015) describe the steps of outdoor mathematics as follows: 1. teachers prepare students to be ready to follow the lesson. 2. teachers express purpose of learning. 3. teacher conveys exactly the material to be learned and how learning will be done outside the classroom. 4. teachers encourage students out of the classroom to the place (an object) that is associated with mathematics. the objects can be objects, phenomena, or form of the game. 5. students observe and manipulate these objects or perform a play. 6. teachers guide students to discuss the mathematical concepts contained in the object being observed or in a game they do. 7. teachers together students concluded various mathematical concepts contained in the objects and the game were done. 8. teachers invite students back into the classroom. 9. teachers clarify and review the mathematical concepts that have been obtained by the students outside the classroom and to associate with learning objectives (competencies) to be achieved. 10. the teacher presents a summary of the lessons that have been done together students. haji, abdullah, maizora & yumiati, developing students’ ability of mathematical … 14 11. teachers assign tasks to students to solidify understanding of the concepts they have learned and provide direction on the material to be studied and activities outside the classroom at the next meeting. method type of research this research is a quasi-experimental design with a nonrandomized control group, pretestposttest design, the following: experiment group : y1 x y2 control group : y1 y2 y1 = pretest, y2 = posttest. x = treatment in the form of outdoor learning mathematics. sample and population these students study population are of state elementaryschool 67 bengkulu city. while the study sample is grade 5 at state elementary school 67 bengkulu city. sampling is done purposefully instrument reseach and development the instrument of this research is questions about the ability of mathematical connections on the topic operation count on integer. on the essay form as much as four grains. the trial results and analysis, obtained a set of questions about the mathematical connections that are valid and reliable. the results of the analysis instrument by using spss 12 (trihendradi, 2004) is presented in table 1 below. table 1.development of research instruments no. variables result 1 average score 66,00 2 standard deviation 10,01 3 correlation xy 0,62 4 reliability test 0,76 5 items 4 6 validity item 1 0,79 7 validity item 2 0,75 8 validity item 3 0,80 9 validity item 4 0,71 10 number of subjects 40 results judging of grains instruments by three experts in the field of mathematics education show that they give the same rating to the four-point test that can be used to measure the ability of mathematical connections, after some repairs. volume 6, no. 1, february 2017 pp 11-20 15 the research was conducted at mathematics education, teacher training and education faculty, widya dharma university, klaten. the research was quasy experimental research because only some of relevant variables that were controlled or manipulated.the sample was the students of mathematics education in even semester who took integral calculus subject with the use of certain integral in accomplishing a problem as the material. cluster random sampling was implemented to find out a representative sample of the population. based on the result of cluster random sampling, the control group that was taught using conventional model consisted of 22 students meanwhile the experimental group that was taught using guided discovery learning consisted of 34 students. data analysis data about upgrading mathematical connections were analyzed by using n-gein. while data on the achievement of mathematical connection capabilities were analyzed using t -test when the data were normally distributed. when the data are not normally distributed, the mannwhitney test was used. results and discussion 1. achieving the ability to mathematical connection students taught through outdoor learning mathematics the test results of pretest and posttest data normality on a mathematical connection capability acquired rejection by ho which means each pretest and posttest data is not normal. this is shown in table 2 and table 3. similarly to the data of n-gein, is not normal. this is shown in table 2. table 2. results of normality pretest data of mathematical connection data group n average dev. stand. kolmogoro v-smirnov z sig. (2way) h0 pretest of experimental group 40 34,25 19,20 0,172 0,004 rejection pretest of control group 40 33,50 15,94 0,158 0,013 rejection based on table 2, the data pretest mathematical connection capabilities in both sets of data are not normally distributed. therefore, to know the difference between the two groups, namely learning mathematics outdoor learning and conventional learning mann-whitney test was used. mann-whitney test to pretest the data connection capabilities of the experimental group and the control group revenue generating ho, so no significant differences prior knowledge mathematical connection between the experimental group and the group control. so initial capabilities of the two groups are the same. this is shown in table 3. haji, abdullah, maizora & yumiati, developing students’ ability of mathematical … 16 table 3. results of mann-whitney test of mathematical connection abilty data group average u mann whitney z sig.(2way) h0 pretest of experimental group 34,25 770,500 -0,289 0,772 accepted pretest of control group 33,50 the data normality test results posttes mathematical connection capabilities of the two groups using the kolmogorov-semirnov indicate that ho is accepted. this means, the two groups are not normally distributed. this is shown in table 4. therefore, different test for two groups using the mann-whitney test. table 4. results normality test data of mathematical connection ability data group n average dev. stand. kolmogorovsmirnov z sig. (2way) h0 posttest of experimental group 40 71,25 6,96 0,354 0,00 rejection posttest of control group 40 66,25 9,66 0,216 0,00 rejection test of differences achievement the students’ability aboutmathematicalconnection atthe experimental group with the control group using the mann-whitney ho generates revenue. this means that there are differences in achievement ability mathematical connection between the experimental group and control group. this is shown in table 5. the mean scores of mathematical connection ability of the experimental group 71.25 greater than the mathematical connection ability control group students at 66.25. this means, the achievement of mathematical connection ability of students taught using the outdoor mathematics better than students taught using conventional learning. table 5. calculation results mann-whitney postest of mathematical connection ability data group average u mann whitney z sig.(1-way) h0 postest of experimental group 71,25 505,000 2,932 0,0015 rejection postest of control group 66,25 students are taught through through outdoor learning mathematics can link between mathematics and the mathematics section with everyday life. such as linking the relationship between the number 10 to number 5. this relationship is as follows (maizora & haji, 2015): a. 10 = 5 x 2 b. 10 = 2 x 5 c. 10 = 15 – 5 d. 10 = 5 + 5 volume 6, no. 1, february 2017 pp 11-20 17 e. 10 is greater than 5. f. 5 is smaller than 10. g. 10 = 50 : 5 as many as 83% of students correctly answered questions about the relevance of the number 10 to number 5. the ability of the student connection because the influence of outdoor learning mathematics that has provided the opportunity for students in linking various objects in everyday life. students relate the number of marbles with numbers and a game of marbles with operations in a game of marbles. purwanto (2008), outdoor learning can bring learners with learning objects. the real learning objects as a context for helping students understand mathematical connections. nuriadin (2015) explains that there are differences in the students ability in mathematical connection taught through contextual learning with conventional. beside, martin, falk, and balling (1981) explain learning by a field trip to a novel setting is a better understanding of the details and nature of the cognition. in this case the game of marbles with the concept of numbers and their operations. this is shown in figure 1 below. figure 1. the students were playing marbles outside the classroom when students learn about the topic operation count on integer in the 'sunday market' in bengkulu. students relate between buyers and sellers. seller is a person who sold the goods to the buyer. while the buyer is a person who buys goods from a seller. similarly, students link the bananas and oranges. students explain the link between bananas and oranges through color. students say that the color of bananas and oranges at the yellow. as many as 47% of students who answered questions correctly about the link between the number 10 to number 5. a total of 53% of students taught by conventional teaching experience difficulties in linking between the number 10 to number 5. according to them, number 10 with number 5 are two different numbers , so it cannot be associated with each other. achievement of mathematical connection abilityin both groups shown in figure 2 below. haji, abdullah, maizora & yumiati, developing students’ ability of mathematical … 18 figure 2. achievement of students ability of mathematical connections through the outdoor learning mathematics 2. to increase the students’ ability ofmathematical connection taught through outdoor learning mathematics results of data normality test n-gain mathematical connection capabilities by using the kolmogorov-smirnov obtained rejection of ho. this means, the two groups are not normally distributed. this is shown in table 6 below. table 6. results normality test data connection n-gain mathematical ability data group n average dev. stand. kolmogorovsmirnov z sig. (2way) h0 n-gain eksperimental group 40 0,53 0,17 0,158 0,014 rejection n-gain control group 40 0,46 0,20 0,145 0,034 rejection different test improvement (n-gain) mathematical connection capability in both groups using the mann-whitney ho generate revenue. this means, by enhancing the mathematical connections of students taught using the outdoor learning mathematics is no different with students taught using conventional learning. this is shown in table 7 below. table 7. calculation results mann-whitney n-gain of mathematical connections ability data group average u mann whitney z sig.(1-way) h0 n-gain of eksperimental group 0,53 652,500 -1,424 0,078 received n-gain of control group 0,46 the magnitude of the increase in the ability students of mathematical connection taught using the outdoor learning mathematics 0.53 greater than the increase in the ability students of mathematical connection taught using the conventional learning 0.46. despite an increase in 0 20 40 60 80 outdoor math konvensional 34.25 33.5 71.25 66.25 s k o r k e m a m p u a n k o n e k si m a te m a ti s pretes postes volume 6, no. 1, february 2017 pp 11-20 19 the mathematical connectionability to the two groups did not differ significantly. this is because the outdoor learning makes students more excited in learning mathematics, thereby increasing their interest in mathematics. widayanti (2003) explains that the method of outdoor study successfully increased student interest. such improvements in terms of linking between sections of mathematics and mathematics relate to everyday life. such as linking between the number of patients treated at the hospital dr. m. yunus bengkulu as a summation and the number of patients who return stay overnight as a reduction on outdoor learning mathematics with the theme 'inpatient hospital of dr. m. yunus bengkulu '. improving the mathematical connection ability in the two study groups are presented in figure 3 below. figure 3. upgrades mathematical connections students through the outdoor learning mathematics conclusion 1. based on the statistical computation result of the data analysis found a score 0,53. it means that mathematical connections whinch taugh outdoor learning is increased significantly. 2. acievement of the students’ ability of mathematical connection taught through outdoor mathematics learning is significantly greater than the students whom taught by using conventional learning. the score of data analysis computation result for out door learning at experimental group is 71.25, while the attainment score found through the conventional learning method at the control group is 66.25. references amin, c. (2008). memupuk tradisi ilmiah siswa sekolah dasar menggunakan metode outdoor learning process. simposium tahunan penelitian. brattorn, c. (2005). learning outdoors. nsa publication. burriss, k., & burris, l. (2011). outdoor play and learning: policy and practice. international journal of education policy and leadership, 6(8), 78-89. daher, w., & bayaa, n. (2011). characteristics of middle school students learning actions in outdoor mathematical activites with the celluler phone. oxford university press. 0.53 0.46 score of n-gain outdoor math haji, abdullah, maizora & yumiati, developing students’ ability of mathematical … 20 husumah. (2013). pembelajaran luar kelas outdoor learning. jakarta: prestasi pustaka. kennard, j. (2007). outdoor mathematics. mathematics teaching incorporating micromath, 201, 16-18. kusuma, d. (2003). meningkatkan kemampuan koneksi matematik siswa sekolah lanjutan tingkat pertama dengan menggunakan metode inkuiri. program pascasarjana upi. bandung: not published. maizora, s., & haji, s. (2015). model pembelajaran outdoor mathematics untuk meningkatkan kemampuan koneksi dan komunikasi matematis siswa sekolah dasar. lppm universitas bengkulu. bengkulu: not published. martin, w., falk, j., & balling, j. (1981). enviromental effecs on learning: the outdoor field trip. science education, 65(3), 301-309. moffett, p. (2010). back in time on a mathematics trail. mathematics teaching, 219, 31-33. nctm. (1989). curriculum and evaluation standards for school mathematics. usa: the national council of teachers of mathematics inc. nuriadin, i. (2015). pembelajaran kontekstual berbantuan program geometer’s sketchpad dalam meningkatkan kemampuan koneksi dan komunikasi matematis siswa smp. infinity journal, 4(2), 168. doi:10.22460/infinity.v4i2.80. purwanto. (2008). penerapan metode partisipatori untuk meningkatkan keterampilan menulis puisi siswa kelas v melalui pembelajaran di luar kelas. retrieved january 5, 2011, from http://purwanto65.wordpress.com ruseffendi, e. (1991). pengantar kepada membantu guru mengembangkan kompetensinya dalam pengejaran matematika untuk meningkatkan cbsa. bandung: tarsito. ruspiani. (2000). kemampuan siswa dalam melakukan koneksi matematika. fps upi. bandung: not published. sumarmo, u. (2000). proses belajar dan pemahaman materi kuliah. lokakarya tpb. bandung: itb. trihendradi, c. (2004). memecahkan kasus statistik: deskriptif, parametrik, dan nonparametrik. yogyakarta: penerbit andi. widayanti, n. (2003). efektivitas pembelajaran geografi melalui metode outdoor study dalam upaya meningkatkan minat belajar siswa. buletin pelangi, 6(1), 125-134. yusepa, b. (2002). penerapan model cooperative learning tipe student-team achievement divisions (stad) dalam upaya meningkatkan kemampuan koneksi matematika siswa. upi. bandung: not published. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p77-86 77 clusters of prevalent patterns of geometric thinking levels among mathematics students joshua edson gorme ordiz*, ghanine rhea mecate southern leyte state university, eastern visayas, philippines article info abstract article history: received oct 18, 2021 revised dec 28, 2021 accepted jan 10, 2022 geometric thinking skills are the perceived abilities of an individual to think and reason in geometric contexts. these skills acquired by students in geometry remain poor and unsettling because of the misconceptions that hinder the students in learning the components of geometry. the study described the common unplaceable patterns in geometric thinking of 153 mathematics education students in a state university in eastern visayas, philippines. frequency analysis was employed in the study to determine the number of occurrences of the patterns stressing the cause for students placed under level 0 or unplaceable. van hiele achievement test was used to gather the students’ performance in geometry at all levels, namely: visualization, analysis, informal deduction, deduction, and rigor. the findings attested that only 13.1% of the students managed the third level of the van hiele levels while 43.1% of them were unplaceable. common patterns were drawn and describe to understand the consequences in geometric thinking ability at level 0. these observable patterns were grouped into core-remedial, topicalcorrective, and close-corrective groups. the clusters will enable educational institutions to address the individual gaps in geometry. keywords: geometry, mathematics performance, proving this is an open access article under the cc by-sa license. corresponding author: joshua edson gorme ordiz, faculty, college of teacher education, southern leyte state university concepcion st, sogod, southern leyte, filipina email: sirjosh.ordiz@gmail.com how to cite: ordiz, j. e. g., & mecate, g. r. (2022). clusters of prevalent patterns of geometric thinking levels among mathematics students. infinity, 11(1), 77-86. 1. introduction improving the students’ performance in mathematics, specifically in geometry, is a challenge by educators because students find it complicated. despite the efforts made by the academic community to address the students’ performance in geometry in the discipline, it remained low and degrading. concurrently, number of students do not attain any of the geometric levels denoting poor achievement in geometry (mullis et al., 2016). the content of the students’ performance calls the understanding of geometric thinking ability to determine the common patterns exist in level 0 or unplaceable and describe its’ consequences. learning geometry measures on the van hiele levels of geometric thinking https://doi.org/10.22460/infinity.v11i1.p77-86 https://creativecommons.org/licenses/by-sa/4.0/ ordiz & mecate, clusters of prevalent patterns of geometric thinking levels … 78 ability of the students which is helpful in analyzing the learners’ performance (alex & mammen, 2012). result of the study of atebe (2008) which indicated 41% of the learners is at level 0. this study showed the difficulty that the learners have in recognizing figures in nonstandard positions. this finding is supported by alex and mammen (2012), presented that majority of the learners were unplaceable which means none of the students acquired any levels. the assignment of learners into levels showed the percentage in level 0 was 48%. this seeks for the need to deliver instruction at a level appropriate to learners’ level of thinking on the one hand and improving the quality of education starting from lower levels on the other hand. besides, marchis (2012) stressed out that students have confusion in geometry due to idea definition. proper idea definition produces a self-idea of the image. this concept image may not develop in a few understudies, and in others, it may not identify with the formal definition. there is the need to address these misinterpretations when educating to enable the learners to consider where the misconception between the verbal meaning and mental image originates from. characterizing and distinguishing shapes inclination is given to a visual model than a formal description (özerem, 2012). students prefer to rely on a visual prototype rather than a verbal definition when classifying and identifying shapes. to obtain the mathematical knowledge required in everyday living, educating the techniques on how to solve problems, and inhibiting reasoning strategies are the objectives of mathematics. troubles in learning geometry clarify cognitive improvement (idris, 2009). individual mental capacity is not just about visual discernment, breaking down components, knowing the connections between properties, building and appreciating proofs but also decision making, which is vital to accomplishing higher-level thinking in learning geometry. an individual with better visual perception has an advantage in geometric reasoning (walker et al., 2011). learners need help to uncover these misconceptions and thus, build on correct perceptions. learners need to develop and build up the proper schema about the previous knowledge before taking the new higher lessons in the upper educational level. teachers must provide learning experiences that fit the level of thinking of the students. concrete experiences of the learner in the primary level help to shorten the gap in abstract concept with the use of solid objects. in addition to that, visual assisted tools are being used to enhance the geometric thinking ability, and it functioned as a mental reference (kamina & iyer, 2009; zanzali, 2000). moreover, giving attention to the application of dialect is one of the pedagogical practices that support the development of the mathematical knowledge of the students (schleppegrell, 2007). one of the main contributors to overall comprehension in many content areas, including mathematics, is vocabulary understanding. these observations were evident in this study to anchor the van hiele theory (van hiele, 1999) after supporting numerous knowledge that emphasizes the mediocre achievement in geometry. the van hiele model considers significant imperative models in educating geometry and geometric thoughts and ideas. this model has five phases in which each level represents the development of the thinking process in geometry. the improvement of the geometric thinking of the students will lead to the summarization of the learning which is vital in using in a real-life situation (pegg & tall, 2010). it is one of the theories that are effective in teaching geometry to students through the school stages (mistretta, 2000). as per van hiele, the five levels of geometric reasoning are visualization, analysis, abstraction, deduction, and rigor (groth, 2005). volume 11, no 1, february 2022, pp. 77-86 79 figure 1. the procedural pattern of learning geometry the illustration displays the levels of van hiele model: the visualization, analysis, informal deduction, deduction, and rigor (see figure 1). furthermore, the patterns are formulated to identify the levels placed by the students, understand and describe the consequences of every unplaceable patterns in geometric thinking ability which is the main goal of the study. these consequences are necessary to address the concern that majority of the students are not able to reach even the first level of the model. these students are referred to be unplaceable into any of the levels of the van hiele model. one of the problems of the teachers is that the ability of teachers to present problems related to geometry is weak. they cannot transfer their knowledge appropriately about geometric thinking levels and claimed that course contents to be designed for practice are thought to improve the subject matter knowledge related to geometry (erdogan, 2020). focusing on the geometric thinking skills of those students who were not able to be classified in any of the levels is indispensable in order to help the struggling educators to identify the possible interventions that is useful in improving the geometric levels by imparting the precise way of presenting knowledge to the learner. levels are hierarchical, it is needed to fully acquire the previous levels in order to reach a higher level. failing to do so leads to not acquiring any of the van hiele model. these levels are associated to geometric experiences (van de walle et al., 2014). exploring the reasons why students failed to attain any of the van hiele model. will result to providing teachers the possible solutions to the main concern by understanding what happened in the levelling of the said ability. teaching geometry is the focus of the improvement of logical thinking and a vital factor of mathematical understanding (van hiele, 1999). educators have a critical role in teaching and learning geometry. ordiz & mecate, clusters of prevalent patterns of geometric thinking levels … 80 2. method this study utilized frequency analysis. it deals with the number of occurrences or frequency of the patterns stressing the cause for students placed under unplaceable. it describes the data set and provide a fair idea of what patterns the students are acquiring. the method used was complete enumeration in the conduct of the study. the locale of the study was all campuses of southern leyte state university. all mathematics students enrolled in academic year 2018-2019 were considered to be the participants of the study towards exploring the placement of learners’ geometric ability (see table 1). table 1. distribution of respondents year level number of students percentage freshmen 46 30% sophomore 36 24% junior 33 21% senior 38 25% total 153 100% distribution of the questionnaires to all students majoring mathematics in the master list provided by the university followed. the achievement test for the van hiele was composed of five questions in each level. in every query, students will choose the best answer from the options. students must reach three points in each level to grasp the level need to attain. however, your previous score in the lower level is less than to 3 points, and you achieved a score that is <3 points in the next level, students can be classified as unplaceable. it is impossible for the students to obtain the higher levels without accomplishing the lower levels. hence, students should have consistent score that is <3 without failing the levels in between to grasp any of the levels. mean, frequency, and weighted average. this was used to determine the mean rating of the sample and the exact number of each pattern exist. it is necessary for analysis and interpretation of any data and it indicates how well the data is. 3. results and discussion 3.1. results 3.1.1. van hiele geometric thinking ability van hiele geometric thinking designates how an individual acquires learning in the field of geometry that proposes five levels of geometric thinking. each level utilizes its symbols and language and students can pass through this level “step-by-step.” table 2. geometric thinking ability according to van hiele levels van hiele model year level freshmen level sophomore level f <cf % f <cf % unplaceable 24 46 52.2 16 36 44.5 1 6 22 13.0 7 20 19.4 2 7 16 15.3 7 6 19.4 3 6 9 13.0 6 3 16.7 4 0 3 0 0 0 0 5 3 3 6.5 0 0 0 volume 11, no 1, february 2022, pp. 77-86 81 van hiele model year level junior level senior level f <cf % f <cf % unplaceable 17 33 51.3 9 38 23.7 1 9 16 27.4 9 29 23.7 2 7 7 21.3 8 20 21.1 3 0 0 0 8 12 21.1 4 0 0 0 0 4 0 5 0 0 0 4 4 10.4 van hiele model over all f <cf % unplaceable 66 153 43.1 1 31 87 20.3 2 29 56 19.0 3 20 27 13.1 4 0 7 0.00 5 7 7 4.5 legend: 0 unplaceable, 1visualization, 2analysis, 3informal deduction, 4deduction, 5rigor table 2 shows the achievement of the students across year levels in terms of their geometric thinking ability. amongst respondents of the study, 20.3% were only level 1; 19.0% are level 2; and 13.1% are level 3. there is a decreasing number od students who were able to reach a higher level in the van hiele model. looking at the distribution of students per geometric level, only 56.9% managed to attain visualization level. in the analysis level, only 36.6% achieved this level. meanwhile, in the informal deduction level, 17.6% were left to reached the third level. among the respondents, there is only 4.5% who were able to developed the two highest level of geometric thinking which is deduction and rigor. on the other hand, a staggering 43.1% of the students were classified to be level 0 or unplaceable. these students were not able to reach visualization level which is the basic foundation of the geometric skills. the huge percentage of students found to be unplaceable initiated the query on what were the achievement made in the long period of time being exposed to geometry and other related areas in mathematics. patterns were observed in the study as presented below. 3.1.2. common patterns of respondents under unplaceable outcomes of the international tests revealed that the leaners' geometry knowledge and skills were not at the preferred level. geometry is the lowest performance area among all fields (mullis et al., 2016). the data below represent the pattern of the scores of the students under the undesirable level called unplaceable. each digit of the pattern represents each of the levels of the van hiele achievement test in an orderly manner. table 3. patterns of geometric thinking skills of unplaceable patterns mathematics students f % 00000 40 26 01000 48 30 ordiz & mecate, clusters of prevalent patterns of geometric thinking levels … 82 patterns mathematics students f % 00100 17 11 01100 15 10 11101 17 11 11001 7 4 10100 7 4 10010 7 4 *0level not attained, 1attained level core-corrective group the core-corrective group include the patterns 00000, 01000, 00100 and 01100 (see table 3). this group barely succeeded the foundation of the geometric skills and abilities. in 00000, it means that none of the levels were partially attained. different aspects of the mathematics achievement of students can cause failure in achieving any of the levels (meng & sam, 2013). this means that students were really having a hard time visualizing concept, reasoning, synthesizing three to two dimensional objects, formulating assumptions, creating logical inferences, and also proof skills in geometry. also, this is affected by the influence of teaching-learning process. researches derive conclusions that students’ level of geometric thinking depends on how the instruction is delivered (abu & abidin, 2013; alex & mammen, 2012). in the 01000 and 00100 patterns (see table 3), respondents succeeded in the second phase but failed in the rest of the levels. it was because students considered the properties of the figures which is not needed in the visualization level and not enough in attaining other levels. students are unable to visualize both solids and positions of solids that they cannot see, neither the pattern of movements. there is also a lack of coordination between geometric presentation and visualization. in the 01100 patterns (see table 3), students attained the mid-levels; however, previous and succeeding levels failed to reach and considered as unclassified. it was because students used the properties possessed by each of the shapes, thus failed in obtaining the first level. considering characteristics acquired by the figures were not enough to reach into the higher levels. this stresses that learners have a hard time to produce a formally deduced definitions and properties in order to prove and provide a logical concept which is necessary to reach in its desired level. topical-corrective group the patterns 11101 and 11001 (see table 3) were classified to be topical-corrective group. this group managed to complete the foundation skills but not the higher levels of the model. in these patterns, 11101 and 11001 (see table 3), students failed between the levels, thus considered it as unplaceable. in the first pattern, failing in deduction level (level 4) but reaching into the higher-level means failing to attain any of the van hiele levels. it was due to lack of knowledge in proving and understanding axioms, corollaries, and postulates. in the second pattern, students used their instincts rather than logic in creating conceptual statements and properties of the geometric figures and not considering the necessary and sufficient conditions in the euclidean and non-euclidean geometry. both patterns failed in attaining the formal deduction level. in this level, students failed to provide logical reasons or proofs of the following mathematical structure of the elements. students’ mathematical volume 11, no 1, february 2022, pp. 77-86 83 understanding of proofs, axioms and postulates and corollaries is at risk. it implies that students are struggling in terms of proving. thus, students need to build a strong foundation of the basic understanding of geometry to achieve the higher levels. recently, a growing number of studies have provided evidence that students have difficulty with mathematical proofs (ko & knuth, 2009). close-corrective group this group of patterns managed to achieved the foundation skills in geometry which is visualization, however, failed to achieved even partially and struggling to complete the levels of the van hiele model. 10100 and 10010 patterns were categorized as unplaceable. students attained the first levels but failed in reaching the second and third level. it was unplaceable since students must achieve the second level to surpass to the next levels. students cannot skip into t he higher levels unless attained the previous levels. it was because students only focused on their visual perception, and nonverbal thinking but neglecting the properties, relationships, proofs and manipulating axioms. in this level, they can measure, fold and cut paper, use geometric software, etc. it implies that undertakers only focused on their visual recognition, considered as difficult in applying the properties and unable to perceive relationship between figures and components of each shape used to justify and support for the students to answer. the pattern reveals that difficulty in understanding the properties and perceiving the relationships between properties and figures can affect the students’ performance in the higher level of mathematics. after identifying patterns, we classified all the existing patterns into three groups, topicalcorrective, close-corrective, and coreremedial groups. 3.2. discussion as a result of the van hiele geometric achievement test, it is concluded that majority of the undertakers are under unplaceable. thus, exploring the patterns existed in the said level give us the keys in observing and providing insights on remedial actions in geometry that assess the geometric thinking skills of the students in order to improve the levels of their geometric ability. topicalcorrective group, these are the group of students that have higher chances in learning the levels of geometric thinking however, teachers should take consideration in providing exercises or activities that can help improve in proving and logical reasoning. the group that already captured visualization level but finds it difficult for them attain the succeeding levels are placed in close-corrective group because they cannot break down information into concepts. under core-remedial group, students need recalibration of the geometrical concepts from simple to complex. students placed in this group failed to visualize geometric figures. thus, students were having a hard time understanding geometry. all these remedial actions provide us analysis of errors observed in every unplaceable patterns. each group introduced differentiated instruction based from the individual group needs to improve the geometric levels. these actions will contribute plenty intervention strategies and techniques to educators that can be integrated in their geometry lessons. 4. conclusion individual gaps in the performance of geometry in the early levels of education is important to nurture because of the relevance of the topics in geometry in higher education. ordiz & mecate, clusters of prevalent patterns of geometric thinking levels … 84 the progress of a student determined the ability of them to perform well in any mathematics related disciple. the levels of van hiele’s geometric thinking showed that many of the student were not able to attain any of the levels namely visualization, analysis, informal deduction, deduction, and rigor. appropriate strategies were needed to ensure full understanding by the students are needed by all mathematics educators especially in the basic education to address the issue of geometric thinking performance. it will showcase the distinction of every student and the required remedial action based from their performance. references abu, m. s., & abidin, z. z. 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(2000). designing the mathematics curriculum in malaysia: making mathematics more meaningful. universiti teknologi malaysia. retrieved from http://math.unipa.it/~grim/jzanzali.pdf https://doi.org/10.1016/j.sbspro.2012.09.557 https://doi.org/10.1007/978-3-642-00742-2_19 https://doi.org/10.1080/10573560601158461 https://doi.org/10.5951/tcm.5.6.0310 https://doi.org/10.4236/ce.2011.21004 http://math.unipa.it/~grim/jzanzali.pdf ordiz & mecate, clusters of prevalent patterns of geometric thinking levels … 86 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 9, no. 2, september 2020 e–issn 2460-9285 https://doi.org/10.22460/infinity.v9i2.p223-246 223 two decades of realistic mathematics education research in indonesia: a survey rully charitas indra prahmana*1, laela sagita2, wahyu hidayat3, niken wahyu utami2 1universitas ahmad dahlan, yogyakarta, indonesia 2universitas pgri yogyakarta, yogyakarta, indonesia 3institut keguruan dan ilmu pendidikan siliwangi, cimahi, indonesia article info abstract article history: received may 26, 2020 revised aug 30, 2020 accepted sep 29, 2020 freudenthal's ideas on mathematics that stated mathematics as a human activity and mathematics must be connected to reality have colored towards learning mathematics all over the world, including indonesia. existing research on this topic in indonesia among two decades is categorized and compared. this paper presents a systematic literature review of empirical studies on this learning approach in indonesian context. the implementation of this approach in indonesia was analyzed comprehensively and divided into several major topics. resources of this research come from 110 articles by ten highest rank accredited journals by the ministry of research, technology, and higher education, the republic of indonesia as an achievement for the peerreviewed journal, which has excellent quality in management and publication. the results show that there are seven categories summarized in this research: the dominance of published rme articles, rme research subjects, mathematical topics, students' abilities, rme terms, and the research method used in rme articles. lastly, the summarized 110 papers are displayed in the table to give essential information as a fundamental idea for further rme research. keywords: freudenthal’s idea, indonesia, mathematics education approach, realistic mathematics education, survey copyright © 2020 ikip siliwangi. all rights reserved. corresponding author: rully charitas indra prahmana, departement of mathematics education, universitas ahmad dahlan, jl. pramuka no. 42, pandeyan, umbulharjo, yogyakarta, indonesia. email: rully.indra@mpmat.uad.ac.id how to cite: prahmana, r. c. i., sagita, l., hidayat, w., & utami, n. w. (2020). two decades of realistic mathematics education research in indonesia: a survey. infinity, 9(2), 223-246. 1. introduction realistic mathematics education (rme) is a specific mathematics instruction theory from the netherlands (van den heuvel-panhuizen & drijvers, 2020b). the basis of it is mathematics often out of the students' thoughts, so mathematics instruction must be connected to reality, or use context problems. context problems in rme are used from the beginning until the end of a learning sequence to add on (gravemeijer & doorman, 1999). therefore, rme can be used to color the approach to learning mathematics. rme is adopted in many countries, including indonesia. indonesian version of rme is known as pendidikan matematika realistik indonesia (pmri) or pendidikan matematika http://dx.doi.org/10.22460/infinity.v6i1.234 mailto:rully.indra@mpmat.uad.ac.id prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 224 realistik (pmr). the history of the development of rme in indonesia began since the publication of articles about rme in the jpm (one of the national journals in indonesia) in 2007. rme continues to grow in indonesia, as seen from the increasing number of rme articles published in various national journals in indonesia. there are multiple variations of rme articles in indonesia, such as literature studies, design research, case studies, classroom action research, and experiments. these various articles are coloring the use of rme on mathematical education research in indonesia since 2007. rme in indonesia for two decades has been discussed by zulkardi, putri, and wijaya (2020). zulkardi, putri, and wijaya (2020) reported the process of adapting, implementing, and developing rme to the indonesian context over the past two decades. their explanation starts from the history of how rme came to indonesia, how rme is learned from the netherlands, how rme is implemented, disseminated, and developed in indonesia. furthermore, the history continues to discuss initiative several activities namely the international master's program on mathematics education (impome), the international conference on design research (sea-dr), the mathematical literacy contest (klm) and the context-based mathematics tasks indonesia (comti) project, the web portal on pmri set up by the p4mris, the course on realistic mathematics education for junior secondary school mathematics teachers in southeast asia (sea-rme course), and the journal on mathematics education (jme). the journal, namely journal on mathematics education as the last initiative activity, is one of this article's resources. however, zulkardi, putri, and wijaya (2020) do not explain the significant topics in the various articles on applying rme into categories to obtain the rme classification in indonesia. this article would comprehensively analyze and divide them into groups to receive the rme classification major topics in indonesia. we can use this classification to add theories related to the development of rme in indonesia in the last two decades. 2. method this paper is a survey paper. we used the sample, which comes from ten highest rank accredited journals by the ministry of research, technology, and higher education, the republic of indonesia, as an achievement for the peer-reviewed journal with excellent management and publication quality. ten journals have been accredited in categories 1 and 2. in the first category, there is the journal on mathematics education (jme). next, there are nine journals in the second category, namely jurnal riset pendidikan matematika (jrpm), infinity journal (infinity), beta: jurnal tadris matematika (beta), international journal on emerging mathematics education (ijeme), al-jabar: jurnal pendidikan matematika (aljabar), jurnal pendidikan matematika (jpm), jurnal elemen (elemen), journal of research and advances in mathematics education (jramathedu), and aksioma: jurnal program studi pendidikan matematika (aksioma). this paper analyzed 110 papers published over the last 14 years from january 2007 to march 2020 in these ten accredited journals. the keywords used by researchers in collecting these reviewed articles are realistic mathematics education, pendidikan matematika realistik indonesia, pendidikan matematika realistik, and indonesian realistic mathematics education. the journal papers were analyzed according to the criterion of term realistic research on mathematics education. we started our research analysis by searching for the word "realistic" and "realistik" within each journal on ten highest rank accredited journals by the ministry of research, technology, and higher education, the republic of indonesia. furthermore, we classified all contributions from these papers, such as titles, years, journal names, topics of the studies, context, research subject, mathematics content, and mathematics ability of the courses. volume 9, no 2, september 2020, pp. 223-246 225 this paper aims to conduct a comprehensive study of the existing research on rme in indonesia among two decades. we analyzed comprehensively, categorized, compared, and divided the implementation of this approach in indonesia into several major topics. we present categories on rme articles based on the dominance of published rme articles, rme research subjects, mathematical topics, students' abilities, rme terms, and the research method used in the rme articles. the categories are obtained from various papers and articles that can be accessed and then affirmed by our interpretations as a team of researchers. the discussion begins with the number of rme articles published in each journal. further studies regarding some of the rme research subjects, mathematical topics, students' abilities, rme terms, and the research method is used in rme articles. in every section, we report the percentage of contributions assigned to each category. in the last section, we write separately on the table of rme research paper resumes in indonesia. 3. results and discussion 3.1. trends of rme articles on various journals in indonesia the trend of realistic mathematics education publication from 10 journals from 2007 to 2013 has increased in the number of articles. both articles published in the jpm became the starting point of the stretching of the publication of rme papers in indonesia. jpm became the first journal to publish two rme articles in 2007. soedjadi (2007), in his article, explained the philosophical basis, theoretical basis (rme principle), applicative basis, and the impact of realistic mathematics education. furthermore, the second article published in jpm journal in 2007 discusses the development of teacher manuals that are equipped with student books, student worksheets, play activities, enrichment activities, and remedial activities (amin, 2007). the number of publications decreased by 54% in 2014. furthermore, the number of articles published has fluctuated until 2017. on the other hand, the number of publications has increased again until 2018 and reached the highest number in 2019 of 15 articles scattered in aksioma, al-jabar, jme, ijeme, infinity, and jramathedu. based on the number of rme article releases each year, jme and jpm are journals that consistently produce papers with rme scope. it can be seen in most jme word count keywords, namely research design, rme, and pmri. the number of rme articles published by jme is 37 articles until 2020. this number is the highest number of articles from other journals in proportion to the age of jme. the first rme article published by jme in 2010 discussed the effectiveness of rme in improving problem-solving ability in yogyakarta for junior high school students (sugiman & kusumah, 2010). jpm, as a pioneer journal of rme articles in indonesia until 2020, has published 26 articles with even distribution each year. the number of rme articles that have been published by each journal is directly proportional to each journals' age. based on the analysis, the dominance of rme articles is in jme and jpm. rme articles in the jme are more than one-third of all rme articles, while rme articles in the jpm are almost a quarter of all rme articles. the number of rme articles in each journal shown in figure 1. prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 226 figure 1. the number of rme articles in each journal from all journals that contain rme articles, in general, the focus and scope of each journal are in the fields of education and mathematics education. based on the information presented on the journal website, there are only four journals that specifically mention rme as one of the focus and scope, including jpm, jme, ijeme, and infinity. 3.2. research subject on rme articles in indonesia the momentum for rme research in indonesia began in 2001 through the experimental learning of six lecturers at the lptk higher education institutions (educational personnel education institutions) as an rme team in 12 elementary schools. this was a form of government intervention at that time to make elementary school teachers the spearhead of the expansion of rme in indonesia as a reform of mathematics learning (sembiring, 2010). since then, rme's progress has been continued by providing rme knowledge for lptk lecturers, especially elementary school teacher candidates. the collaboration is also with universities to establish an international master's program on mathematics education (impome), holding rme workshops with teachers, and continuous dissemination with stakeholders. based on the history of rme entry into indonesia, which began with trials at the elementary level, up to two decades of rme in indonesia, more than ⅖ rme research in indonesia took the subject of elementary school. slowly but surely, the application of rme is not only at the elementary level. more than ⅖ the application of rme learning at the secondary school level has been carried out, as shown in figure 2. expansion of this subject is very likely to occur because the main objective of rme's entry into indonesia is to bring about fundamental changes in the teaching and learning process of mathematics in indonesia (julie, suwarsono, & juniati, 2013), besides that, it is also because the mathematics curriculum in indonesia at the secondary school level is mostly applied mathematics material. so that the principle of rme as a human activity using a real context volume 9, no 2, september 2020, pp. 223-246 227 in learning can be done. the research subject on rme articles in indonesia has shown in figure 2. figure 2. the research subject on rme articles in indonesia figure 2 shows that the most rme research studies were carried out in elementary schools and junior high schools. almost half of the research was conducted in elementary school, and nearly half of the other research was conducted in junior high school. studies conducted in senior high school, teachers or pre-service teachers, universities, and others tend to be a little. the application of rme at the university level is still low, with a proportion of less than 10% or consisting of 5 well-known publications in three journals, including jme, aljabar, and aksioma. the small number of rme publications at the university level is interesting for us to discuss. the question that arises is, "can the basic principles of rme developed for elementary school mathematics be adapted for mathematics topics in university?". the answer, indeed, is that rme can be adapted for learning at the university level. several previous studies have applied rme at the university level, such as calculus (kwon, 2002; rasmussen, kwon, allen, marrongelle, & burtch, 2006), algebra (larsen, 2013), geometry (larsen & zandieh, 2008). these studies show that the presentation of problems in mathematics learning at the university level is not only done with textbooks but can be more varied with the interaction model between lecturers and students, as well as the presentation of the problems used by students to learn. in line with the results of research conducted in the usa, rasmussen, and king (2000) emphasized that mathematics learning reforms that have been implemented at the primary and secondary school levels can be expanded at the university level. one of the keys is the preparation of a local instructional theory before the lecture process based on the three basic principles of rme according to gravemeijer (1994) where learning accommodates guided reinvention and progressive mathematizing, didactical phenomenology, and selfdeveloped models (larsen, 2013; larsen & zandieh, 2008; rasmussen et al., 2006). become an opportunity for the development of rme in indonesia through the expansion of the rme subject at the university level. thus, the reform of mathematics learning through rme is becoming more extensive and comprehensive. prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 228 3.3. mathematical topic on rme articles in indonesia mathematics has many topics or materials that must be studied. however, some of them are quite difficult or cannot be taught using the rme approach. applied mathematics topic is easier to design on rme learning, which uses 6 principles in rme: activity principle, reality principle, level principle, intertwinement principle, interaction principle, and guidance principle (van den heuvel-panhuizen & wijers, 2005). as we know, there are various mathematics topics in schools. likewise, rme has various topic content. based on the analysis of various rme articles in various journals in indonesia, the topics discussed is quite diverse, including combinatorial, integral, circle equation, ratio, linear equation and inequality, relation and function, equation of two-variable linear, trigonometry, social arithmetic, lcm, gcd, mathematics logic, linear programming, length and time measurement, algebra, 2d shapes, 3d shapes, number pattern, decimals, geometry, percentage, set, number, area and perimeter, fractions, integer, speed, table and graphs, time measurement, statistics, reflection, and symmetry, etc. however, we can classify the topic so that it is easier to analyze. at outline, five topic content standards throughout mathematics class are number and operations, algebra, geometry, measurement, and data analysis and probability (nctm, 2000; walle, karp, & bay-williams, 2013). the classification of the topic on the topic content standards shown in figure 3. figure 3. mathematical topic in indonesian journal of rme a number of rme research on the topic of “numbers and operations'' are in elementary and junior high schools, not in senior high schools or even universities. the widest research subjects on the numbers and operations topic are on elementary school students because the numbers and operations is the most heavily emphasized strand up to grade 5 (walle, karp, & bay-williams, 2013). hence, numbers and operations are the most widely discussed in the rme article published in various journals in indonesia. nearly 40% of the topics discussed in the rme articles are numbers and operations (figure 3). coincidentally, we found an rme research on "numbers and operation" in the jme journal, in which the research subject was not students of elementary or junior high school. however, the research topic material is also related to numbers and operations in elementary schools. volume 9, no 2, september 2020, pp. 223-246 229 meanwhile, geometry is also widely discussed as material in rme research. nearly one-third of the material discussed in rme research is geometry (figure 3). it is because there are a lot of objects around us that contain geometric values that are suitable for use in learning mathematics at an elementary or junior high level. many research subjects of rme are also in junior high school. therefore, algebra is on the third-ranks in the number of articles discussing rme on the rme article. almost one-sixth of all rme articles are about algebra (figure 3). algebra heavily emphasized a strong focus in junior and senior high school, and a lesser emphasis in early grades (walle, karp, & bay-williams, 2013). mathematical topics on data analysis and probability, measurement, and others (integral, trigonometry, mathematics logic) are rarely discussed in the rme research. 3.4. mathematics ability on rme articles in indonesia the goals of the research on mathematics education have various types of mathematical abilities. there are many categories on it, i.e. taxonomy bloom: remembering, understanding, applying, analyzing, evaluating, and creating (bloom, 1956); problem solving, reasoning & proof, communication, connections, representation (nctm, 2000); etc. this paper categorizes the goal of the research on rme as mathematics ability and non-mathematics ability based on taxonomy bloom and nctm. mathematics abilities are mathematical abilities obtained in learning with rme and non-mathematics abilities, namely attitudes or abilities that are not mathematical abilities. mathematics ability is divided into 11 parts: mathematical understanding, communication, connections, representation, reasoning and proof, mathematical literacy, critical thinking, creative thinking, problemsolving, disposition, and intuitive skills. furthermore, non-mathematics ability consists of eight items: character, student activities, learning independence, self-efficacy, student performance, learning interest, achievement, and intrapersonal intelligence. the categories created to position mathematics ability in various rme articles are presented in figure 4. figure 4. the categories of mathematics ability on rme articles rme research related to mathematical understanding amounted to more than ¼ of the total sample used, ¼ the rest was mostly on communication skills, and problem-solving was in the third-largest order (figure 4). revina and leung (2018) explain that the goal of prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 230 learning mathematics in the curriculum in indonesia is to equip students with mathematical understanding skills and to develop communication and problem-solving. this is a reference for researchers to conduct research in achieving the goals of the indonesian curriculum. if we analyze more deeply, the topic of mathematical understanding used by researchers is mostly in the number and operations group of material; the rest is on measurement and geometry with a small portion. it is presented in figure 3, where numbers and operations are the most material used as rme research topics in the last two decades. nursyahidah, putri, and somakim (2013) explained that the number and operations material is one of the prerequisite knowledges for learning mathematics on other topics. so that students must have the correct and robust concept on number and operations material. in addition to mathematical understanding and other mathematical skills, as in figure 4, the author performs another grouping using the term "others." in this group, several researchers conducted research by looking for the impact of rme learning on students. such as student character (palinussa, 2013), student personality (kowiyah, mulyawati, & umam, 2019; amin, 2007), gender (hasratuddin, 2010), and students' intuitive abilities (hirza et al., 2014). for example, research on the four personality types of students after participating in rme learning found students with rational personality types were very good at identifying and making examples, traits, and conditions. students with idealistic personality types are very good at identifying verbally and in written form. the conceptual understanding of the four personality types when identifying and differentiating concepts is still weak. meanwhile, for the aspect of mathematical representation, the four personality types are excellent in visual, accurate representation in the form of numbers and tables. there is an opportunity for future rme research to follow up on the results of the classifications that have been made by these researchers, namely, by designing an rme lesson that is tailored based on student character, student personality, gender, and student cognitive level. 3.5. various terms used on rme articles in indonesia the adoption of rme in indonesia conducted in various studies refers to the netherlands. however, the use of the term at rme in indonesia is varied, starting from using the original term "rme" and using another term, i.e. pendidikan matematika realistik indonesia (pmri), pendidikan matematika realistik (pmr), indonesian mathematics realistic education (irme), and realistic, educational, contextual, cognitive, and evaluation (recce). variations of the terms used in various articles can be seen in the figure 5. the term most commonly used is the original term, rme, and the least used is the term recce. the term which is widely used, namely pmri and pmr, occupies an almost similar position. the term rme has become very popular in publications in indonesia, where nearly half of the 110 samples used in this article come from eight journals. this is commonplace if most authors choose to use the term rme because this term is commonly used throughout the world. zulkardi et al. (2020) use an alternate term between rme and pmri in an article entitled "two decade of realistic mathematics education," depending on the setting of the story being used. another term that is also used in almost a quarter of the part is pmr. this term first appeared in 2010 through the writings of sugiman and kusumah (2010). despite the different terms used, these 20 articles refer to the definitions and principles of rme. this can be seen from the reference sources used mostly come from the theory of gravemeijer. another small section of the article uses the term irme which is the english equivalent or modification of pmri. two authors, namely assiti, zulkardi, and darmawijoyo (2013), as volume 9, no 2, september 2020, pp. 223-246 231 well as risdiyanti and prahmana (2020), have previously emphasized that the term irme used is the adoption of rme both the definition and principle used. figure 5. variations of the terms of “rme” on rme articles the interesting thing from the analysis of the term rme used in 10 sample journals is the emergence of the term recce published in jme by chong, shahrill, and li (2019) and only one of the entire samples. as shown in figure 5, it becomes a tiny part of the orange. this framework was created and given the term "recce-model," which emphasizes understanding and thinking with a view of mathematics embedded in real life. "recce," which stands for realistic, educational, contextual, cognitive, and evaluation. this framework includes the principles underlying mathematics learning through activities that use real-life in developing students' problem-solving competencies. seeing this phenomenon, it is possible that in the future, there will be a development of an approach or model that adopts the rme principle so that other terms will appear. for example, the terms for rme that are online based, rme based on culture, or rme for children with special needs. 3.6. research methods on rme articles in indonesia the articles on rme in 10 sample journals were mostly the result of research with various methods. the research methods used include literature review, design research, experiment, classroom action research (car), causal-comparative research, research and development (rnd), mix method, and descriptive qualitative. the data shows that more than a quarter of the rme research uses the design research method. almost all sample journals have articles with the design research method except jrpm, al-jabar, and axioms. research with the design research method is 71% in the jme journal, about 10% of design research articles at jpm, and the rest is in other journals. the second-largest research method used by rme researchers is the experimental method. the number of articles that contain experimental methods in the jme journal and in jpm is the recce 0.9% prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 232 same, namely about 23%, respectively. other journals that contain articles on rme with innovative methods are jrpm, al-jabar, jramathedu, and aksioma, which are about 7% each, and element around 3%. in other words, almost all journals contain rme with experimental methods, except ijeme and beta journals. the third-largest research method used by researchers is research and development (rnd). research with the rnd method is mostly at jpm. there are 50% of articles with the rnd method in jpm. axioma also has a lot of rnd research methods, which is around 30%, while the rest are in jrpm, al-jabar, and jramathedu. another research method used is descriptive qualitative. qualitative descriptive methods are present in almost all journals, except ijeme and axioms. apart from these various methods, other research methods are prevalent and not widely used in rme research. for example, classroom action research methods only exist in beta, element, and jpm, respectively 1. meanwhile, comparative causal research and mix methods occupy a minority position. there are not many rme articles in various journals in indonesia that use these two methods. based on the analysis, the dominant research methods of the articles from all journals that contain rme articles use design research, followed by experiment, research and development, and descriptive qualitative. the rarely used research methods are causalcomparative research and mix methods between quantitative and qualitative research. of all the research methods used in the rme article in indonesia can be seen in figure 6. figure 6. the research methods on rme articles in indonesia although design research dominates rme research for up to two decades of rme in indonesia, the author predicts that research with the rme research design method will still be in demand by rme researchers. based on the opinion of van den heuvel-panhuizen and drijvers (2020a) that a teacher must have a proactive role in preparing rme learning scenarios and designs. through design, research will produce a local instruction theory in the learning process. the process of finding the theory of local instruction is long and continuous, even with the same material. volume 9, no 2, september 2020, pp. 223-246 233 3.7. rme research paper in indonesia in the last two decades the question posed by zulkardi, putri, and wijaya (2020) in two decades of rme in indonesia is, "is pmri still alive in mathematics education in indonesia?" in his paper, zulkardi explained about the untold story of the development of rme in indonesia for two decades. table 1 is a classification based on the subject level starting from elementary school, junior high school, senior high school, undergraduate, and others for those not included in formal education. the data presented in table 1 generally answer this question, that rme in indonesia is still alive, this can be seen from the number of rme publications scattered in ten journals that are accredited by the ministry of research, technology and higher education, which varies greatly from a subject, topic, term, and method used. jpm, jme, ijeme, and infinity are journals that consistently produce publications with one focus and scope on rme. publications in journals are proof that rme activities as an innovation in mathematics learning are still ongoing. journals can be used as a medium for stakeholders such as teachers, lecturers, students, students, teacher educators, researchers, and book writers to accept and share their findings. the next question in zulkardi, putri, and wijaya (2020) are "what are the activities or developments of pmri currently?". this article found the distribution of rme subjects that did not exist in previous years. research conducted by risdiyanti and prahmana (2020) at the yogyakarta mathematics study club (ymsc), a club consisting of several mathematics education graduates who are involved in innovating mathematics learning. another point of view, the researcher found the background of conducting research on mathematical understanding, including the learning process at the primary and secondary school levels still using conventional teacher-centered learning and the delivery of material that is too formal (pramudiani, zulkardi, hartono, & amerom, 2011; kesumawati, 2015; zabeta, hartono, & putri, 2015; nursyahidah, putri, & somakim, 2013, sari, juniati, & patahudin, 2012; rifandi, 2016), the learning process of mathematics is not sufficient to help develop students' mathematical thinking (anwar, budayasa, amin, & de haan, 2012; lismareni, somakim, & kesumawati, 2015), learning is less meaningful and focuses on procedures (khairunnisak, maghfirotun, juniati, & de haan, 2012), tends to focus on procedures (rianasari, budayasa, & patahuddin, 2012). with the learning process that has occurred so far, students do not understand the concept of numbers and operations (rianasari et al., 2012; nursyahidah et al., 2013; julie et al., 2013), students' understanding of the concept of prism volume material is still lacking. (fadlilah, 2014), and students do not have individual readiness to understand the concept in depth (sari, 2014). in the last two decades, more than ¼ of the rme research is design research. more than ½ of the design research is the material number and operation. the more so is the material on algebra, data analysis, and probability, and measurement, each with the same amount. these materials are dominated by material at the elementary level. products from development research produce rme learning tools including teachers and students manual books (diba, zulkardi, & saleh, 2009), teaching materials (misdalina, zulkardi, & purwoko, 2009; fuadiah, zulkardi, & hiltrimartin, 2009; taufik, zulkardi, & somakim, 2012; syutaridho, zulkardi, & hartono, 2012; simanulang, 2014), student worksheet (komsiatun, 2018; wewe & juliawan, 2019), assessment (rahayu, purwoko, & zulkardi, 2008; utami, & nirawati, 2018 ), it-based media (wahyuni, masykur, & pratiwi, 2019; pratama, yahya, & pratiwi, 2019). the rme experimentation shows the effectiveness of learning using rme compared to conventional, ctl, scientific, and expository learning. several researchers used the one-group experimental method, including testing the implementation of realistic mathematics education using geometer's sketchpad (dhayanti, johar, & zubainur, 2018), implementing rme based on adobe flash professional cs6 (umbara & nuraeni, 2019). prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 234 lastly, table 1 summarizes the grouping of sample articles in this paper based on the mathematical content topic by subject level. therefore, we hope that this table could be used as a guide to examine the distribution of rme studies over the two decades in indonesia by researchers. table 1. resume of rme research paper in indonesia no level mathematical content topic references 1 elementary algebra (jupri et al., 2020; apsari et al., 2000; nursyahidah, saputro, & rubowo, 2018; lestariningsih et al., 2015; assiti et al., 2013; dhayanti et al., 2018) geometry (komsiatun, 2018; fuadiah et al., 2009; taufik et al., 2012; syutaridho et al., 2012; alkusaeri, 2014; palupi, 2017; rizkianto et al., 2013; winarti et al., 2012; yuberta et al., 2011; haris & ilma, 2011; alriavindrafunny, 2014; wewe & juliawan, 2019; helsa & hartono, 2001) measurement (amin, 2007, jaelani et al., 2013; khikmiyah et al., 2012) number and operation (muncarno & astuti, 2018; afriyansyah & putri, 2013; edo et al., 2015; johar et al., 2016; yusmanita et al 2018; julie et al., 2013, nursyahidah et al., 2013; murdiyani et al., 2013; triyani et al., 2012, khairunnisak et al, 2012; sari et al., 2012; anwar et al., 2012; rianasari et al., 2012; shanty et al., 2011; pramudiani et al., 2011; putra et al., 2011; astuti, 2014; shanty, 2016; nasution et al., 2018; saleh et al., 2018; ardiyani et all., 2018; putri & zulkardi, 2019; rifandi, 2016; diba et al., 2009) others (zulkardi & ilma, 2010; hirza, kusumah, darhim, & zulkardi, 2014; haji, yumiati, & zamzaili, 2019) 2 junior high school algebra (septianawati, 2014; wahyuni et al., 2019; sulastri et al., 2017; kusumaningsih et al., 2018; nuraida & amam, 2019; simanullang, 2013) data analysis and probability (putria et al., 2015; muttaqin et al., 2017; sari, 2014) volume 9, no 2, september 2020, pp. 223-246 235 no level mathematical content topic references geometry (utami & nirawati, 2018; putra, 2016; fadlilah, 2014; yunisha et al., 2016; maisyarah & prahmana, 2020; sari, 2016; nurdiansyah & prahmana, 2017; haji & abdullah, 2015; khaerunisak et al., 2017; fitriani et al., 2018; fahrurozi et al., 2018; rahayu et al., 2008; farhatin, 2012) number and operation (sari & sari, 2019; rahmawati & amah, 2018; simanulang et al., 2013; lismareni et al., 2015; zabeta et al., 2015; meryansumayeka et al., 2011; saleh & isa, 2015; suherman, 2015) others (paradita, vahlia & es, 2019; hasratuddin, 2010; kesumawati, 2014, 2015; palinussa, 2013; sugiman & kusumah, 2010; abdurahim, 2016; wibowo, 2017; habsah, 2017; haji & abdullah, 2016; herawaty & rusdi, 2016; muhtarom, nizaruddin, nursyahidah & happy, 2019; umbara & nuraeni, 2019) 3 senior high school geometry (dwijayani, 2018; krismiati, 2013) algebra (hidayat & iksan, 2015; niswarni, 2012) data analysis and probability (meika et al., 2019) others (zulkardi, 2019; budi, 2008, chong, shahrill & li, 2019) 4 undergraduate geometry (kowiyah et al., 2019) others (pratama et al., 2019; farida et al., 2019, julie et al., 2013, 2014; yilmaz, 2020) 5 others number and operations (risdiyanti & prahmana, 2020) others (setyaningsih et al., 2019) 4. conclusion we have summarized 110 papers displayed in table 1 and categories the published rme articles on seven categories, i.e. the dominance of published rme articles, rme research subjects, mathematical topics, students' abilities, rme terms, and the research prahmana, sagita, hidayat, & utami, two decades of realistic mathematics education … 236 method used in rme articles. rme was adopted from the netherlands, so the most used term is "rme," which is the original term, followed by pmri. on the other hand, most articles were in the jme and followed by the jpm, with the most widely used design research method. the most topic of rme material studied is numbers and operations, followed by geometry. it is an elementary level of school mathematics. hence, rme research is mostly in elementary schools and followed by junior high schools. we encourage rme researchers to conduct the study at other levels, such as higher education. the mathematics materials using rme is needed to be more diverse and comprehensive, not only on early levels or elementary students but also on the advanced level mathematics subject. moreover, most of the mathematics ability in rme research is mathematical understanding, followed by mathematics achievement. whereas based on the characteristics of pmri, it is not only mathematical understanding or achievement but also many other mathematical abilities that can be developed. therefore, we encourage the rme researchers to study the effect of rme on different mathematics abilities. furthermore, rme researchers not only can research mathematics abilities but also include affective abilities. overall, we encourage the community on the teaching and learning of mathematical modeling and applications to submit their papers more often to high impact journals, to improve the visibility of modeling in the mathematics education community. furthermore, more quantitatively oriented studies seem to be necessary for the future teacher to come to more general results on the effects of teaching and learn mathematical modeling, focusing on cognitive aspects and including significant elements. acknowledgements the author would like to all authors whose papers are cited in this paper as a valuable reference and resource for this paper. furthermore, thanks to all reviewers for their helpful feedback. references abdurahim, a. 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(2019). celebration of a decade of jme. journal on mathematics education, 10(1), v–vi. https://doi.org/10.22342/jme.10.1.6916.v-vi https://doi.org/10.1007/978-3-030-20223-1_18 https://doi.org/10.22342/jme.10.1.6916.v-vi infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p7-14 7 a comparison between stad-type and tps-type cooperative learning in middle school students’ geometry learning asri ode samura iain ternate, jl. lumba-lumba, marikurubu, ternate city, north maluku, indonesia asri22samura@gmail.com received: march 21, 2017 ; accepted: july 11, 2017 abstract this study aims to describe and compare the effectiveness of the application of stad and tps types in cooperative learning to the students. this study was a quasi-experiments research in which students were designed into a form of the experimental group (pre-test and post-test group designs), where the cooperative learning of stad and tps types will be applied in learning mathematics. the population in this study were junior high school students in ternate, while samples were taken by 60 random students who had the same characteristics as the population. the result of the study shows that cooperative learning model of both stad and tps types are very effective to be applied on studying mathematics. however, between the stad and tps types of cooperative learning, the stad types are more effective than tps, seen from accomplishing standards of competence, mathematical communication skills, and mathematical thinking ability of junior high school students. keywords: logical reasoning, stad, tps. abstrak penelitian ini bertujuan untuk mendeskripsikan perbandingan keefektifan penerapan pembelajaran kooperatif tipe stad dan tipe tps pada siswa. penelitian ini merupakan penelitian quasi-eksperimen yang didesain menggunakan bentuk kelompok eksperimen pretes-postes (pretest-postest group design) kepada siswa yang akan diterapkan pembelajaran kooperatif tipe stad dan tipe tps pada pembelajaran matematika. populasi dalam penelitian ini adalah siswa smp di ternate, sedangkan sampel diambil 60 orang siswa secara acak yang memiliki karakteristik yang sama dengan populasi. hasil penelitian menunjukkan bahwa: (1) model pembelajaran kooperatif learning tipe stad dan tipe tps sangat efektif jika diterapkan pada pembelajaran matematika; (2) kedua model pembelajaran kooperatif learning kelas tipe stad dan tipe kelas tps dilihat keefektifannya, lebih efektif kelas tipe stad dibandingkan dengan kelas tipe tps dilihat dari ketercapaian standar kompetensi, kemampuan komunikasi matematis, dan kemampuan berpikir matematis siswa smp. kata kunci: penalaran logis, stad, tps. how to cite: samura, a.o. (2018). a comparison between stad-type and tps-type cooperative learning in middle school students’ geometry learning. infinity, 7 (1), 7-14 doi:10.22460/infinity.v7i1.p7-14 samura, a comparison between stad-type and tps-type … 8 introduction advances in sciences and technology requires that one has the ability to master the available information and knowledge (herman, 2007), which necessitates the ability to obtain, sort and manage information. this ability must be based on critical, systematic and logical thinking as it is material to the analysisand evaluation of arguments in order to take rational and accountable decisions. therefore, educational programs that can promote critical, systematic and logical thinking skills are needed, one of which is mathematics. the development of the world of mathematical education nowadays contitutes the connection between mathematics’ function as a didactic science and as educational psychology. in fact, the position of mathematics as a science is mutli-interpretative in nature. mathematics is one of the subjects taught at schools, including elementary schools, middle schools, high schools and college or university. since mathematics taught at schools is part of mathematics, different characteristics and interpretations of mathematical contents from various points of view play a significant role in the development of sciences and technology. mathematics is an exact science and it is organized in a systematic manner. as a science, mathematics has logical and critical aspects that have consistent order. the knowledge of mathematics is formed through the thought of the experience of certain objects or events. mathematical objecs are intended to motivate every level of education to learn mathematics. regarding the existence of mathematics, hidayat (2017) describes the reason why it is important to learn mathematics. according to him, mathematics emphasizes more the activities in the rational world (reasoning) instead of experiments. in other words, mathematical observation is formed due to human thoughts which are related to ideas, processes and reasoning. consistent and logical mathematical structure is essential for everyone to develop their critical thinking skill for fulfilling his or her needs in life. this consistency requires mathematics teachers’ professionalism in order to be able to act appropriately in selecting learning strategies or in explaining mathematical contents for the learning process at schools (hidayat, wahyudin, & prabawanto, 2018). competency in mathematics refers to expertise performance capability, which is built through mathematical knowledge, skills and attitude fostering. one’s mathematical competency can be seen from his or her capability in meeting the specification of the job, or performance capability in handling the job in mathematical activities. competency and measurement of learning outcomes are closely related to learning process, either for teachers’ purpose in improving the quality of the learning or for students’ purpose in improving the learning outcomes. to put it simply, they want to know the grade that they have achieved. to answer this question, teachers need to develop a measurement instrument that can measure the mastery of competency referred to in the learning objectives. by communicating ideas when learning, students will be able to use and control mathematical concepts with higher confidence than they have presently. students have to assume a different role in the mathematics class, and so do the teachers. students must be involved and responsible for their own learning, whereas the teachers must help them do so. teachers may perform their task by: (1). changing the way students interact with their works and with each volume 7, no. 1, february 2018 pp 7-14. 9 other; (2). providing them with more challenging problems to be solved, and (3). asking them to express mathematical ideas in writing. los angeles county office of education (mahmudi, 2009) mentions several forms of mathematical communication, namely (a) reflecting and clarifying thoughts on mathematical ideas; (b) linking everyday language with mathematical language, which uses symbols; (c) using the skills of reading, listening, interpreting, and evaluating mathematics ideas, and (d) using mathematical ideas to make conjectures and to convey convincing arguments. mathematical communication covers written communication and spoken or verbal communication. written communication can be in the form of words, figures, tables or other forms which depict the thinking process of the students. communication can be in the form of problem solving or mathematical proving, which describe the ability of the students in organizing various concepts when solving problems. meanwhile, spoken communication can be in the form of verbal expression and explanation of a mathematical idea. spoken communication may take place through interaction between students, for example when learning in a group discussion setting (rahmi, nadia, hasibah, & hidayat, 2017). the learning pattern developed in indonesia nowadays requires students’ activeness in the learning and teaching process, and students’ creativity in processing the materials provided by the teacher. this is intended to allow meaningful reasoning construction and critical thinking in analyzing and solving mathematical problems faced by students. according to hidayat (2012), students who think critically are those who are able to identify, evaluate and construct arguments, and who are able to solve problems correctly. students who think critically will be able to help his or herself and others solve the problems encountered. creating learning situation that is fun, filled with cooperative spirit, wise and creative can be done through cooperative learning. hidayat (2012) states that cooperative learning allows for fun learning situation, in which students will have equal opportunity, competition is turned into friendship, cooperation and participation spirit is reinforced, and all students have the right to be wise and creative. cooperative learning is developed into a number of types. among those types, the types of cooperative learning to be studied in this research are student teams achievement division (stad) and think pair share (tps). method the method of this research is a quasi-experiment research. it is designed using experimental groups (pretest and post-test) to the group of students who applied the cooperative learning, both stad and tps types. the research data analysis was focused on describing comprehensively the mean, standard deviation, variants, minimum score and maximum score of the data before and after treatment, the standard achievement improvment, mathematical communication skills and mathematical thinking skills after the stad type and tps type cooperative learning was implemented. the quantitative research data were obtained from pre-test and post-test given to 60 students before and after treatment using stad type and tps type cooperative learning. the qualitative research data were obtained from observation and interview. observation included: the activities of students to whom cooperative learning were implemented, the activities of samura, a comparison between stad-type and tps-type … 10 students and teacher (the researcher) in the implementation of stad type and tps type cooperative learning in learning cube and rectangular prism. the interview data were obtained from some students to whom the cooperative learning had been implemented in accordance with the problem characteristics shown by students in answering and completing the research instrument. furthermore, in order to test the difference of competency standard achievement data, matheatical communication skills and mathematical thinking skills between those from pretest and those from post-test, statistic test: t, and mann-whitney u were used. to meet the testing requirements, data normality and variants homogenity tests were conducted to every data pair. the normality test used shapiro-wilk (sw) statistics and the homogenity test for every data pair used levene test. the data were analyzed using software spps for windows version 20. results and discussion the description of competency standard achievement data, either for the class employing stad or the class employing tps, can be seen in table 1. table 1. description of competency standard achievement data description stad tps beginning end beginning end mean 50.23 97.09 49.79 92.67 theoretical score 100 100 100 100 maximum score 68.54 98.79 62.72 99.29 minimum score 39.34 71.95 32.91 57.89 standard deviation 8.72 8.19 7.78 8.94 the data above revealed that the mean score of students competency standard achievement, either in class that employs stad technique or class that employes tps technique, before the treatment has yet to achieve the mean score of 75, while after the treatment, the mean score exceeds 75. the description of mathematical communication skills data, for stad and tps techniques, are presented in table 2. table 2. description of mathematical communication skills data description stad tps beginning end beginning end mean score 50.23 97.09 49.79 92.67 theoretic maximum score 69.54 67.98 65.45 99.75 theoritic minimum score 7.89 79.68 24.72 62.63 standard deviation 11.62 7.94 13.43 12.76 variants 149.09 68.74 199.87 165.71 based on table 2, the mean score of students’ mathematical communication skills, either in the class emloying stad technique or the class employing tps technique, before the volume 7, no. 1, february 2018 pp 7-14. 11 treatment has yet to achieve the mean score of 75, while after the treatment, the mean score exceeds 75. the description of the data of students’ mathematical thinking skills in the mathematical learning process for classes that employed stad and tps can be seen in table 3. table 3. description of mathematical thinking skills data description stad tps beginning ending beginning ending mean score 39.45 87.54 41.10 85.42 theoretical maximum score 60.12 95.52 70.94 97.15 theroritical minimum score 19.84 65.58 18.62 62.63 standard deviation 10.83 12.79 15.65 12.57 variants 125.15 105.17 176.86 98.17 based on table 3, the mean score of students’ creative thinking before the treatment is under 75. after receiving treatment using stad and tps techniques, the mean score exceeds 75. now we are discussing the data normality and homogenity. the results of normality and homogenty tests of students’ competency standard achievement, mathematical communication skills and mathematical thinking skills data before and after the treatment using stad and tps techniques are presented in table 4 and 5. table 4. normality test data class before the treatment after the treatment stad 65.75 % 55.35 % tps 56.54% 60.15% table 4 shows that about 50% of the data has value . this means that the data of competency standard achievement, mathematical communication skills and mathematical thinking skills before and after the treatment using stad technique in the class had met the normality assumption. table 5. homogenity test data before after treatment treatment box”s m 5.785 10.569 f 1.132 1.765 sig 0.812 0.187 table 5 shows that the f significancy value is above 0.05.in other words, the competency standard achievement, mathematical communication ability and mathematical thinking values before and after treatment have met the homogenity assumption. samura, a comparison between stad-type and tps-type … 12 the results of the test of cooperative learning effectiveness (stad and tps techniques) in the competency standard achievement, mathematical communication ability and mathematical thinking aspects are presented in table 6. table 6. one sample t-test result data aspects stad tps sig sig standard achievements 10.532 0.000 10.357 0.000 mathematical communication skills 8.923 0.000 3.352 0.000 mathematical thinking skills 7.632 0.000 5.324 0.000 mathematical thinking skills questionnaire 4.513 0.000 3.132 0.000 table 6 shows that the t significancy values of all aspects are lower than 0.05. this means that h0 is rejected. in other words, the stad type and tps type cooperative learning is effective viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills aspects. the results shown above conform with the theoretical review about both types of cooperative learning viewed from the three aspects being measured. this is due to the fact that in the stad type cooperative learning, students were active in the discussion in order to solve the problem, in which positive interdependence were established between students who continuously tried to improve their achievement in order that their groups receive reward. in stad type cooperative learning, students were directly involved in the learning, starting from understanding the problem until finding out the concept contained in the problems being faced. the involvement did not stop when they found the concept. it continued in the class discussion activity, either the discussion on concept finding or discussion on the result of working on the example or excercise in front of the class. students were allowed to give responses, questions or even answers related to what had been delivered by other groups or particular students in front of the class during the discussion. such activity made students not only skillful in answering questions, but also skillful in providing rationale related to the answers they offered/had. in every meeting in which stad type cooperative learning was employed, rewards were given to groups, where students took part in contributing points to their groups. the group that achieved the highest score would be rewarded. the teacher told the students that whoever contributed the highest score/point to his or her group would boost his or her group’s performance and make the group the best group. such motivation will motivate students to make their respective group achieve the highest scores. besides, in the tps type cooperative learning, students were given opportunities to solve problem-natured questions. students were given time to solve their own problems (think), in which they could have a better understanding on the problems given. at least, in this stage, students would have transient understanding and answers for the problems given. afterwards, students were given an opportunity to sit in pairs to discuss the problem given, unify opinions/thoughts and find the concept. after finding the answer, students presented the results of thier discussion in front of the class (share). in this stage, students’ understanding volume 7, no. 1, february 2018 pp 7-14. 13 could increase because they had the chance to ask about what they did not understand to their friends or the group that presented their answers and shared ideas in front of the class. the results that show whether there is any difference in the ability of both classes before and after the treatment, and whether there is any difference in the effectiveness of both cooperative learning types (stad type and tps type) viewed from the competency standard achievement, mathematical communication ability and mathematical thinking are presented in table 7. table 7. manova result data before and after treatment f sig. class (before treatment) 0.635 0.655 class (after treatment) 9.322 0.000 table 7 shows that the data before treatment has f significance value greater than 0.05, while the data after the treatment has f significance value lower than 0.05. thismeans that there is no difference in the initial ability between stad type and tps type class before the treatment viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills aspects. after the treatment, there is a difference in the effectiveness between stad type and tps type cooperative learning viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills aspects. after finding out that there is a difference in the effectiveness between both learning methods, t-benferroni test was conducted to see whether stad type cooperative learning is more effective than tps type cooperative learning viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills aspects. the results of t-benferroni test are presented in table 8. table 8. t-benferroni test results aspects tbenferroni ( ) competency standard achievement 3.75 2.27 mathematical communication skills 3.10 2.27 mathematical thinking skills 2.45 2.27 table 8 describes that t-benferroni > . in other words, it can be said that stad type cooperative learning is more effective than tps type cooperative learning, which is viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills aspects. these results are also in line with the theoretical review which revealed that stad type cooperative learning is more effective than tps type cooperative learning viewed from the three aspects that had been measured. this is the case in the learning process that used stad model. it was also found that the students are not only involved in finding concept and class discussion, but also motivated to improve their groups’ scores in order that their group gain greater reward than other groups. samura, a comparison between stad-type and tps-type … 14 if this research is compared to other relevant studies, the results of this research are in line with those of the other studies. this can be seen from the results of other relevant studies that suggest that stad type cooperative learning is more effective than tps type cooperative learning viewed from the competency standard achievement, mathematical communication skills and mathematical thinking skills. the results of the study have been described above, supported by a theoretical review of relevant studies such as hendriana, hidayat & ristiana (2018), but as mentioned earlier, some limitations that hindered this research were still found. considering these weaknesses, some suggestions were offerred: this research is only limited to eight meetings to make it easy for assessing the competency standard achievement, mathematical communication ability and mathematical thinking. besides, it takes a considerable time to be able to find out how well the three aspects develop. the researcher selected only materials about cube and rectangular prism on flat sides in this research, which limits generalization in relation to the research results. conclusion according to the discussion above, based on the competency standard achievement, mathematical communication skills and creative thinking ability skills in the stad type and tps type cooperative learning, stad type cooperative learning is more effective and efficient than tps type cooperative learning. references hendriana, h., hidayat, w., & ristiana, m. g. (2018). student teachers’ mathematical questioning and courage in metaphorical thinking learning. in journal of physics: conference series (vol. 948, no. 1, p. 012019). iop publishing. herman, t. (2007). pembelajaran berbasis masalah untuk meningkatkan kemampuan penalaran matematis siswa smp. cakrawala pendidikan, 1(1), 41-62.. hidayat, w. (2012). meningkatkan kemampuan berpikir kritis dan kreatif matematik siswa sma melalui pembelajaran kooperatif think-talk-write (ttw). in seminar nasional penelitian, pendidikan dan penerapan mipa. hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. hidayat, w., wahyudin, & prabawanto, s. (2018). improving students’ creative mathematical reasoning ability students through adversity quotient and argument driven inquiry learning. in journal of physics: conference series (vol. 948, no. 1, p. 012005). iop publishing. mahmudi, a. (2009). komunikasi dalam pembelajaran matematika. jurnal mipmipa unhalu, 8(1), 1-9. rahmi, s., nadia, r., hasibah, b., & hidayat, w. (2017). the relation between self-efficacy toward math with the math communication competence. infinity journal, 6(2), 177182. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 1, february 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i1.p133-148 133 marginal region mathematics teachers’ perception of using ict media trisna roy pradipta*1, krisna satrio perbowo1, afifah nafis1, asih miatun1, sue johnston-wilder2 1universitas muhammadiyah prof. dr. hamka, indonesia 2university of warwick, england, uk article info abstract article history: received aug 24, 2020 revised jan 29, 2021 accepted jan 31, 2021 the article presents a marginal region mathematics teachers' perception of ict as learning media and the type of ict media used in mathematics classrooms. a survey was designed including two domains: the usability and the importance of ict. a questionnaire was administered to 84 mathematics teachers in marginal regions. the respondents were chosen conveniently based on their accessibility and availability. about 50% of marginal region mathematics teachers do not use ict in teaching. the most common media used is microsoft office software to present teaching materials from internet or digital sources. using the rasch model, the data show that mathematics teachers' perception of the usage and importance of ict as a media of mathematics learning can be categorized as 'medium' level. mathematics teachers in the marginal regions consider ict as mathematics learning media to be fairly important. keywords: ict, marginal region, perception copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: trisna roy pradipta, department of mathematics education, universitas muhammadiyah prof. dr. hamka, jl. tanah merdeka, pasar rebo, jakarta timur, jakarta, indonesia. email: troymath@uhamka.ac.id how to cite: pradipta, t.r., perbowo, k.s., nafis, a., miatun, a., & johnston-wilder, s. (2021). marginal region mathematics teachers’ perception of using ict media. infinity, 10(1), 133-148. 1. introduction only a few published articles were found talking about mathematics education in marginalised contexts (howley, howley, & huber, 2005). it means, more attentions are needed to improve the quality of mathematics education in marginal regions. more collaboration is needed, both practically and academically, in mathematics education in marginal contexts (bush, 2005; nicol, archibald, & baker, 2013). a small part to contribute in solving mathematics education issue in the marginal region is to depict mathematics teachers’ perception of using ict as learning media in their classroom. learning media is a tool to deliver learning information that is designed so that it can help students understand the subject (muhson, 2010), as well as a communication tool for interaction between teachers and students in the learning process (ahern, 2016). the purpose of learning media use is to stimulate the mind and attention of students (triyanto, anitah, & suryani, 2013). while, the function of learning media is to accelerate the learning process effectively, improve quality and concretise abstract material (nurseto, 2011). the use of https://doi.org/10.22460/infinity.v10i1.p133-148 pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 134 varied and innovative learning media can increase the joy and love of learning mathematics (safitri, hartono, & somakim, 2013). the rapid development of technology in the modern age has become the latest innovation in the learning process. ict media is one of the learning media besides teaching aids that are often used in learning mathematics. the rapid development of technology has an impact on the world of education, creating expectations to integrate ict into the classroom; this can change conventional learning styles (directs-teaching & learning) into constructivist learning (donnelly, mcgarr, & o’reilly, 2011; hismanoglu, 2012; lin, wang, & lin, 2012; smith, shin, & kim, 2017). ict-assisted learning is an effective and efficient way of building knowledge so that it helps teachers create a modern learning context (eyyam & yaratan, 2014). learning by involving ict can facilitate students in understanding concepts in theoretical and practical approaches by using technology such as led tv, lcd projectors, laptops, tablets, computers, mobile phones, internet, etc. (gautam & agrawal, 2012). the forms of ict media used in the learning process are text, audio, video or images (blum & parette, 2015). ict plays the most important factor in creating enjoyable and efficient learning to achieve learning effectiveness so that it can improve learning outcomes. ict media software in learning mathematics includes geogebra, sketchpad, microsoft mathematic, autograph, and others (bakar, ayub, & tarmizi, 2010). learning via the internet is rampant nowadays (limayem & cheung, 2011) because the internet is a place of learning to obtain information and knowledge anywhere and anytime, following various forms of media via the internet such as ruang guru, zenius, youtube, brainly, and edmodo. there are many benefits obtained by using ict (ali, haolader, & muhammad, 2013), most teachers recognise the importance of using ict in learning so training is needed so that teachers can integrate ict effectively in the teaching and learning process. ict becomes one of the important factors in a country's development process depending on the readiness of infrastructure and the spread of ict (alghamdi, goodwin, & rampersad, 2011). in indonesia, the progress of ict has developed rapidly. the tim indikator tik pusat litbang ppi (2015) surveyed conditions within households, in indonesia, with the use of ict as follows: (1) have computers in urban regions 41.9% while rural 19.9% (2) use the internet in urban regions 47.9% while rural 24.7% (3) have mobile phones in urban regions 90.9 % while rural 78.8%. based on the survey results, the use of ict was still relatively low in rural regions compared to urban regions. this finding has been supported by several studies; ict can be experienced throughout indonesia. amin (2016) found the use of the internet in the education sector in the eastern border region using mobile broadband is as much as 50.4%, while 40.2% of schools are using fixed broadband. based on the ministry performance report (afidah, doom, & putri, 2017), a strategy for the availability of ict infrastructure and the development of ict ecosystems has been worked with villages in the border areas, disadvantaged areas, including local transportation services available in 4.02% of 222 locations and 3 piloting villages. the development of ict in the marginal regions is one of the active missions carried out by the ministry of research and technology since 2000 among others in the form of warung informasi dan teknologi (warintek), community access points (cap), and mobil pusat layanan internet kecamatan (mplik), which by 2013 had a number of 84 stalls scattered in 28 of 34 provinces in indonesia (kusnandar, 2013). all statements above show that only some marginal regions are occupied with information and technology services. however, the use of ict as a learning media in marginal regions has not been realised comprehensively in almost schools in indonesia. this is because human resources and marginal region governance have not contributed fully in the context of developing their regions (chaerul & aisyah, 2014), so that there are many limitations of facilities and infrastructure owned by marginal regions including the difficulty of getting access to basic volume 10, no 1, february 2021, pp. 133-148 135 services, such as education, health, water, infrastructure, transportation, electricity, and telecommunications (kementerian ppn/bappenas, 2016). despite the problems of the marginal regions, there are very few marginal regions that have experienced the use of ict. limited access to ict in marginal areas raises a question about mathematics teachers' perception of the use of ict media in learning mathematics in marginal regions. 2. method this research was conducted by a survey with a quantitative descriptive approach. data were collected by a questionnaire adapted and translated from albalaw (2017). the questionnaire was translated and checked by an expert from english department of university of muhammadiyah prof. dr. hamka, indonesia. the instrument contained statements of perception of mathematics teachers toward the use of ict. it consists of two domains or variables: the use of ict as a learning media (19 items) and the importance of ict as a learning media (27 items). the validity and reliability of the instrument were checked with the rasch model. the instrument is valid due to the high ptmea corr score for each item (x > 0.2), and raw-variance more than 40% and unexplained-variance less than 15% for each domain (linacre, 2011). the reliability of the instrument is high with person reliability 0.93, item reliability 0.98 and alpha cronbach 0.98 (bond, yan, & heene, 2020). data from the questionnaire were processed using the rasch model assisted with winstep application to convert ordinal data into interval data (sumintono & widhiarso, 2014). the respondent perception and the item responses are represented on the wright map logit (log odds unit) scale such as those given in figure 1 and figure 2. the results of respondents' responses are converted into the form of measure scale values with standard deviation calculations so that each domain has a different scale as seen in table 1. table 1. item logit categories domain 1: the use of ict domain 2: the importance of ict mean measure range decision mean measure range decision 0.765 < 𝑥 never existed 0.75 < 𝑥 unimportant 0.255 < 𝑥 ≤ 0.765 rarely 0.25 < 𝑥 ≤ 0.75 fairly important −0.255 < 𝑥 ≤ 0.255 sometimes −0.25 < 𝑥 ≤ 0.25 somewhat important −0.765 < 𝑥 ≤ −0.255 mostly −0.75 < 𝑥 ≤ −0.25 important 𝑥 ≤ −0.765 always 𝑥 ≤ −0.75 very important due to the limitation to reach population of mathematics teachers in marginal regions of indonesia, this research was using convenience sampling that consisted of 84 mathematics teacher respondents in the marginal regions with levels of schools ranging from elementary (d), junior high (p), to high school (a) spread from western to eastern indonesia; the number of respondents in each region can be seen in table 2. the geographical locations in western indonesia include sumatra, java, kalimantan, and bali province. geographical locations in eastern indonesia include sulawesi island, maluku province, west nusa tenggara province, east nusa tenggara province, and papua island. pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 136 table 2. grouping data based on respondent geographical location and level of school geographical location level total d p a east 2 24 8 34 west 27 17 6 50 total 29 41 14 84 3. results and discussion 3.1. results 3.1.1. types of ict as a learning media the type of ict used in mathematics learning activities in marginal regions based on geographical location in table 3 shows that microsoft office is most used in western indonesia with 11 people compared to eastern indonesia with four people with a percentage of 17.86% users. this is different from eastern indonesia, where the type of ict media most used is mathematics software with a total of 8 people while in western indonesia there are 5 people with a percentage of users of 15.48%. however, there are still many mathematics teachers in marginal areas who did not use ict media in mathematics learning, with a percentage of 50.00% of the total respondents. the majority of marginal regions mathematics teachers are not using any ict or even digital media during teaching mathematics. this seems due to the constraint of facility and communication access. table 3. types of ict media based on geographical location media location total percentage east west electronic book 1 0 1 1.19% social media 0 2 2 2.38% microsoft office 4 11 15 17.86% online media 3 8 11 13.10% mathematics software 8 5 13 15.48% not using 18 24 42 50.00% total 34 50 84 100.00% the most common media used within the regions is microsoft office, especially in the western part of indonesia. the teachers use powerpoint to present their works, or they use word to show the teaching materials they wrote or took from internet/digital sources to their students. meanwhile, mathematics software is more prevalent in eastern indonesia. the type of ict based on the level of education used in learning mathematics in the marginal regions in table 4 shows that the use of ict media in mathematics learning for the elementary school level (d) is widely used, namely six people use online media. for the junior high school level (p), the use of ict media which is widely used in learning mathematics is mathematics software, as many as 10 out of 13 users. whereas for the high school level (a) there are two ict media which are widely used in mathematics learning, as many as 3 out of 15 users of microsoft office and 13 users of mathematics software. volume 10, no 1, february 2021, pp. 133-148 137 table 4. types of ict media based on school levels media level education total percentage d p a electronic book 0 1 0 1 1.19% social media 1 1 0 2 2.38% microsoft office 4 8 3 15 17.86% online media 6 4 1 11 13.10% mathematics software 0 10 3 13 15.48% not using 18 17 7 42 50.00% total 29 41 14 84 100.00% the types of ict used in mathematics learning are grouped based on geographical location and education levels in table 5. it can be seen that the ict media that are widely used in western indonesia are online media with 6 users found in d. while for eastern indonesia, mathematics software is most widely used with 6 users in p. table 5. types of ict media based on geographical location and school level media ict east west total d p a d p a electronic book 0 1 0 0 0 0 1 media social 0 0 0 1 1 0 2 microsoft office 0 3 1 4 5 2 15 online media 0 3 0 6 1 1 11 mathematics software 0 6 2 0 4 1 13 not using 2 11 5 16 6 2 42 total 2 24 8 27 17 6 84 3.1.2. the usage of ict as a learning media the questionnaire data distribution consisted of 19 items of statements regarding the use of ict as a learning media. the variable map in figure 1 shows the item that was most difficult to be agreed upon by respondents is item number 6. in comparison, the items that were most approved were item numbers 17 and 3. pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 138 figure 1. variable map of the use of ict as a learning media table 6 shows the logit standard deviation values according to the item logit category of respondents' responses in table 1. from table 6, it is known that there are six items in the category of ‘rarely’ use (numbers: 4, 6, 12, 13, 14 and 16). while the ‘always’ usage category contained three items (no: 3, 5 and 17). in addition, it can be seen that there are 14 items that are categorised as ‘rarely’ and ‘sometimes’ and five items that are categorised as ‘always’ and ‘mostly’. table 6. domain 1: the use of ict as a learning media no items logit value most responses 1. i use ict as one of my methods in teaching mathematics -0.17 sometimes 2. i give students my social media contact information at the beginning of the semester -0.17 sometimes 3. i encourage students to learn mathematics through ict -0.83 always 4. i encourage students to follow an online learning forum 0.60 rarely 5. i give students an idea about the necessary websites and apps for learning mathematics -0.77 always h ig h m id l o w volume 10, no 1, february 2021, pp. 133-148 139 no items logit value most responses 6. i recognise students’ academic improvement through ict 0.71 rarely 7. by the use of ict, i implement the concept of cooperative learning 0.15 sometimes 8. i deliver the content of the courses to my students through ict 0.19 sometimes 9. i give students a chance to cooperate learning through the use of ict 0.06 sometimes 10. i solve students’ learning problems through ict -0.04 sometimes 11. i teach some parts of the math course using ict -0.23 sometimes 12. i give students extra-curricular activities assigned through ict 0.32 rarely 13. i assign students in groups to discuss and solve math problems through ict 0.57 rarely 14. i prepare quizzes for students and tell them to do it through ict 0.64 rarely 15. i design courses using ict 0.25 sometimes 16. i train students to discuss and explore the concept of mathematics through ict 0.60 rarely 17. ict gives me other ways of teaching math -0.89 always 18. i give students the chance to search for information using ict -0.38 mostly 19. i use social media in exchanging mathematics teaching strategies with my colleagues -0.62 mostly the value distribution in general of the perception of mathematics teachers in the marginal area towards the use of ict media in figure 1 shows that by using ict the way of teachers teaching became more varied and not monotonous. the distribution of data based on school levels shows that those who have high perceptions of ict are found in junior high schools (p) by 61.54% of a total of 13 people. at the most moderate perception, there is 54.55% of the total of 44 people in junior high school (a). whereas, the lowest perception is at elementary school (d) level that is 48.15% of 27 teachers, as shown in table 7. table 7. perception of the use of ict based on school level category level total d p a high 4 8 1 13 15.48% medium 12 24 8 44 52.38% low 13 9 5 27 32.14% total 29 41 14 84 100.00% the perception of mathematics teachers in the marginal area towards the use of ict media based on geographical location is that those who have a high perception are located pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 140 in western indonesia (w) by 69.23%. likewise, for medium and low perception is also the case in by western indonesia (w) by 61.36% and 51.85%. this can be seen in table 8. table 8. perceptions of the use of ict based on geographical location category location total e w high 4 9 13 15.48% medium 17 27 44 52.38% low 13 14 27 32.14% total 34 50 84 100.00% the perception of mathematics teachers in the marginal area towards the use of ict media based on school levels and geographical location (see table 9) illustrates that high perceptions were found in junior high schools (p) in western indonesia (w) by 38.46% of respondents. the medium perception category lies in junior high school (p) level in eastern indonesia (e) by 34.09%. whereas for the low category owned by elementary school (d) respondents in western indonesia (w) as much as 40.74%. table 9. perception of the use of ict based on school level and geographical location category d p a total e w e w e w high 0 4 3 5 1 0 13 15.48% medium 0 12 15 9 2 6 44 52.38% low 2 11 6 3 5 0 27 32.14% total 2 27 24 17 8 6 84 100.00% 3.1.3. the importance of ict as a learning media questionnaire data distribution consisted of 27 items of statements regarding the importance of using ict as a learning medium, and there were 84 respondents of mathematics teachers in the marginal regions ranging from elementary school (d), junior high school (p) and high school (a) scattered from western indonesia (w) to eastern indonesia (e). the variable map in figure 2 shows that the items that are most difficult to be agreed upon by respondents are items number 17 and 2. in contrast, item that is highly approved is item number 9. volume 10, no 1, february 2021, pp. 133-148 141 figure 2. variable map the importance of ict as a learning media while table 10 shows the standard logit deviation values according to the item logit category of respondents' responses in table 1. from table 10, it is known that there are three items in the ‘unimportant’ category (no. 2, 17 and 18). while the category of usage is ‘very important’ there is 1 item (no. 9). in addition, it can be seen that there are 15 items that are categorised as ‘fairly important’ and ‘somewhat important’ and nine items that are ‘important’ and ‘very important’. table 10. domain 2: the importance of using ict as a learning media no items logit value most responses 1 i use ict in teaching math -0.67 important 2 i give students my social media contact information at the beginning of the semester 1.09 unimportant 3 i encourage students to use ict more than the other ways 0.44 fairly important h ig h m id l o w pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 142 no items logit value most responses 4 ict makes it easy for students to communicate within groups 0.31 fairly important 5 i give students an idea about necessary apps and math learning website -0.04 somewhat important 6 some students advance/improve because of their use of ict -0.22 somewhat important 7 ict is important for students’ learning advancement/improvement -0.33 important 8 i encourage students to seek inspiration when they used ict -0.15 somewhat important 9 using ict makes learning to be more amusing -0.99 very important 10 using ict helps to build students’ collaborative methods and skills -0.36 important 11 ict helps students by delivering the content of math’s courses -0.55 important 12 ict helps me to facilitate the presentation of information -0.67 important 13 ict helps in considering individual differences among students 0.17 somewhat important 14 ict gives students the chance to cooperate in learning 0.14 somewhat important 15 ict gives me a chance to solve students’ problems about learning mathematics -0.04 somewhat important 16 i teach the content of some courses through ict 0.07 somewhat important 17 i give students extracurricular activities through ict 1.09 unimportant 18 i assign students in groups to discuss and solve mathematics problems through ict 0.90 unimportant 19 ict helps me in preparing quizzes and conducting them through ict 0.54 fairly important 20 i designed the math course using ict 0.38 fairly important 21 using ict helps students gain more confidence 0.07 somewhat important 22 using ict gives students important life experiences -0.36 important 23 using ict equips students with the skills for discussion and debate -0.07 somewhat important 24 ict equips students with the skill of self-learning -0.25 important 25 ict helps in creating a variety in math-teaching methods -0.33 important 26 ict pushes students to learn -0.18 somewhat important 27 ict helps students gain social skills 0.03 somewhat important based on responses shown in table 10, it can be seen that the teachers mostly agree that ict plays an important role to make the learning process to be more interesting and enjoyable for students. volume 10, no 1, february 2021, pp. 133-148 143 the distribution of data based on school levels shows that those who have a high perception of the importance of using ict as a learning media are found at the junior high school level (p) by 72.73% of a total of 11 people. at the most moderate perception, it has at the junior high school level (p) 46.43% of the total of 56 people. likewise, the lowest level of perception was in junior high school level (p), namely 41.18% of 17 people, as shown in table 11. table 11. perceptions of the importance of ict based on school category level total d p a high 3 8 0 11 13.10% medium 21 26 9 56 66.67% low 5 7 5 17 20.24% total 29 41 14 84 100.00% the perception of mathematics teachers in the marginal area towards the importance of ict media based on geographical location is that those who have a high perception are located in western indonesia (w) by 72.73%. likewise, for moderate perception found in western indonesia (w) which is 67.86%. whereas in the low perception owned by eastern indonesia (e) was 76.47%. this can be seen in table 12. table 12. perceptions of the importance of ict based on geographical location category location total e w (n) (%) (n) (%) (n) (%) high 3 27.27% 8 72.73% 11 13.10% medium 18 32.14% 38 67.86% 56 66.67% low 13 76.47% 4 23.53% 17 20.24% total 34 100.00% 50 100.00% 84 100.00% mathematics teachers’ perception in the marginal area on the importance of ict media based on education level and geographical location as seen in table 13 illustrates that high perceptions were found in junior high schools in western indonesia by 45.45%. on the perception at the elementary school level in western indonesia by 35.71%. whereas for low perception, there is a junior high school level in eastern indonesia that is 41.18%. table 13. perceptions of the importance of ict based on education level and geographical location category d p a total e w e w e w high 0 3 3 5 0 0 11 13.10% medium 1 20 14 12 3 6 56 66.67% low 1 4 7 0 5 0 17 20.24% total 2 27 24 17 8 6 84 100.00% pradipta, perbowo, nafis, miatun, & johnston-wilder, marginal region mathematics teachers’ … 144 3.2. discussion this study found that the majority of marginal regions mathematics teachers are not using any ict or even digital media during teaching mathematics. the constraint of facility and communication access are common limitations in marginal regions (kementerian ppn/bappenas, 2016). this can be understood by looking at the situation in the regions that many mathematics teachers do not utilise ict in mathematics classrooms. in addition, a study by perbowo, maarif, and pratiwi (2019) shows that the perception of marginal regions mathematics teachers on the use of manipulative tools is higher than on the use of ict. most marginal region mathematics teachers are choosing microsoft office to help their teaching since microsoft holds the biggest office software market over the globe (gandal, markovich, & riordan, 2018); obviously, microsoft office software becomes the most common bundle for a personal computer or laptop unit and does not require internet access to be operated. this software is sometimes considered as the standard software for the requirements for successful e-learning (waterhouse & rogers, 2004). in line with lawrence and tar (2018), the utilisation of ict provides more opportunities for teachers to improve the quality of teaching and learning environment. in addition, the findings show that the teachers would promote the use of ict to students to help them learn mathematics. the use of ict media as a medium of mathematics learning is dominated by western indonesia because there has been more developement and improvement for the access of the community to ict, compared to eastern indonesia, especially in garut district and riau province (djuwendah, hapsari, renaldy, & saidah, 2013; syahza & suarman, 2013). according to tossavainen and faarinen (2019), ict adds more positive value in mathematics classroom while traditional teaching of mathematics is more or less boring. on the contrary, teachers tend to mind giving students their social media contact information. they see this as an unimportant matter in mathematics teaching and learning. in addition, teachers also assume that giving extra activities for students to discuss and doing mathematical activities through ict as another unimportant case. teachers must be aware that ict not only can increase student motivation (tossavainen & faarinen, 2019), but also can enhance students’ understanding and mathematics proficiency (drijvers, boon, & van reeuwijk, 2011; widodo et al., 2019). the rapid development of digital media, especially ict, is affecting the way in which humans live and learn (voogt & roblin, 2012). mathematics teachers, especially in marginal regions, need to continuously transform and adapt in order to comply with the competencies needed for industrial revolution 4.0 in the 21st century. the findings of this study show that marginal regions mathematics teachers do not have a high perception of adopting ict in their teaching. in addition, they do not have enough concerns about the role of ict in promoting students' performance and proficiency in mathematics. we assume that it is because of all barriers and limitations they face in their region. yet, they must be able to deal with the situation. the challenge for the future is to give a handful of designs to teach mathematics in marginal regions; a design that can optimise the learning environment within the regions. thus, further studies for teaching and learning designs that are suitable in marginal regions are needed; whether it involves digital media, hands-on manipulative tools or any media that can help teachers to teach mathematics as concrete as possible. thus, at least there are two factors that needed to be considered in order to create the most suitable teaching design for mathematics classrooms in marginal regions which are learning environment and realistic mathematics context. volume 10, no 1, february 2021, pp. 133-148 145 4. conclusion the study findings indicate that most marginal regions mathematics teachers are still not utilising ict or any other digital media in their teaching. most teachers who use ict are using microsoft media since this software is common and does not require internet access to be operated. in specific, online media is mostly used in teaching mathematics for primary school, while mathematics software for secondary school and microsoft office for senior high school. the perception of mathematics teachers in the marginal regions towards the domain of the use of ict can be categorised as ‘medium’ which is dominated by secondary school mathematics teachers in marginal regions of eastern indonesia. the teachers are used to adopting ict in teaching mathematics regularly. in comparison, the perception of mathematics teachers in the marginal area on the importance of ict can be categorised as ‘medium’ which is dominated by elementary mathematics 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(2019). visual media in team accelerated instruction to improve mathematical problemsolving skill. in proceedings of the 1st international conference on science and technology for an internet of things. european alliance for innovation (eai). http://doi.org/10.4108/eai.19-10-2018.2281297 https://doi.org/10.1080/00220272.2012.668938 http://doi.org/10.4108/eai.19-10-2018.2281297 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p149-156 149 experimentation of spices learning strategies with the method of problem based learning (pbl) to build motivation and the ability to think logically for vocational school students anggita maharani 1 , laelasari 2 1,2 swadaya gunung djati, jl. pemuda no.32, sunyaragi, kesambi cirebon, west java, indonesia 1 anggi3007@yahoo.co.id, 2 lala.mathunswagati@gmail.com received: november 11, 2016 ; accepted: july 11, 2017 abstract this study aims to expose the interactions that occur in the process of mathematics learning and describe a method of learning motivation and logical mathematical thinking of students in classes. the method used in this research was a quasi-experimental design group post-test. in its implementation, this study uses a class experiment. qualitative data are obtained in the form of motivational learning scale, whereas the quantitative data in the form of logical thinking ability score using normalized gain. the results showed the existence of a positive interaction during learning. however, learning motivation of students didn’t show a significant improvement. while the logical thinking ability of students after learning of mathematics using the strategy of spices with pbl method is in the category is fair. in its implementation, the strategy of spices needed support some teachers from several fields of science. keywords: logical thinking, problem based learning, spices, motivation. abstrak penelitian ini bertujuan untuk menelaah interaksi yang terjadi dalam proses pembelajaran matematika menggunakan strategi spices dengan metode pbl dan mendeskripsikan motivasi belajar dan berpikir logis matematis siswa pada kelas yang menggunakan strategi spices dengan motode pbl. metode yang digunakan dalam penelitian ini adalah kuasi eksperimen dengan disain kelompok postes. dalam implementasinya, penelitian ini menggunakan satu kelas eksperimen. data kualitatif yang diperoleh berupa skala motivasi belajar matematika siswa, sedangkan data kuantitatif berupa skor kemampuan berpikir logis dihitung gain ternormalisasinya. hasil penelitian menunjukkan ad anya interaksi yang positif selama pembelajaran. namun demikian, motivasi belajar siswa tidak mengalami peningkatan yang signifikan. sedangkan kemampuan berpikir logis siswa setelah mendapat pembelajaran matematika menggunakan strategi spices dengan metode pbl termasuk ke dalam kategori cukup. dalam implementasinya, strategi spices memerlukan dukungan beberapa guru dari beberapa bidang ilmu. kata kunci: berpikir logis, problem based learning, spices, motivasi. how to cite: maharani, a., & laelasari (2017). experimentation of spices learning strategies with the method of problem based learning (pbl) to build motivation and the ability to think logically for vocational school students. infinity, 6 (2), 149-156. doi:10.22460/infinity.v6i2.p149-156 mailto:anggi3007@yahoo.co.id mailto:lala.mathunswagati@gmail.com maharani & laelasari, experimentation of spices learning strategies … 150 introduction a graduate expected by the industry is to have basic mathematical skills with the ability of concept that can apply to practices and used in any workplace. however, corporate leaders often find that young people are unable to apply math concepts they have learned in school to the new problems in the workplace. based on the analysis of the automotive business in west java, the industry needs the level of intelligence, namely the ability to think logically, ability to solve problems, the ability to use logic techniques and reasoning techniques. skills required by the industry are mathematical ability of students. but the survey results from several automotive business places (atpm) conducted by researchers in the area of west java, indicating that the test results of mathematics from vhs graduates were far from expected. from the results of the interviews with the teachers of the vhs, the main cause of the low ability of vocational students is the lack of motivation in learning, especially for math lessons. it is necessary to use an effective learning method that capability for increase student learning motivation so that vhs students can be able to have the skills expected by the curriculum and industry like ability of logical thinking,creative ability, high work motivation, perseverance, and the disposition of mathematics. one of the ways to involve students actively in the learning process, student-centered, integrated, problem-based, community-based, elective and systematic study of the concept is to use spices. pbl uses the issue as a focal point for the purpose of investigation and research students. the essence of pbl involves the presentation of authentic situations and meaningful, which serves as the foundation for the investigation and the inquiry. spices learning strategies spices was born from the world of medical education. an article written by dent (2014) writes that spices provides opportunities for student-centered, integrated learning and problem-solving. some phenomena that occur in the learning process, has become the reason to change the paradigm of learning through various innovations. orientation of learning goals is now not only for students to remember for the short term, but the results of the learning process should be able to equip students to solve problems in the long term. in 1984, harden et al. initiate the concept of spices. at first, this learning concept was practiced and developed in medical education. spices are an acronym for (1) studentcentered, (2) problem-based, (3) integrated, (4) community-based, (5) elective, (6) systematic. it illustrates learning components that are presented in the spices strategy. 1. student-centered means learning is oriented on the student’s activity. the student is a subject who actively learn to build his understanding through experience that has owned as well as the experience of the recently found. student-centered learning can be done by students in confronts to the real world through learning resource that can encounter. 2. problem-based. learning starts with the actual problems and authentic, will give meaning to students. through a problem, students will learn the concepts/theories at the same time solve the problem. thus, learning not only produce (answer) but also generating process (how to solve the problem). 3. integrated-based. an integrated approach based on the view that the learner or student builds their understanding of topics they have to learn rather than recording a lesson in the form is arranged systematically. learning objectives on integrated learning is to help students achieve the learning objectives are interrelated. volume 6, no. 2, september 2017 pp 149-156 151 4. community-based. spices learning strategy takes the problems that are happening in the community as the "starter" to obtain a meaningful learning. learning with this strategy also invites students to be able to implement what he have learned into the context of a society. 5. electives. every student has diverse characteristics. innovative learning should pay attention to the characteristic on each student. as a subject can determine when students want to learn and how to learn. the teacher acts as a source of learning, tutors, counselors, evaluators, and motivational speaker. 6. systematic. the substance of the subject matter, generally the hierarchical. a material sometimes required other materials as prerequisites. each procedural step is a prerequisite for the next step. thus, the study should be done systematically. views of the components contained in the acronym spices, this strategy offers several advantages as a follow, (1) foster the student's motivation and activeness in his education, (2) develop the skills of solve the problems creatively and comprehensive, (3) develop analytical and logical thinking skills sharper and comprehensive, (4) build social skills, (5) provide opportunities for students to learn in accordance with his interest, (5) gives the feel of an orderly and effective learning. problem based learning (pbl) pbl have the following characteristics: (1) learning begins with a problem, (2) ensures that problems are related to the real world of students, (3) organizes lessons around issues, not around disciplines, (4) give students full responsibility for experiencing their own learning process directly, (5) using small groups, and (6) demanding students to demonstrate what they have learned in the form of a product or performance. problem-based learning is a learning in the classroom to organize learning around problemsolving activities through the delivery of arguments and mathematical ideas, and communicating to peers through the interaction of various components in the classroom (soekisno, 2015) according to forgarty (rusman, 2010), problem-based learning starting with issues that are not structured (something that messed up). five stages of pbl as follows: 1. provide orientation about the issue to students based on its structure, problems in learning can be classified into two types, namely the problems defined clearly (well-defined) and the problems that are not clearly defined (illdefined). in pbl, students are given a structured problem ill-defined context of daily life. 2. organizing students to do research the results of the research on problem-solving that are practiced in class with something ill-defined structure problem, impacts as follows (1) discovery problem can increase creativity; (2) motivate students to make learning feels good; (3) problems with ill-defined structure requires skills different from standard-problem (4) encourage students to understand and acquire relations problems with specific disciplines (5) the information entered into the long-term memory is more amplified by using ill-defined structured problem (krulik & rudnick, 1995). maharani & laelasari, experimentation of spices learning strategies … 152 3. helping independent investigation and groups students prepare temporary answers against the problems involving logic -mathematical intelligence. students conduct an investigation of the data and information obtained is problems oriented. 4. develop and present the results students revise the formulation of the problem through the real picture that they understand. students involve verbal-linguistic intelligence fix the problem formulation statements wherever possible to use a more appropriate word. a reformulation of problems is a more focused investigation, and clearly shows the facts and information that need to be searched, as well as providing a clear objective in analyzing the data. 5. analyze and evaluate the process of solving the problem learning assessment according to constructivist paradigm to be an integral part of learning itself. starting from this view, the assessment of learning pbl implemented integrated with the learning process. spices strategies using pbl in learning mathematics of vhs the principle of learning to do, learning to be, and learning together implies that mathematics learning should be based on the idea that learning should be comprehensive and integrated. the learning materials according to dewey (fip-upi, 2007) should be prepared by considering two main requirements as follows: 1. prepared concretely and in detail, been useful for life and truly is a necessary learning material. 2. the knowledge gained as a result of learning, is placed in a position that meaningful, which allows doing new activities is the development of previously acquired. in order for the learning of mathematics vhs line with the needs, the context chosen is related to the field of vocational correspond to areas of expertise by way of design learning integrated material vocational through strategies to encourage students to actively search for information related to the context of everyday life. motivation to learn motivation will arise if a person has a purpose to be an interest. by having a goal, someone will have the energy to achieve it. sardiman (2012) says that motivation is the driving force that leads to learning activities. the characteristics of motivation according to sardiman (2012) is (1) persevering in the face of the task (work continuously for a long time); resilient in the face of adversity (not easy to despair); (3) does not need a push from the outside to perform as best as possible (not easily satisfied with the accomplishments that have been achieved); (4) shows interest in all kinds of things; (5) prefer to work alone; (6) quickly bored against the routine tasks; and (7) happy searching and solving problems. the motivation of the students in the learning process will be very influential towards the achievement of learning objectives. as expressed by elliott and dweck (2005) that motivation is a very important factor affecting the learning and achievement of learning ranging from childhood to adolescence. the incidence of motivation, can be influenced by external factors (influence from outside) and internally (from within own influences). external influences such as praise teachers, get value or graduation, or desire to compete with his friend. while internal factors can be a form of interests, talents, or ideals. in line with volume 6, no. 2, september 2017 pp 149-156 153 that, slavin (2008) also states that the motivation encourages students to do the learning activities. it is very important to motivate students. motivation as the process has some function especially in the achievement of learning objectives. the motivational function of hamalik (2009) includes: (1) encouraging the emergence of behavior or an act; (2) motivation serves as directors; and (3) the motivation serves as a driving force. the teacher has responsibility for creating a classroom environment that can foster the learning motivation of students. a variety of mathematical ability is expected to appear in accordance with the indicator nctm, will not be achieved without the motivation of the student. in building conceptual understanding for example. motivation to learn to be critical in the achievement of understanding in all areas. a study of motivation in students of vhs conducted by bakar (2014) in the region of west sumatra, indicating that there was significant influence between the motivation of learning with the competence of the productive. logical thinking ability think logically refers to understanding (understand), application capabilities, the ability of analysis, synthesis capability, even the ability to form evaluation skills (a process). logical thinking can also be interpreted as measures. logical thinking is an important basic skill of mathematics. logical thinking is the key to drawing inferences and solve problems that are complex. logical thinking is not the same as the logical reasoning, but both contain some similar activities. it is expressed by sumarmo, hidayat, zukarnaen, hamidah & sariningsih (2012) that in some discussion of the term logical thinking is often interchangeable with the term logical reasoning. further, sumarmo, et. all (2012) said that indeed, the term logical thinking has a wider coverage of logical reasoning. in the process, logical thinking ability is measured based on the results of the test of logical thinking (tolt) which was initiated by capie and tobin (sumarmo, et. all, 2012) is based on the theory of the mental development of piaget. the question of the test of logical thinking (tolt) which includes five components: control variables (variable controlling), proportional reasoning (proportional reasoning), probabilistic reasoning, correlational reasoning, and combinatorial thinking. the question of the test consists of 10 item in the form of multiple choice with five answer choices are accompanied by a selection of the reason. in addition to the test of logical thinking (tolt), there is also a longeot test developed by the sheehan (sumarmo, et. all, 2012). sheehan classifies mental development of the child into 26 rounds of test items which include formal logic components, a combination of formal, formal and proportions. in a test of longeot, sublogic tests or formal reasoning proposition presented in the form of a series of statements, followed by a choice answers as logical conclusions based on inference rules. further, reasoning that inference is called logical reasoning. the operational definition of logical thinking that is used in this study is the definition of logical thinking as revealed by sumarmo, et. all (2012) i.e., 1. draw conclusions or making the forecasts and interpretations based on the proportion of the corresponding maharani & laelasari, experimentation of spices learning strategies … 154 2. draw conclusions or making the forecasts and the predictions based on opportunities 3. draw conclusions or make estimates and predictions based on the correlation of two variables 4. set the combination of multiple variables 5. analogy is drawing conclusions based on the likeness of the two processes 6. do proofs 7. arrange the analysis and synthesis of several cases method this is quasi-experimental research with the pretest-postest design. in its implementation, this research uses experimental group i.e. one class with learning strategies using the spices with the methods of pbl. during the learning process, the motivation of student learning was observed. implementation of the research consisted of two stages. the first stage is an introduction which consists of problem identification, preparation of instrument, and learning device with the development, continued with the test validation and reliability of learning device. next is the determination of the control and experiments classes from school to be a place of experimentation. the next stage is the implementation of the research. data collection is done by administering a test question the ability of logical thinking and observations about the motivation of learning that occurs during the learning process. the logical thinking ability tests had previously tested for the validity, reliability, power of distinction, and index of difficulty. data processing is carried out to know the existence of a correlation between learning motivation caused from the use of the strategy of spices with pbl method against the logical mathematical thinking ability of students. results and discussion observations on student activities are conducted by the observer for each meeting. the format of the observation is filled with the aim to know the extent of student activity during the learning process takes place. observations on student activities were conducted by three observers for each meeting. the focus of observation consists of 8 observed aspects. while the indicators were observed based on the learning steps by using spices strategy with pbl method. tabel 1. recapitulation results of learning activity variable day average activity 1 66,37 2 71,46 3 75,88 4 81,14 the motivation analysis of students learning based on the results obtained after the student's mathematical learning experience using the strategy of spices with pbl method. the results are the following; there are increased on each of his encounters, though not too significant, the average learning motivation of students at the first meeting is 5.03 with percentage is 55.87% by category is enough, while the average of student learning motivation at the end of the meetings reached only 58.73% with category is enough. volume 6, no. 2, september 2017 pp 149-156 155 tabel 2. recapitulation results of learning motivation variable week average % category motivation 1 5,03 55,87 enough 4 5,28 58,73 enough the average results of pre-test from the ability of logical thinking are 29,43 with the minimum value is 13 and the maximum value is 38. while the average results of the post-test are 68.11 with a minimum value is 16 and a maximum is 85. the results from the calculation of the correlation are 0.704 which means in the category of enough. by taking the real extent of α = 5%, n = 35, and the value of df = (35-2) = 33 so retrieved 0.344 rtabel. because the rser > rtabel that is 0.704 > 0.344 then it can be inferred that there was a significant positive relationship between learning motivation caused from the use of the strategy of spices with pbl method against the logical mathematical thinking ability of students. tabel 3. recapitulation results of logical thinking ability variable test average deviasi standart varians logical thinking ability pretest 29,43 7,322 53,605 postest 80,11 12,774 163,163 conclusion the results showed the existence of a positive interaction during learning. nevertheless, learning motivation of students did not experience a significant improvement. while logical thinking ability of students after learning of mathematics using the strategy of spices with pbl method is included in the category is enough. nevertheless, we can say that the use of the strategy of the spices with the methods of pbl can grow learning motivation of students that is expected to give significant effects against mathematical ability improvement. in its implementation, the strategy of spices needed support some teachers from several fields of science. this is intended so that the given problem on students can be served authentically and integrative in accordance with characteristics of spices and pbl. application of spices strategy with pbl method of student-centered learning activities, so students participate in the learning process, learn how to develop independent and group learning, and discuss what they have thought and make students more active during the learning process. with the onset of student activeness, students can develop their potential. activity of students in learning can stimulate and develop the talents they have, critical thinking, and can break the problems in everyday life. stages in the spices strategy with pbl method in addition to increasing student activity, the stage can also increase students' learning motivation in the classroom. it can be seen from the result of observation analysis of students' learning motivation during the learning process by applying spices strategy with pbl method take place. the result of the observation analysis shows that the students' learning motivation has improved in every learning process. this shows that by applying spices strategy with pbl method in the classroom, it can grow student's learning motivation. the result of data analysis is known that there is significant influence on spices strategy implementation with pbl method to students' logical thinking ability. maharani & laelasari, experimentation of spices learning strategies … 156 references bakar, r. (2014). the effect of learning motivation on student’s productive competencies in vocational high school, west sumatra. international journal of asian social science, 4(6), 722-732. dent, j. a. (2014). using the spices model to develop innovative teaching opportunities in ambulatory care venues. korean journal of medical education, 26(1), 3-7. fip-upi, t.p.i.p. (2007). ilmu dan aplikasi pendidikan i: ilmu pendidikan teoretis. pt imperial bhakti utama. elliott, a.j., & dweck, c.s. (2005). handbook of competence and motivation. new york: guilford press. hamalik, o. (2009). metode evaluasi dan kesulitan-kesulitan belajar. bandung: tarsito. krulik, s., & rudnick, j. a. (1995). the new sourcebook for teaching reasoning and problem solving in elementary school. a longwood professional book. boston: allyn & bacon. rusman (2010). model-model pembelajaran mengembangkan profesionalisme guru. jakarta: raja grafindo persada. sardiman, a.m. (2012). interaksi dan motivasi belajar mengajar. jakarta: rajawali pers. slavin, r. e. (2008). cooperative learning, success for all, and evidence-based reform in education. éducation et didactique, 2(2), 149-157. soekisno, r. b. a. (2015). pembelajaran berbasis masalah untuk meningkatkan kemampuan argumentasi matematis mahasiswa. infinity journal, 4(2), 120-139. sumarmo, u., hidayat, w., zukarnaen, r., hamidah, m., & sariningsih, r. (2012). kemampuan dan disposisi berpikir logis, kritis, dan kreatif matematik (eksperimen terhadap siswa sma menggunakan pembelajaran berbasis masalah dan strategi thinktalk-write). jurnal pengajaran mipa, 17(1), 17-33. infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p1-6 1 the strategy of formulate-share-listen-create to improve vocational high school students’ mathematical problem posing ability and mathematical disposition on probability concept tina rosyana 1 , m. afrilianto 2 , eka senjayawati 3 1,2,3 ikip siliwangi, jl. terusan jenderal sudirman, cimahi, west java, indonesia 1 tinarosyana@ikipsiliwangi.ac.id, 2 muhammadafrilianto1@ikipsiliwangi.ac.id, 3 ekasenjayawati@ikipsiliwangi.ac.id received: march 27, 2017 ; accepted: january 29, 2018 abstract this study aims to examine the improvement of students’ mathematical problem posing ability and mathematical disposition through the strategy of formulate-share-listen-create (fslc) on probability concept. the method used in this research is the experimental method, with the design of pretest-posttest control group. the population is all students of the vocational high school in cimahi, while the sample was selected two classes from one of the vocational high school selected at random. the instrument of a test in the form of description is used to measure students’ mathematical problem posing ability, while the non-test instrument is questionnaire of mathematical disposition scale. the results showed (1) the mathematical problems posing of the students who obtained fslc learning strategy is better than that of those who obtained conventional one; (2) the improvement of mathematical problems posing of the students who obtained fslc learning strategy is better than that of those who obtained conventional one; (3) the mathematical disposition of students who obtained fslc learning strategy is better than that of those who obtained conventional learning. keywords: disposition, formulate-share-listen-create, problem posing. abstrak penelitian ini bertujuan untuk menelaah peningkatan kemampuan problem posing dan disposisi matematis siswa dengan strategi formulate-share-listen-create (fslc) pada konsep peluang. metode yang digunakan dalam penelitian ini adalah eksperimen, dengan desain kelompok kontrol pretes-postes. populasinya adalah seluruh siswa smk di kota cimahi, sedangkan sampelnya dipilih dua kelas dari salah satu smk yang dipilih secara acak. instrumen penelitian ini yaitu tes bentuk uraian dalam kemampuan problem posing matematis, skala disposisi matematis, dan pedoman observasi. hasil penelitian menunjukkan bahwa (1) kemampuan problem posing matematis siswa yang memperoleh pembelajaran dengan strategi fslc lebih baik daripada yang memperoleh pembelajaran biasa; (2) peningkatan kemampuan problem posing matematis siswa yang memperoleh pembelajaran dengan strategi fslc lebih baik daripada yang memperoleh pembelajaran biasa; (3) disposisi matematis siswa yang memperoleh pembelajaran dengan strategi fslc lebih baik daripada yang memperoleh pembelajaran biasa. kata kunci: disposisi, formulate-share-listen-create, problem posing. how to cite: rosyana, t., afrilianto, m., & senjayawati, e. (2018). the strategy of formulate-share-listen-create to improve vocational high school students’ mathematical problem posing ability and mathematical disposition on probability concept. infinity, 7 (1), 1-6. doi:10.22460/infinity.v7i1.p1-6 mailto:tinarosyana@ mailto:2%20muhammadafrilianto1@ikipsiliwangi.ac.id rosyana, afrilianto, & senjayawati, the strategy of formulate-share-listen-create … 2 introduction in mathematics learning, problem posing process is very important, especially in the middle school. nctm (2000) recommended that mathematical problem formulate based on many situational, whether outside or inside mathematics, arranging and finding conjecture, also learning to generate and to extend problems through problem posing. kilpatrick, swafford, & findell (2001) stated, "problem posing is an essential content in mathematics and nature of mathematical thinking, as well as an important part of mathematical problem solving. according to da ponte & henriques (2013), "investigation of mathematics affords a great opportunity to bring up the problem posing". it is based on the view that the problem posing can trigger the on-going of mathematical activities through the process of asking questions. kilpatrick, swafford, & findell (2001) stated the quality of the questions students describes their abilities in solve the problem. in fact, according to da ponte and henriques (2013), "at the heart of mathematics is to pose a problem and solve it". mayadina (2012) stated that mathematical problem posing consist of two aspect are accepting and challenging. however, according to sumarmo (2015), in contrast to the large attention to the discussion of mathematical problem solving, the mathematics curriculum has not paid much attention to the discussion of mathematical problem posing (mpp). other than, the reality on the ground shows that vocational high school students are more geared to master certain applied skills, so the ability of problem posing is appropriate to be trained to assist them in solving mathematical problems. besides demanded to have the mathematical problem posing ability, students are expected also to make improvement of their performance in learning through the positive behavior as part of the soft skills. in connection with students’ affective, sumarmo (2013) argued, "mathematical soft skills as components of mathematical thinking process in the affective domain are characterized by affective behavior shown by someone when executing mathematical hard skill. the affective behavior is associated with the term disposition showing a tendency to behave with a strong impetus. "mathematical disposition is also demonstrated through strong dedication to positively thinking. mathematical disposition is the correlation and appreciation of mathematics that is a tendency to think and act in a positive way (bernard, 2015). then, according to polking (hidayat, 2012; sumarmo, hidayat, zukarnaen, hamidah, & sariningsih, 2012), “mathematical disposition indicates: 1) confidence in using mathematics; 2) flexibility in solving problems; 3) persistence in working on mathematical tasks; 4) interest, curiosity, and discovery power in performing mathematical tasks; 5) monitoring and reflecting their own performance and reasoning; 6) assessment of the application of mathematics to other situations in mathematics and everyday experience; 7) appreciation of the role of mathematics in culture and values, mathematics as a tool, and as a l anguage. however, according to sugilar (2013) state that this moment, the students' mathematical power and disposition has not been fully achieved. one of the effort that is expected to improve the student’s mathematical problem posing and mathematical disposition by applying learning strategies with grouping. kilpatrick, swafford, & findell (2001) stated that problem solving can be done easily through discussions in large groups, but the problem-solving process will be more practical when done in small groups working together. one of the learning strategies that can be applied is the formulate-sharelisten-create strategy. volume 7, no. 1, february 2018 pp 1-6. 3 for the sake of students’ character development, sumarmo (2013) stated that mathematical learning can help students to form their character or personality in various ways. selection of strategies in mathematics learning can form students’ characters. therefore, we need a learning strategy to improve students’ mathematical problem posing ability and mathematical disposition. this strategy can make them active, train them to collaborate and help each other in solving a given problem and provide opportunities find themselves and understand the material more deeply. fslc is a form of cooperative learning in small groups and is a modification of the thinkpair-share (tps) strategy. fslc which includes the steps as follows: a) formulate: the activity of recording information related to the duties and making plans for settlement; b) share: students share their opinions with their partner; c) listen: each pair mutually hear from other couples, and note the differences and similarities of the opinions; d) create: students discuss to reach a conclusion. method the method used in this study is experimental method, with the design of pretest-posttest control group. in this type of design there is a grouping of randomized subjects (a), the pretest (o), and their posttest (o). the research design is like the followings: a o x o a o o notes: a : the selection of a random sample of classes at population o : pretest = posttest (test of mathematical problem posing and mathematical disposition ability) x : the application of fslc learning strategy the population is students in one of vocational high school in kota cimahi. the samples in this study are two classes randomly selected from class xi smk. students in the experimental class who obtained fslc learning strategy, while students in control class who obtained conventional learning. the instrument used in this research are: 1) mathematical problem posing anality test, 2) mathematical disposition scale, and 3) student observation guidelines. results and discussion results the data were analyzed by descriptive and inferential statistical analysis. all data is processed by microsoft excel 2007 and spss 17. here are described the results of research and its discussion. before performing data analysis, first is presenting the data descriptive statistics of pretest ability of mathematical problem posing (mpp). the descriptive data of students’ mathematical problem posing are presented in the following table 1. rosyana, afrilianto, & senjayawati, the strategy of formulate-share-listen-create … 4 table 1. descriptive statistics mathematical problem posing ability (mpp) test class statistic statistical values pretest fslc ̅ 7.13 s 1.33 cl ̅ 7.15 s 1.81 posttest fslc ̅ 15.73 s 2.59 cl ̅ 14.26 s 3.69 n-gain fslc ̅ 0.51 s 0.15 cl ̅ 0.42 s 0.21 disposition fslc ̅ 98.53 % 82.11 cl ̅ 91.57 % 76.31 the data analysis of posttest results aims to test the first hypothesis, which is to find out the mathematical problem posing ability between the fslc learning strategy and the conventional learning. the statistic used is t-test. the result of statistical t-test are presented in the following table 2: table 2. t-test results of posttest data of mathematical problem posing ability asymp.sig. (2-tailed) asymp.sig. (1-tailed) conclusion 0.054 0.027 reject h0 according to the table above, it is obtained that the value asymp.sig (one-tailed) is 0.027 which is less than mathematical problem posing of students who obtained fslc learning strategy is better than that of those who obtained conventional learning on probability concept. n-gain data analysis aims to test the second hypothesis, which is to find out the improvement of mathematical problem posing ability between the fslc learning strategy and the conventional learning on probability concept. the statistic used is t-test. the result of statistical t-test are presented in the following table 3: volume 7, no. 1, february 2018 pp 1-6. 5 table 3. t-test results of n-gain data towards the ability of mathematical problem posing asymp.sig. (2-tailed) asymp.sig. (1-tailed) conclusion 0.053 0.0265 reject h0 according to the table above, it is obtained that the value asymp.sig (one -tailed) is 0.0265 which is less than α = 0.05, so h0 is rejected and h1 accepted. this means that the improvement of students’ mathematical problem posing ability who obtained fslc learning strategy is better than that of those who obtained conventional learning. the analysis of disposition aims to test the third hypothesis, which is it to examine the mathematical disposition between the fslc learning strategy and the conventional learning. the statistic used is t-test. the result of statistical t-test are presented in the following table 4: tabel 4. the results of t-test of mathematical disposition data asymp.sig. (2-tailed) asymp.sig. (1-tailed) conclusion 0.048 0.024 reject h0 based on the table above, it is obtained that the value asymp.sig (one-tailed) is 0.024 which is less than α = 0.05, so h0 is rejected and h1 accepted. this means that the mathematical disposition of students who obtained fslc learning strategy is better than that of those who obtained conventional learning. discussion fslc is a strategy of learning in small groups in pairs which contains steps: formulating their own opinion, sharing opinions with other couple friends, and deducing by combining the best ideas. this research aims to examine: 1) the improvement students’ mathematical problem posing ability through fslc learning strategy compared to those who obtained conventional learning on probability concept; and 2) the students’ mathematical disposition through fslc learning strategy compared to those who obtained conventional learning on probability concept. in general, the implementation of fslc learning strategy has run well and been in line with expectations. statistical tests conducted towards the posttest data showed that the students’ mathematical problem posing ability who obtained fslc learning strategy is better than that of those who obtained the conventional learning. then, the statistical tests conducted towards n-gain data showed that the increased students’ mathematical problem posing ability who obtained fslc learning strategy is better than those who obtained conventional learning. furthermore, the analysis of mathematical disposition showed that students who obtained fslc learning strategy are better than that of those who obtained the conventional learning. the results of this research are equal with the results of the research by anggraeni (2013) showed that implementing fslc was able to improve students’ mathematical understanding and mathematical communication abilities better than that of conventional approach. students’ mathematical understanding and communication abilities were classified as mediocore. rosyana, afrilianto, & senjayawati, the strategy of formulate-share-listen-create … 6 conclusion the conclusions of this research are: (1) the mathematical problems posing of the students who obtained fslc learning strategy is better than that of those who obtained conventional one; (2) the improvement of mathematical problems posing of the students who obtained fslc learning strategy is better than that of those who obtained conventional one; (3) the mathematical disposition of students who obtained fslc learning strategy is better than that of those who obtained conventional learning. references anggraeni, d. (2013). meningkatkan kemampuan pemahaman dan komunikasi matematik siswa smk melalui pendekatan kontekstual dan strategi formulate-share-listencreate (fslc). infinity journal, 2(1), 1-12. bernard, m. (2015). meningkatkan kemampuan komunikasi dan penalaran serta disposisi matematik siswa smk dengan pendekatan kontekstual melalui game adobe flash cs 4.0. infinity journal, 4(2), 197-222. da ponte, j. p., & henriques, a. (2013). problem posing based on investigation activities by university students. educational studies in mathematics, 83(1), 145-156. hidayat, w. (2012). meningkatkan kemampuan berpikir kritis dan kreatif matematik siswa sma melalui pembelajaran kooperatif think-talk-write (ttw). in seminar nasional penelitian, pendidikan dan penerapan mipa. kilpatrick, j., swafford, j., & findell, b. (2001). adding it up: helping children learn mathematics. washington, dc: national academy press. mayadina, s. (2012). meningkatkan kemampuan problem posing matematika mahasiswa calon guru sd melalui model pembelajaran scpbl. eduhumaniora, 3(1). nctm. (2000). principles and standards for school mathematics. usa: the national councils of teachers of mathematics. sugilar, h. (2013). meningkatkan kemampuan berpikir kreatif dan disposisi matematik siswa madrasah tsanawiyah melalui pembelajaran generatif. infinity journal, 2(2), 156-168. sumarmo, u. (2013). kumpulan makalah berpikir dan disposisi matematik serta pembelajarannya. bandung: fakultas pendidikan matematika dan ilmu pengetahuan alam universitas pendidikan indonesia bandung. sumarmo, u. (2015). mathematical problem posing: rasional, pengertian, pembelajaran, dan pengukurannya. retrieved from stkip siliwangi: http://utarisumarmo.dosen.stkipsiliwangi.ac.id/files/2015/09/problem-posing-matematikpengertian-dan-rasional-2015.pdf. sumarmo, u., hidayat, w., zukarnaen, r., hamidah, m., & sariningsih, r. (2012). kemampuan dan disposisi berpikir logis, kritis, dan kreatif matematik (eksperimen terhadap siswa sma menggunakan pembelajaran berbasis masalah dan strategi think-talk-write). jurnal pengajaran mipa, 17(1), 17-33. http://utari-sumarmo/ http://utari-sumarmo/ infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.243 51 senior high school teachers’ mathematical questioning ability and metaphorical thinking learning heris hendriana mathematics education, stkip siliwangi, cimahi, indonesia herishen@yahoo.com received: october 30, 2016 ; accepted: january 28, 2017 abstract this study is designed in the form of experiment with the design of control group and posttest only aimed at investigating the role of learning that teaches metaphorical thinking in improving senior high school teachers’ mathematical questioning ability. the population of this study was senior high school teachers of mathematics in west java province and the samples were 124 senior high school teachers of mathematics set purposively and randomly to be included into the experimental class and control class. based on the results and discussion, it is concluded that: (1) mathematical questioning ability of the teachers who received metaphorical thinking learning is better than those who received conventional learning; (2) learning factors and kam (kemampuan awal matematis = prior mathematical ability) affect the achievement of teachers’ mathematical questioning ability. in addition, there is an interaction effect between the learning and kam in developing teachers’ mathematical questioning ability; (3) teachers’ mathematical questioning ability has not been achieved optimally on the indicators submitting problems in the form of non-routine questions and open-ended questions. keywords: kam, mathematical questioning ability, metaphorical thinking abstrak studi ini dirancang dalam bentuk eksperimen dengan disain kelompok kontrol dan postes saja yang bertujuan menelaah peranan pembelajaran yang mengajarkan berpikir metaforik terhadap kemampuan bertanya matematis guru sma. populasi dalam penelitian ini adalah guru sma mata pelajaran matematika di provinsi jawa barat, sedangkan sampel penelitian ini adalah 124 orang guru sma mata pelajaran matematika yang ditetapkan secara purposif kemudian ditetapkan secara acak yang termasuk ke dalam kelas eksperimen dan kelas kontrol. berdasarkan hasil dan pembahasan diperoleh kesimpulan: (1) kemampuan bertanya matematis guru yang memperoleh pembelajaran metaphorical thinking lebih baik daripada yang memperoleh pembelajaran biasa; (2) faktor pembelajaran dan kam masing-masing mempengaruhi ketercapaian kemampuan bertanya matematis guru. selain itu, terdapat efek interaksi antara pembelajaran dan kam secara bersama-sama dalam mengembangkan kemampuan bertanya matematis guru; (3) ketercapaian penguasaan kemampuan bertanya matematis guru masih belum tercapai dengan baik pada indikator pengajuan permasalahan berupa pertanyaan non-rutin dan pertanyaan terbuka. kata kunci: kam, kemampuan bertanya matematis, metaphorical thinking how to cite: hendriana, h. (2017). senior high school teachers’ mathematical questioning ability and metaphorical thinking learning. infinity, 6 (1), 51-58. hendriana, senior high school teachers’ mathematical questioning ability … 52 introduction conversations in daily life can be done both in formal and informal situations. understanding a conversation is necessary in order to response interactions that occurs with each other. hutchby & wooffitt (rezvani & sayyadi, 2015) stated that conversations in real life are organized to achieve the goals that can be understood by each other. yule (rezvani, &sayyadi, 2015) stated that the ability to ask questions not only allows students to create structured and natural conversations. asking a question requires good communication skills so that the message or the questions can be understood as well, and the elements required in asking a question are the mastery of good grammar, that of the material, and communicative skills. high school students with their capacity can ask a question properly because they already have high thinking level. thomas & thorne (2010) suggested "higher order thinking occurs when a person takes new information and information stored in memory and interrelates and / or rearranges and extends this information to achieve a purpose or find possible answers in perplexing situations". stimuli given to students to ask a question are highly important because the stimuli can motivate students to be curious about something. hendriana (2012) suggested that a question is any sentence that has interrogative form or function. the purpose of asking questions is as follow: (1) to develop interest and motivation; (2) to evaluate preparation; (3) to develop thinking skills; (4) to foster insight; (5) to assess achievement of objectives; (6) to enhance knowledge. students’ success in learning mathematics is not only seen by their work on problems or tests given. the ability to ask helps them uncover what has been learned so it can comprehensively explore their ability in posing questions. this is in line with the opinion of tofade, elsner & haines (2013) who suggested that questions initiated by students can improve high level learning because the questions require them to analyze, connect and illustrate information. a question can be classified good or less good. less good questions can make learning process to be not conducive and create confusion. the ability to ask good question covers aspects of quality, language, relevance and frequency. this is in line with widodo (hendriana, rohaeti & hidayat, 2016) who suggested that about half questions posed by teachers regarding with the material are mostly closed-ended questions that require short answers and are in the forms of memorization and understanding. understanding a mathematical problem must be significant to enable students to express meaningful statements based on the data, warranties, ideas, ideas that can generate good solutions. elder & paul (2002) stated that thinking without statement is not an intellectual one. when facing complex problems, students often have difficulties in finding solutions because the problems are not routinely found by students. piatek-jimenez (2010) suggested that students often have difficulties in interpreting and proving a statement. the quality of mathematical learning is dependent on how students understand the problems and solve them using innovative solutions. zhang, wang & li (2012) suggested in their study that students’ ability is not only seen from their grade school, but also from how they are taught to make solution they can understand so it becomes an innovative one. hendriana (2012) stated that, in the learning process, students simply model and record how to resolve an exercise item that has been done by teachers. if this happens continuously, volume 6, no. 1, february 2017 pp 51-58 53 students will not get completion as expected. minarni, napitupulu & husein (2016) explained that students’ mathematical representation and understanding ability is still relatively low. the study by hendriana, rohaeti & hidayat (2016) suggested that this lack ability is caused by the fact that the students are rarely given the opportunity to think openly due to the fact that they rarely ask questions of the problems being faced. students’ ability to pose mathematical questions is always based on cognitive ability. rahman (2013) argued in his research that students who have cognitive style of independent field are able to propose a mathematical problem which can be solved and load new data, with highquality math problem category. students who have the cognitive style of field dependent are able to ask mathematical problems that can be solved but it does not contain new data with category of medium quality math problems. the results of research by hendriana (2013) about the communication skills in improving mathematics teachers’ ability to pose questions in effect to the students’ learning outcomes in elementary schools in bandung shows that teachers who have communication skills in terms of asking that fall into the low category likely to result in student learning outcomes that is also low, while teachers communication skills in terms of asking who belong to good category tend to produce student learning outcomes that are classified as moderate and high. basically, both teachers and students already have the ability to ask, but it is not explored well yet. therefore, motivating the students to develop mathematical questioning ability needs strategies that must be done by teachers by providing structured learning so that students can learn independently, at least in posing some problems encountered, both in the form of questions and statements. one of the strategies is to teach students to think metaphorically (metaphorical thinking). students’ ability to ask questions is closely related to think metaphorically or clarify one’s thinking. meij & dillon (1994) argued that a question can be classified into necessary question and unnecessary questions. according to holyoak & thagard (hendriana, 2012), metaphor is originated from a concept known by students going to another concept that is unknown or is being learned. metaphor is dependent upon a number of concepts and properties of objects being metaphorized. relations regarding to the development of the ability to ask and think metaphorically (metaphorical thinking) are: (1) students are able to connect the problems of a given statement into a question posed in order to dig deeper information, (2) the students are able to find new concepts that are not already known yet such as conjectures which are expected to become the basis of their question, (3) the students are able to create creative ideas coming from the problems faced, and (4) the students are able to apply the results of their thinking in the form of a question of a problem given statement. in facing mathematics problems, it is expected that students are able to change the paradigm so it will be assumed that learning math is fun and easy. if previous learning only focused on cognitive aspects alone but must now be seen in other aspects as well. besides, the phenomena on happening now is the relationship between real conditions that must be associated with mathematics in their daily lives; it certainly can be a very good reference, because math is not only counting or discovering formulas but also can be applied in their daily lives. carreira (hendriana, 2012) suggested that finding a link between mathematical and real phenomenon is a process and effort to play an important model. hendriana, senior high school teachers’ mathematical questioning ability … 54 metaphoric thinking in mathematics must be started by adding a model of the situation systematically, so that the models are interpreted from the viewpoint of semantics. both teachers and students have the ability to ask questions, but it is not explored well yet. developing the ability to ask in the subjects of mathematics requires strategies that must be done by teachers to support the learning process. one of them is by thinking metaphorically (metaphorical thinking). metaphorical thinking is a process that uses metaphors to understand a concept. in line with holyoak & thagrad (hendriana, 2012), metaphor is originated from a concept known by student going to another concept that is unknown or is being learned. all this depends on the number and nature of the concept of object. to clarify one's thinking of mathematical activity relationship can be track by metaphorical thinking. hendriana (2012) described that the conceptual forms of metaphorical thinking include: (1) understanding of basic mathematical ideas, (2) building relations, (3) redefining. the relationship between learning which teaches metaphorical thinking is the ability to connect problem from questions to become a referred statement to deeper information. hendriana, rohaeti & hidayat (2016) suggested that there are four links between the ability to ask and to think metaphorically, among which are: (1) students are able to connect the issue of the statement, (2) students are able to find a new concept that they have not known yet, (3) the students are able to create creative ideas derived from problems, (4) students are able to apply the results of their thinking into question. in regards with the ability to ask and to think metaphorically in math, assessment needs to be done through further research on high school mathematics teachers' ability to ask in west java province. the definition of the ability to ask in this study is the ability of teachers to connect, discover, create and apply the mathematical concept of statement to generate a problem in the form of the question in line with the concept. based on the background above, the problem and the purpose of this research is to investigate and examine: (1) whether the ability to ask of high school mathematics teacherswho obtain metaphorical thinking learning is better than those who obtain conventional usual learning. (2) whether there is an interaction effect between learning and kam (kemampuan awal matematis/early mathematics ability) in developing the ability to ask of high school mathematics teachers. (3) how is the achievement of the ability to ask of high school mathematics teachers? method this study is designed in the form of experimental with the design of only posttest control group whose purpose is to examine the role of learning that teaches metaphorical thinking towards the ability to ask of high school mathematics teachers. the population in this study is high school mathematics teachers in west java province, while the sample is 124 high school teachers of mathematics courses who are set purposively and then determined randomly into the experimental class and control class. the tests of mathematical questioning proficiency in this study are compiled by referring to the characteristics of the ability to ask and guidelines of good test. data is analyzed by using statistical tests of two-way annova to see the differences and the effects of interaction between the learning and kam in generating high school mathematics teachers’ questioning ability. volume 6, no. 1, february 2017 pp 51-58 55 results and discussion findings that are related to high school mathematics teachers’ questioning abilityare presented in table 1. table 1. mathematical questioning ability prior mathematical ability mathematical questioning ability mtclass (n = 62) conventional learning (n = 62) mean sd mean sd good 8,64 (86,40 %) 1,07 8,16 (81,60 %) 1,22 moderate 6,83 (6,83 %) 0,78 6,37 (63,70 %) 1,06 less 6,21 (62,10 %) 1,10 5,76 (57,60 %) 0,86 total 7,48 (74,80 %) 1,08 7,07 (70,70 %) 1,14 notes: ideal score is 10 based on the results of the above description, interpretations obtained are as follows: a) when viewed totally, the ability to ask of high school mathematics teachers who obtain metaphorical thinking learning is better than the conventional class. the teachers’ ability to ask mathematical question teachers in both classes (conventional class and metaphorical thinking class) is still in high category (74.80%> 70.70% of the ideal score). b) based on the level of early mathematics ability (kam), the ability of mathematical questioning of teachers who acquire metaphorical thinking learning also looks different and shows that the ability of mathematical questioning of teachers who acquire metaphorical thinking learning is better than those in the conventional class. mathematical questioning ability of good level on both class (mt and conventional) belongs to the high category (86.40%> 81.60%), while for medium level and less classified as moderate category (68.30%> 63.70% and 62.10%> 47.60%). c) in terms of factors which affect the development of mathematical questioning ability, then, based on the description in table 1, it looks that both factors (both learning and kam) affect the development of mathematical questioning. in addition, there is no interaction effect between learning and kam jointly in developing teachers’ ability in posing mathematical questioning. to support the description of high school mathematics teachers’ questioning ability above, then, data analysis of high school mathematics teachers’ questioning ability is done through statistical tests of mean difference. after conducting normality test of the data distribution of high school mathematics teachers’ questioning ability, then, it is found that the data normally distribute. based on these findings, the test of ability mean difference above is done by using two-way annova. (see table 2) table 2. summary of two-way annova test development of teachers’ mathematical questioning ability based on the factors of learning and kam sources jk dk rjk f-test sig learning approach (a) 17,808 1 17,808 17,020 0,000 kam (b) 11,446 2 5,723 5,162 0,017 a x b 6,284 2 3,142 3,045 0,042 inter 76,833 119 1,113 (taken from spss. 22 output) hendriana, senior high school teachers’ mathematical questioning ability … 56 learning approach h0 : e  = k  ha : ke   testing criteria: if sig > 0,05 then h0 is accepted based on table 2, it is obtained that sig = 0,000; or in other words sig <0.05. it can be concluded that, at the significance level of 5%, there are significant differences between mathematical questioning ability of teachers whose learning use metaphorical thinking and those who use conventional class. kam h0: ha: there is at least one kam which differ significantly from the other kam testing criteria: if sig> 0.05 then h0 is accepted based on table 2 it was obtained that sig = 0,017; or in other words sig <0.05; it can be concluded that, significance level of 5%, at least there is one particular group of kam whose mathematical questioning ability is significantly different from the other kam. to find out which kam that is significantly different, the scheffe test is conducted. the results of the calculations are presented in table 3. table 3. scheffe test of development of mathematical questioning ability based on kam tkas (i) tkas(j) mean difference (i – j) sig interpretation good moderate 0,240 0,585 not different moderate moderate 0,344 0,343 not different good low 0,762* 0,037 different (taken from spss.22 output) according to table 3 it is concluded that there are significant differences between the mathematical ability to ask at the kam good with low than the kam good and moderate with kam moderate and low on the significance level of 5%. the implication is that mathematical ability to ask of the teachers for kam good and low is more developed than kam good and moderate with kam moderate and low. effect interaction between learning approach and tkas h0: there is no interaction between the effects of approach and kam ha: at least there is one deviation that differs significantly from the others. based on table 2 it was obtained that sig = 0.042 is less than 0.05; it can be concluded that there is a significant interaction effect between learning approach (mt, and basic) with kam to produce teachers’ mathematical ability to ask at the 5% significance level. based on the findings in the field, the achievement of teachers’ mathematical questioning ability has not been as expected. the achievement of the results is attached in table 4. volume 6, no. 1, february 2017 pp 51-58 57 table 4. achievement of mathematical questioning ability indicators of mathematical questioning ability kam mt learning conventional class problems posed are connected with statement contexts given good 100% 100% moderate 100% 100% low 90% 100% total 97% 100% posing problems is in the form of non-routine questions of given statements good 75% 60% moderate 67% 53% low 58% 52% total 68% 56% posing problems is in the form of open-ended questions of given statements good 71% 63% moderate 67% 62% low 66% 59% total 69% 63% the results, shown in table 4, concludes that the achievement of mastery of mathematical ability to ask of high school teachers still has not been achieved well on indicators of problems in the form of a question submission of non-routine and open questions. based on observation, the constraints faced by high school level mathematics teachers are difficult to make non-routine questions with open answers expect from students are as follow: the atmosphere of learning in the classroom in applying the learning activities that leads students to think abstractly is caused by the initial conditions of students who are accustomed to only receive learning material without thinking of the material context in more depth. 1) teachers’ experience which make them not to keep up to date with the relevance of the times in the field of education, so in teaching, the teachers still use models, methods and strategies monotonous. 2) the demands of curriculum in realizing the learning outcomes in a learning atmosphere in accordance with the development of mathematical ability to ask because of the necessity of conformity plans and targets in implementing the curriculum. 3) outputs of student learning outcomes are more quantity-oriented rather than quality in the form of understanding the subject matter. based on this, then we need learning innovation efforts which have the purpose of projecting the development of teachers and students’ mathematical ability. these obstacles are in line with what hendriana, rohaeti & hidayat (2016) proposed that teachers still have difficulty in asking non-routine and open-ended questions at the problems of given statement. conclusion based on the results and discussion, it can be concluded that: (1) the ability to ask mathematical questions of high school teachers who obtain metaphorical thinking is better than those with conventional learning. overall, the ability to ask mathematical questions of teachers with metaphorical thinking and conventional learning falls into the category of good. however, if the review is based on the ability of early mathematics (kam), both classes which obtained metaphorical thinking learning and conventional one at the good kam fall into the category of high and kam moderate and low is classified as a category of moderate. (2) the factors of learning and kam affect the achievement of teachers’ ability to ask mathematical questioning. in addition, there is the effect of the interaction between the hendriana, senior high school teachers’ mathematical questioning ability … 58 learning and kam together in developing high school mathematics teachers’ ability to ask mathematical question. (3) achievement mastery of the high school teachers’ ability to ask mathematical questions still have not achieved well on indicators of problems in the form of a question submission of non-routine and open-ended questions. this makes the urgency of the problems that are treated with the existence of an undertaking in the form of innovative learning such as metaphorical thinking that can be applied from elementary to secondary education. so, that both teachers and students will be accustomed to think more comprehensively from all directions of thought in solving problems of mathematics education. references elder, l., & paul, r. a. (2002). the miniature guide to the art of asking essential questions. santa rosa, ca: fondation for critical thinking. hendriana, h. (2012). pembelajaran matematika humanis dengan metaphorical thinking untuk meningkatkan kepercayaan diri siswa. infinity, 1(1): 90-103. hendriana, h. (2013). analisis hasil belajar matematika siswa sd berdasarkan kemampuan komunikasi guru dan tingkat kemampuan awal matematika siswa. seminar nasional matematika, universitas katholik parahyangan. bandung: universitas katholik parahyangan. hendriana, h., rohaeti, e. e., & hidayat, w. (2016). junior high school teachers’ mathematical questioning ability and metaphorical thinking learning. journal on mathematics education, 8(1), 55-64. meij, v. h., & dillon, j. t. (1994). adaptive student questioning and students' verbal ability. the journal of experimental education, 62(4), 277-290. minarni, a., napitupulu, e. e., & husein, r. (2016, january). mathematical understanding and representation ability of public junior high school in north sumatra. journal on mathematics education, 7(1), 45-58. piatek-jimenez, k. (2010). students’ interpretations of mathematical statements involving quantification. mathematics education research journal, 22(3), 41-56. rahman, a. (2013). pengajuan masalah matematika di tinjau dari gaya kognitif dan kategori informasi. jurnal ilmu pendidikan, 19(2): 244-251. rezvani, r., & sayyadi, a. (2015). instructors’ and learners’ questioning: a case of efl classroom discourse in iran. journal of teaching language skills, 34(3), 141-164. thomas, a., & thorne, g. (2010). how to increase higher-order thinking. center for development and learning metairie, louisiana. tofade, t., elsner, j., & haines, s. t. (2013). best practice strategies for effective use of questions as a teaching tool. american journal of pharmaceutical education, 77(7), 155. zhang, c., wang, h., & li, w. (2012). the research of the relationship between university mathematics learning and quality education and enforce of human’s ability. engineering education and management, 437-441. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p319-330 319 students' perceptions of using e-comics as a media in mathematics learning yulinar safitri*, mailizar, rahmah johar universitas syiah kuala, indonesia article info abstract article history: received feb 23, 2021 revised july 13, 2021 accepted july 14, 2021 the development of technology in education greatly influences learning strategies. thus, teachers must adapt and present interesting and technologybased learning, such as e-comics. therefore, the teacher must see in advance the extent to which students will accept e-comics for use in learning mathematics. this research aimed to determine students' perceptions of the use of e-comics as a media in mathematics learning. this research implemented a quantitative approach with a survey method. the samples were 124 students of junior high schools (smp / mts) in aceh. the research data were obtained from questionnaires filled by students which were collected through the tam (technology acceptance models) framework which was distributed online. the results showed that students used e-comic as a learning media influenced by their perceived benefits and attitudes towards the use of e-comic. the perceived benefits of students' attitudes have a significant role in their behavioral intention to use e-comic in learning mathematics. this research implies that e-comics has the potential to be used as a media in mathematics learning, especially on material that is considered difficult so that it can attract students' attention. keywords: e-comic, media, students perceptions, tam copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: yulinar safitri, departement of mathematics education, faculty of teacher training and education, universitas syiah kuala jl. tgk. hasan krueng kalee, kopelma darussalam, banda aceh, aceh 24415, indonesia email: yulinar.s@mhs.unsyiah.ac.id how to cite: safitri, y., mailizar, m., & johar, r. (2021). students' perceptions of using e-comics as a media in mathematics learning. infinity, 10(2), 319-330. 1. introduction mathematics is the science that deals with the logic, problem-solving, applying logic to real-life situations. problem-solving in real situation involves rational, exponential, and logarithmic functions. however, achieving learning objectives can pose challenges for teachers, especially concerning learning duration. therefore, teachers must find strategies to optimize the available duration. as stated by mamolo (2019), teachers can find more interesting ways when dealing with students, one of which is by developing learning media suitable for students of the 21st century. the topic will be easier for students to understand if they are presented attractively, making students more enthusiastic, motivated, and willing to participate actively in learning. the use of media in learning can help in describing a concept better and easier. the learning https://doi.org/10.22460/infinity.v10i2.p319-330 safitri, mailizar, & johar, students' perceptions of using e-comics as a media … 320 media used must be following the overall learning objectives so that they can play their roles properly and effectively in the learning system. one of the media that can support mathematics learning activities is comics. comics can be defined as a cartoon that expresses characters and implements a story in a sequence that is closely related to images designed to entertain the readers. comics has a storyline that expresses an imagination and tells stories in an interrelated sequence, by combining pictures and writing designed to provide entertainment to readers. comics has an instructional role in raising student’s interest. the use of comics can help students capture abstract things or formulas. along with current technological development that influences the planning of learning strategies, teachers can present their learning in electronic form. electronic-based learning media are expected to meet the needs of students in the 21st century who are considered residents of the digital era. as expressed by kristanto, mustaji, and mariono (2017), there are already many that implement electronic-based learning media because they are the current global issue. some people have developed comics as a medium of teaching for different subjects, levels of education, and educational objectives (lazarinis et al., 2015). comics used in the teaching and learning process can increase motivation and stimulation of learning activities, and it has a psychological effect on students (buchori & setyawati, 2015). besides, comics can also increase student motivation and interest (toh et al, 2017) and creates fun learning (özdemir, 2017). the use of cartoons can help to decrease anxiety in learning mathematics (clair, 2018). the use of cartoons in teaching encourages students to express problems experienced and makes students realize that mathematics instruction is not only about giving "short correct answers" but also involves rich dialogues. their imagination and creativity are set free (clair, 2018). therefore, e-comics are needed to achieve learning objectives and make learning more interesting's so that students are more active and motivated in learning. introducing technology to students is an important aspect to see the extent of acceptance and use of technology, the success of technology-based learning systems will also not be achieved if students cannot use the system (mcfarland & hamilton, 2006; pituch & lee, 2006; yeou, 2016). hence, it is very urgent to know the of students so that the use of technology is successful, such as the use of e-comic in learning mathematics. many studies on the use of electronic learning have been applied, one of which is about e-learning. the use of e-learning must consider intentions and behavior (cigdem & topcu, 2015). in research on the use of technology-based systems, the technology acceptance model (tam) popularized by davis (1989) is most often cited. tam is specifically designed to describe and predict user acceptance of certain types of technology. tam suggests that perceived usefulness (pu) and perceived ease of use (peu) are important factors in influencing individual attitudes toward using technology (nagy, 2018). in the same model, user attitudes are hypothesized to influence behavioral intentions to use the technology. in addition, external factors also affect the usability and ease of use of the system. according to davis (1989), various external factors or stimuli can influence students’ behavior. in this research, there were four factors or indicators of tam which were used, without expanding tam by adding external factors. this effort would increase the effectiveness of tam and its relevance to the indicators of peu, pu, at, and iu to use the system. based on previous explanations, this research aimed to determine the students' perceptions of the use of e-comics as a media in mathematics learning. the technology acceptance model (tam) is based on the theory of reasoned action (tra) in psychological research (fishbein & ajzen, 1975). according to tra, individual behavior to do something is influenced by intention. tra is presented in figure 1. volume 10, no 2, september 2021, pp. 319-330 321 figure 1. tra (theory of reasoned action) according to the flowchart in figure 1, tra has six variables. among them are believers and evaluations, perceived usefulness, normative beliefs and motivation to comply, perceived ease of use, behavior, and behavior intention. the principal variable is behavior, which is influencing by behavior intention. while behavior intention influenced by pu and peu, as well as pu originally based on the beliefs and evaluations and peu by normative beliefs and motivation to comply. meanwhile, tam argues that perceived ease of use and perceived usefulness were the drivers of user attitudes, while perceived ease of use was also considering to affect perceived usefulness. figure 2 the original version of tam (davis, 1989). figure 2. tam (technology acceptance model) figure 2 is the framework used to investigate how and when users adopt emerging technologies. tam has proven appropriate to show the relationship between peu, pu, at, and bi. accord to davis (1989), bi is influenced by at and is also influenced by peu and pu, but not vice versa. tam has been applied in various research to test user acceptance of information technology, such as mobile library (rafique et al., 2020), m-learning (al-emran, mezhuyev, & kamaludin, 2018), e-books (liao et al., 2018), online teaching (wingo, ivankova, & moss, 2017), statistical learning platform (song & kong, 2017), and e-service (taherdoost, 2018). in this research, e-comics is considered as a media that utilizes internet and web technology to convey information to achieve learning objectives in which students can interact directly through computers or smartphones. safitri, mailizar, & johar, students' perceptions of using e-comics as a media … 322 in the acceptance and use of technology, tam is considered the most popular inquiry model (abdullah & ward, 2016). legris, ingham, and collerette (2003) stated that the use of the tam framework can be added by other variables to provide a better explanation of the use of technology. research has expanded tam by adding external variables, such as motivation (zain et al., 2019), teacher self-efficacy (teeroovengadum, heeraman, & jugurnath, 2017), self-efficacy and enjoyment (abdullah, ward, & ahmed, 2016), selfesteem (cheng, 2019), and information quality, system quality, and service quality (chi, 2018). the variables used in this research were perceived ease of use (peu), perceived usefulness (pu), attitude towards using (at), and intention to use (iu), all based on the tam framework. figure 3. structural model and hypothesis based on the flowchart in figure 3, the hypothesis to be tested is to see whether pu is influenced by peu, at is influenced by pu and peu, and iu is driven by peu, pu, and at. perceived ease of use (peu) is defined as the extent to which technology is easy to use (moore & benbasat, 1991). in this research, peu is defined as the extent to which users believe that e-comics is easy to use in mathematics learning. the findings showed that technology acceptance increases along with the increase of peu (huang, 2017). nikou and economides (2017) found that peu affects pu, and both factors can cause iu. there is a relationship between peu and pu, and both have an impact on at (revythi & tselios (2019), but not on iu (huang, 2016). another study investigating the relationship between peu and pu showed that peu had no direct effect on pu (tsai et al., 2017). this research decided to use peu indicator to see the extent students used e-comics in mathematics learning and the influence of peu on pu, at, and iu. the researchers hoped that peu would influence pu, at, and iu, and students would continue to use e-comics in mathematics learning in the future. then, the following hypothesis 1, hypothesis 2, and hypothesis 3 were proposed. h1: peu has a significant influence on pu in using e-comics in mathematics learning. h2: peu has a significant influence on at in using e-comics in mathematics learning. h3: peu has a significant influence on iu in using e-comics in mathematics learning. perceived usefulness (pu) is defined as the level at which users believe that a particular system will improve its performance (davis, 1989). in this research, pu is defined as the extent to which users believe that using e-comics in mathematics learning will improve their learning performance. from the perspective of tam theory, it shows that when someone intends to do something, pu is a precursor to a person's iu before they commit volume 10, no 2, september 2021, pp. 319-330 323 their act. there are some literature reviews in different academic fields (khor, 2014; tarhini et al., 2013; 2014; wu & zhang, 2014) regarding the importance of pu in the adoption of new technologies. users who believe in the benefits of using technology will accept the technology (davis, 1989) and consequently influence the intention to use technology (joo, park, & shin, 2017). in addition, when the use of technology is considered useful, users will use the technology (gao, krogstie, & siau, 2014; saroia & gao, 2019). this research implemented pu indicator to see the extent to which students use ecomics in mathematics learning with the influence of pu on at and iu. the researchers hope that pu could influence students' at and students' intention to continue using e-comics in mathematics learning in the future. therefore, hypothesis 4 and hypothesis 5 were proposed. h4: pu has a significant influence on at in using e-comics in mathematics learning. h5: pu has a significant influence on iu in using e-comics in mathematics learning. attitude towards using (at) is acceptance or rejection when someone uses technology. the results showed that attitudes affect individual behavior. this attitude consists of cognitive, affective and behavioral components. many studies have investigated at in technology acceptance and findings suggest that iu can be increased by at (ibili et al., 2019). in this research, at indicator is to measure students' acceptance or rejection of the use of e-comics in mathematics learning. the researcher hoped that at could influence students' iu to use e-comics in mathematics learning. hence, hypothesis 6 was proposed. h6: at has a significant influence on iu in using e-comics in mathematics learning. intention to use (iu) is a behavioral tendency to continue using technology in the future. many studies have investigated iu regarding technology acceptance such as nikou and economides (2019) which investigated technology in science and iu of mathematics teachers to use mobile-based assessment in their teaching environment. the findings showed that peu is the most important determinant of teacher iu to use mobile-based assessment. a study by sánchez-prieto, olmos-migueláñez, and garcía-peñalvo (2017) said that pu on iu is influenced by peu, and that the two variables have a direct influence on iu. in this research, the researchers measured students' iu for using e-comics in mathematics learning. 2. method the data for this study were collected using a survey method in collecting qualitative data which comprise statements on students' perceptions about e-comic in learning mathematics. descriptive statistics were used to analyze the responses on the questionnaire. the questionnaire was used to answer research questions. statistical package for the social sciences (spss) was used to analyze the different indicators for students’ acceptance of ecomic in learning mathematics and the interrelationships between these indicators. in this study, 124 secondary school students in aceh from over 10 schools were surveyed. of the participating students, 52.4% were female (n= 65) and 47.6% were male (n= 59). students who fill out the questionnaire are students who never have used mathematics e-comic in learning. however, students have been introduced to e-comic which will be used as a medium in learning mathematics. the content of mathematics learning in the e-comic has gone through the validation stages by several experts and has been declared valid. likewise, e-comic has also been validating by several experts who are experts in the field of comics. safitri, mailizar, & johar, students' perceptions of using e-comics as a media … 324 the instrument used to measure student perceptions was questionnaires with tam (technology acceptance model) framework, adapted from masrom (2007). the measured aspects were perceived ease of use (peu), perceived usefulness (pu), attitude towards using (at), and intention to use (iu). the questionnaire was a closed-questionnaire form. the validity of the questionnaire has been tested with cronbach's alpha as presented in table 1. the research data were obtained online. each student was asked to complete an online questionnaire showing their agreement or disagreement with each statement on a 7-point likert scale with the final points at both ends being "strongly disagree" and "strongly agree". in this research, spss was used to analyze various indicators of students' perceptions of using e-comics in mathematics learning. table 1. cronbach alpha indicator items 𝜶 perceived ease of use (peu) 4 0.89 perceived usefulness (pu) 4 0.89 attitude towards using (at) 3 0.85 intention to use (iu) 3 0.85 total 14 3. results and discussion 3.1. results table 2 presents the correlation matrix between indicators under discussion. the results indicated that the correlation between all indicators was significant (p <0.05) and positive. they also showed that the majority of the correlation coefficient (r) was greater than 0.5 which means that the correlation between the two variables was very strong. among all indicators, it was found that the peu→pu showed a high r-value of 0.690 (p <0.01) while the r-value of at→iu was the lowest 0.445. this indicated that the correlation between the two indicators was very strong. table 2. the correlation matrix indicator peu pu at iu peu 1.000 pu 0.690** 1.000 at 0.481** 0.593** 1.000 iu 0.450** 0.522** 0.445** 1.000 **. correlation is significant at the 0.01 level (2-tailed) table 3 presents the hypotheses testing of the structural model. the results showed that among the three independent variables, pu indicator was the largest contributor to the dependent variable peu (pu→peu; β = 0.581). while for the independent variable, the at indicator was the biggest contributor to the dependent variable pu (at→pu; β = 0.450). on the other hand, the iu indicator was the only contributor to the independent variable at (iu→at; β = 0.445). volume 10, no 2, september 2021, pp. 319-330 325 table 3. multiple regression analysis dv r2 iv beta () standard error of β tstatistic significance hypothesis peu 0.492 pu 0.581 0.082 6.709 0.000 (p<0.001) h1 at 0.089 0.103 1.080 0.282 (p>0.05) h2 iu 0.108 0.091 1.384 0.169 (p>0.05) h3 pu 0.434 at 0.450 0.101 5.894 0.000 (p<0.001) h4 iu 0.322 0.094 4.213 0.000 (p<0.001) h5 at 0.198 iu 0.445 0.075 5.485 0.000 (p<0.001) h6 figure 4 presents the results of the hypotheses model, while the results of testing the hypothesis are presented in table 4. for peu, the results showed that pu indicator (h1: peu→pu; β = 0.581, p < 0.001) significantly influenced peu. meanwhile, at indicator (h2: peu→at; β = 0.089, p > 0.05) and iu indicator (h3: peu→iu; β = 0.108, p > 0.05) did not significantly influence peu. these three indicators represented 49.2% of the peu variant. pu indicator (h4: pu→at; β = 0.450, p < 0.001) showed a positive significant correlation with at and iu indicators (h5: pu→iu; β = 0.322, p < 0.001). these two indicators explained 43.4% of the variant in pu. meanwhile, the at indicator (h6: at→iu; β = 0.445, p < 0.001) significantly influenced iu with a positive influence, as well as a 19.8% variant in at. table 4. results of hypothesis hypothesis effects direction path coefficient result (support to hypotheses) h1 peu→pu positive 0.695 supported h2 peu→at negative 0.096 not supported h3 peu→iu negative 0.169 not supported h4 pu→at positive 0.647 supported h5 pu→iu positive 0.226 supported h6 at→iu positive 0.268 supported figure 4. hypotheses of the structural model the flowchart shows that peu has a positive impact on pu. pu also has a positive effect on at and iu. however, at also has a positive influence on iu (see figure 4). safitri, mailizar, & johar, students' perceptions of using e-comics as a media … 326 3.2. discussion the main objective of this study is to examine the factors that influence high school students to use e-comic in learning mathematics. this study focus on students' perceptions of using e-comic in learning mathematics. the tam model (davis, 1986) was adopted without adding any external variables in this study. the results of this study show there are four points. first, this study shows the perceived ease of use (peu) of e-comic has a significant direct effect on the perceived usefulness (pu) of e-comic. this finding is consistent with other studies regarding the significant relationship of peu to pu. in terms of perceived usefulness, as research by park (2009) shows that perceived usefulness is one of the strongest determinants of the tam model. therefore, students who find this system useful in their education are likely to adopt the system. therefore, improving the quality of e-comic media in learning mathematics is very necessary so that students use it. during learning mathematics, students who used e-comic wil have many opportunities to learn because it can be used anytime and anywhere. for example, you don't need to carry books where students go, but can be accessed directly via smartphones. this study shows that the use of e-comic provides an opportunity for student to improve learning achievement. therefore, the use of e-comics in learning mathematics, in particular, has a positive impact on the achievement of learning objectives. second, regarding perceived usefulness (pu), the results show that perceived usefulness has a significant relationship to user attitudes (at). in this study, students' attitudes to using e-comic in learning mathematics were influenced by the perceived usefulness of using comics. this finding is following the hypothesis in this study that pu has a positive influence on students' at to use e-comic. previous studies have shown that perceived usefulness (pu) is important for predicting attitudes towards technology use (atu). this result also agrees with that reported by davis (the predecessor of tam theory), who assumes that perceived usefulness (pu) is one of the top indicators of user attitudes (atu) (davis, 1989; davis, bagozzi, & warshaw, 1989; venkatesh & davis, 2020). the third is regarding the behavior (ui) of students to use e-comic in learning mathematics. the results show that (ui) has a significant effect on pu and at. previous studies have also proven attitude (at) is very important for behavioral intention (hussein, 2017; taat & francis, 2020). other results show that peu is not significantly and positively related to ui. previous research showed a similar finding that peu (lew et al., 2019) was not significant for the prediction of iu's behavioral intention to use e-learning. the findings of this study indicate that the ease of using e-comic does not guarantee students to use e-comic during mathematics learning. in this case, attitude is an important factor that determines the use of e-comics in learning mathematics. students will use e-comic if e-comic has great benefits for learning mathematics while they are studying. 4. conclusion this study reveals how students perceive to use e-comic in learning 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(2019). investigating student's acceptance of an edmodo content management system. international journal of instruction, 12(4), 1-16. https://doi.org/10.29333/iji.2019.1241a https://doi.org/10.7763/ijiet.2013.v3.233 https://doi.org/10.1016/j.chb.2014.09.020 https://doi.org/10.1080/02188791.2017.1339344 https://doi.org/10.1371/journal.pone.0180102 https://doi.org/10.1287/mnsc.46.2.186.11926 https://doi.org/10.24059/olj.v21i1.761 https://doi.org/10.1080/0144929x.2014.934291 https://doi.org/10.1177/0047239515618464 https://doi.org/10.29333/iji.2019.1241a sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p285-300 285 mathematics teacher’s engagement and students’ motivation to learn mathematics mary juliet adapon doño, benjamin baguio mangila* josefina h. cerilles state college, philippines article info abstract article history: received may 23, 2021 revised july 10, 2021 accepted july 11, 2021 effective teachers are those who are highly engaged and who have an essential role in promoting student motivation and achievement. thus, this study was conducted to ascertain the engagement of mathematics teachers and its relation to the learning motivation of students in a state college in the philippines. it employed the mixed methods, specifically creswell’s (2014) sequential explanatory approach, with the survey-questionnaire, interview, and focus group discussion as data collection techniques. the findings of the study revealed that teacher’s engagement in mathematics in terms of “body language and behaviors,” “consistent focus,” and “individual attention,” were “very high” while “rigorous thinking,” “meaningfulness of work,” “verbal participation,” “clarity of teaching,” “performance orientation,” “interest and enthusiasm,” and “confidence,” were only “high.” meanwhile, students’ motivation to learn mathematics as to “relevance,” “interest,” “satisfaction,” and “confidence” were also “high”. the test of hypothesis on significant correlation showed that there was a close association between teacher’s engagement in mathematics and students’ motivation to learn mathematics. there was also a corroboration between the quantitative data obtained from the survey and the qualitative data acquired during the interview and focus group discussion. the result further implied that teacher’s high engagement contributes positively to students’ willingness to learn essential concepts and skills in mathematics. keywords: mathematics, students’ motivation, teacher’s engagement copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: benjamin baguio mangila, school of teacher education, josefina h. cerilles state college caridad, dumingag, zamboanga del sur, philippines email: benman1586@gmail.com how to cite: doño, m. j. a., & mangila, b. b. (2021). mathematics teacher’s engagement and students’ motivation to learn mathematics. infinity, 10(2), 285-300. 1. introduction several researchers have affirmed the importance of people who are engaged at work (harter et al., 2020; krueger & killham, 2005; wagner & harter, 2006) and the effectiveness of talented teachers to meaningful school outcomes, specifically student achievement (long & hoy, 2006; sanders & rivers, 1996). gordon and crabtree (2006) proclaimed a need to ensure that teachers work in an environment that promotes his or her engagement to tap students' potentials fully. teacher engagement refers to the individual teacher's involvement in and enthusiasm for teaching students in schools and reflects how well teachers are known and how often they get to do https://doi.org/10.22460/infinity.v10i2.p285-300 doño & mangila, mathematics teacher’s engagement and students’ motivation … 286 what they do best. gordon and crabtree (2006) also expressed the importance of valuing teacher talent and engagement more so than any other factor that leads to student success. he emphasized, "identifying and leveraging the underutilized talent of students and teachers... should be the first consideration in improving outcomes for students". yet, researchers still see instructional methods and attitudes today that parallel what teachers provided to their students’ grandparents (gardner, 2000). deal and peterson (2002) noted that too many reform efforts focus on steering improvement from the outside through mandates and policies, and too few look at changing schools from within. improving the working environment and increasing the potential for teacher engagement requires leaders to investigate from the unfamiliar territory: from the inside out (gordon & crabtree, 2006). coffman and gonzalez-molina (2002) concluded that great organizations look inward to move forward. the same conclusion may be appropriate for public schools and for individual school district campuses. teacher engagement is related to teachers' commitment and investment in student learning. it can be manifested through various classroom behaviors, including lesson plan development, the employment of specific teaching strategies, and student evaluations (louis & smith, 1992). marks (2000) found that teachers' engagement "centered on the work they do with students in classrooms, or as more than one participant described it, 'the teaching part of teaching' (that was) essential to their professional motivation." in turn, teachers who were not focused on their work may not have had opportunities to engage with other professionals, or they may lack support from administrators in their school or school system (kirkpatrick, 2007). the operational definition of engagement used in this study is the "interest in, enthusiasm for and investment in teaching; centered on the work (teachers) do with students in classrooms" (kirkpatrick, 2007). the role of effective teachers is fundamental in promoting student motivation and achievement. effective teachers are described as possessing those dispositions that are recurring patterns of thoughts, feelings, or behaviors that result in higher levels of performance as a teacher (hutajulu, wijaya, & hidayat, 2019; mccune & entwistle, 2011). accomplished teachers exhibit an awareness of and attention to content and students, affecting classroom achievement (long & hoy, 2006). sanders and rivers (1996) agreed that teachers were potent influence affecting academic achievement. the effects teachers have on student achievement were both cumulative and additive. darling-hammond and snyder (2000) reported that teacher engagement explained 40 to 60% of the total variance in student gains in mathematics and reading. because of the expectations of preparing students for the 21st century, the attention required of educators to master state-mandated high-stakes testing and federal no child left behind (nclb) accountability, and studies pointing to a need for effective teachers to affect student learning, local school districts are challenged with the task of selecting "competent, caring, and qualified teachers... who can help all students learn" (national council for accreditation of teacher education, 2002) and can have a "positive impact on student learning" (national council for accreditation of teacher education, 2002). theoretical frameworks like self-determination and flow theories point to causal links between teacher engagement and actions, and student engagement and actions. for example, klem and connell (2004) have examined the use of selected educational variables and psychological requisites necessary to facilitate effective engagement. connell's model of motivation (skinner & belmont, 1993; klem & connell, 2004) described how the behaviors of the teacher influenced student engagement. combinations of carefully employed educational variables have been successful in increasing student engagement. these variables include quality teacher and student interaction (kelly, 2007), high levels of student efficacy (linnenbrink & pintrich, 2003), volume 10, no 2, september 2021, pp. 285-300 287 appropriate instructional methods (johnson, 2008), higher teacher expectations (tyler & boelter, 2008), and establishing a supportive and caring classroom community (walker & greene, 2009). the study's findings conducted by shernoff et al. (2014) indicate that challenging tasks produce positive emotions, thereby creating the best opportunity for engagement. effective classrooms reflect academically intense lessons charged with relevant activities, fostering feelings of student control in their learning environment and building self-confidence in their academic ability. in these classrooms, students concentrate, experience enjoyment, and secure immediate intrinsic satisfaction, which creates a foundation of future interests (shernoff et al., 2014). introducing the factors affecting academic achievement determines the quality of the education system (alnabhan et al., 2001). motivation is an essential factor in this sense. it means that motivation is accepted as a critical element of students' academic achievements (freedman, 1997). motivation is taken as a tool that affects the creativity of students' learning styles and academic achievements (kuyper, van der werf & lubbers, 2000). as a result, it is possible to argue that if motivation is ignored, teaching will be ineffective. because motivation is so important in elementary school, cavallo, miller, and saunders (2002) stated that teachers must plan lessons with engaging activities to capture the students' attention. like other disciplines, motivation has a significant effect on mathematics lessons. moreover, since motivation guides students, it can help them predict procedure and result of activities. the willingness, need, desire, and compulsion of a student to participate in and succeed in the learning process is motivation (bomia et al., 1997). middleton and spanias (1999) viewed it as reasons individuals have for behaving in a given situation. ames (1992) stated that motivation exists as part of one's goal structures, one's beliefs about what is essential. according to skinner and belmont (1993), motivated students "select tasks at the edge of their competencies, initiate action when given the opportunity, and exert intense effort and concentration in the implementation of learning tasks; they generally exhibit positive emotions during ongoing action, such as enthusiasm, optimism, curiosity, and interest”. mathematics success has a powerful influence on motivation to achieve (middleton & spanias, 1999). also indicated by dickinson and butt (1989), students will find a task more enjoyable when they have a moderately high probability of success than one with a lower chance of success. motivation is defined as "the reasons underlying behavior" (guay et al., 2010). paraphrasing broussard and garrison (2004) defined motivation as "the attribute that moves us to do or not to do something." motivation also entails a web of interconnected beliefs, perceptions, values, interests, and behaviors. as a result, various motivational approaches can concentrate on cognitive behaviors, non-cognitive aspects, or both. academic motivation, for example, was defined by gottfried (1985) as "enjoyment of school learning characterized by a mastery orientation; curiosity; persistence; task-endogeny; and the learning of challenging, difficult, and novel tasks." meanwhile, turner (1995) defined motivation as "voluntary uses of high-level self-regulated learning strategies, such as paying attention, connection, planning, and monitoring." the students' motivation to learning mathematics concept was employed to determine students' motivation level towards mathematics based on four dimensions, namely interest, relevance, confidence, and satisfaction (burden, 2000; seifeddine, 2014). interest is the first dimension of motivation which refers to whether students' curiosity is aroused and whether that passion is maintained over time. this area depends a great deal on whether the learner's curiosity has been engaged. according to motivational studies, people tend to be more interested in 1) things they already know something about or believe in, although the unexpected and unfamiliar can be intriguing within reason, 2) real people and events doño & mangila, mathematics teacher’s engagement and students’ motivation … 288 involving humanity as opposed to abstract or hypothetical events, 3) anecdotes and other devices in which a personal, emotional element is injected into an otherwise purely intellectual or procedural material. relevance refers to the learner's perception of whether instruction meets personal needs or goals. it relies upon three motives: achievement, affiliation, and power. achievement refers to the desire to overcome obstacles, accomplish goals and tasks, and to succeed at things. affiliation is the desire to have close personal relationships with other people that are two-way while power is the ability to influence people. the term "relevance" in education refers to learning experiences that are either directly applicable to students' personal aspirations, interests, or cultural experiences (personal relevance) or are linked to real-world issues, problems, and contexts (real-world relevance) (life relevance). confidence is something that is related to the probability of success that the learner feels and how much control the learner has over that success. expectations of oneself are more self-directed and include locus of control, personal causation, and learned helplessness. locus of control is either internally oriented, whereby the person believes that individual effort brings about advantages, or externally oriented, where the person feels that consequences are not under their control. personal causation is the idea that a unique attempt will lead to positive results. learned helplessness develops when an individual who wants to and is expected to succeed finds success impossible. learned helplessness negatively correlates to effort in that as effort lags, learned helplessness generally would increase. satisfaction can come from a sense of accomplishment, praise from superiors, or simply entertainment. feedback and reinforcement are essential elements, and when learners appreciate the results, they will be motivated to learn. it also refers to intrinsic motivations and reactions to extrinsic rewards. student satisfaction is defined by wiers-jenssen, stensaker, and grogaard (2002) as students' evaluations of the services provided by universities and colleges. due to repeated interactions in the higher education environment, student satisfaction is a constantly changing construct (elliott and shin, 2002). because an institution listens to its students, it is a dynamic process that necessitates clear and effective action. student satisfaction is a complex construct influenced by a variety of student and institution characteristics (thomas & galambos, 2004). student satisfaction refers to a student's overall reaction to his or her learning experience (wiers-jenssen et al., 2002). given the preceding situation, this study ascertained the association between the engagement of a mathematics teacher and the learning motivation of high school students in the sole state college of zamboanga del sur, philippines. specifically, it determined the teacher's engagement in mathematics, the students' motivation to learn mathematics, as well as the significance of the correlation between teacher’s engagement and students’ motivation to learn mathematics. 2. method this study utilized the mixed methods of research, particularly creswell’s sequential explanatory approach, in gathering and analyzing the data on teacher’s engagement and students’ motivation to learn mathematics. creswell (2014) stated that sequential explanatory approach is described by the collection and analysis of qualitative data in order to help explain the findings of the quantitative study. in this study, the quantitativecorrelational method was firstly used and then supported by the qualitative data which were obtained through the interview and focus group discussion (fgd). a total of 41 grade 7 students and their mathematics teacher from the high school department of a state college in a philippine province were involved as participants who volume 10, no 2, september 2021, pp. 285-300 289 were determined using the purposive sampling method. before they were included, the participants were required to accomplish a written informed consent form (icf) to make their involvement/participation in the study proper and ethical. standard questionnaires were distributed by the researchers to the participants to gather relevant information about teacher’s engagement and students’ motivation to learn mathematics. the teacher engagement measurement tool by jones (2008) was used to ascertain teacher’s engagement in terms of body language and behaviors, consistent focus, verbal participation, confidence, interest and enthusiasm, individual attention, clarity of teaching, meaningfulness of work, rigorous thinking, and performance orientation. the students’ motivation to learn mathematics concept was employed to determine students’ motivation level toward mathematics as to interest, relevance, confidence, and satisfaction (burden, 2000; seifeddine, 2014). both these questionnaires used the four-point hypothetical mean range from very high, high, low, and very low. guide questions, meanwhile, were used by the researchers during the teacher’s personal interview and students’ focus group discussion. before using them in the interview and group discussion, the guide questions were scrutinized and underwent pilot testing to avoid validity and reliability problems. both interview and focus group discussion were recorded and transcribed using conventions. both the descriptive (frequency counts and percentage) as well as inferential statistics (pearson product moment correlation coefficient) were used by the researchers in analyzing the quantitative data. on the other hand, content analysis was employed in order to reveal the dominant themes present in the qualitative data. furthermore, anonymity, objectivity, and accuracy were the ethical issues ultimately considered by the researchers. 3. results and discussion 3.1. results 3.1.1. level of teacher’s engagement in mathematics table 1 shows the data which reflect teacher’s level of engagement in mathematics. based on the table 1 presented, “body language and behaviors” ranks first as it obtained the highest weighted mean of 3.47; followed by “consistent focus,” 3.33; and “individual attention,” 3.31. these indicators yield varied weighted mean but all receive the same verbal interpretation of "very high." analysis of the findings reveals that teacher’s engagement in mathematics teaching is "high" as strongly supported by the overall weighted mean of 3.23. the findings imply that the teacher highly demonstrates her enthusiasm in teaching the students to learn mathematics and her willingness to provide and be involved in different classroom activities in order to promote student learning. table 1. level of teacher’s engagement in mathematics items weighted mean interpretation 1. body language and behavior 3.47 very high 2. consistent focus 3.33 very high 3. verbal participation 3.21 high 4. confidence 3.09 high 5. interest and enthusiasm 3.12 very high 6. individual attention 3.31 high doño & mangila, mathematics teacher’s engagement and students’ motivation … 290 items weighted mean interpretation 7. clarity of teaching 3.17 high 8. meaningfulness of work 3.22 high 9. rigorous thinking 3.24 high 10. performance orientation 3.14 high over-all weighted mean 3.23 high 3.1.2. level of students’ motivation to learn mathematics 3.1.2.1.level of students’ motivation to learn mathematics in terms of interest table 2 displays the data that reveal students' motivation to learn mathematics in terms of interest. among the statements, statement 1, “i love learning mathematics” yields the highest weighted mean which is pegged at 3.10; closely followed by statement 4, “i am highly motivated to learn mathematics” 3.05; statement 3, “the hours i spend doing mathematics are the ones i enjoy most” 2.92; and statement 2, “learning mathematics is not frustrating” which earns the lowest weighted mean of 2.61. although the given statements have obtained varied weighted mean, they all receive the same verbal interpretation of "high." analysis of the results entails that students' interest to learn mathematics is "high" as strongly evidenced by the overall weighted mean of 2.92. the results further imply that students highly demonstrate their curiosity to learn, respond, and attend to the subject matter taught by their teacher in their mathematics subject. table 2. level of students’ motivation to learn mathematics in terms of interest 3.1.2.2.level of students’ motivation to learn mathematics in terms of relevance table 3 shows the data which reflect students' motivation to learn mathematics in terms of relevance. among the statements, statement 5, "mathematics is relevant to my needs and goals both in school and at home," ranks first as it has garnered the highest weighted mean of 3.26; followed by statement 4, “mathematics subject matter is related to my daily experiences” 3.18; and statements 6, "mathematics gives me opportunities for choice, responsibility and interpersonal influence" and 7, "mathematics lessons give me opportunities for cooperative social interaction" which earned the same weighted mean of 3.13. other statements yield varied weighted mean but have been interpreted as "high." analysis of the findings reveals that students’ perception about the relevance of the mathematics subject is "high" as confirmed by the overall weighted mean of 2.96. the items weighted mean interpretation 1. i love learning mathematics. 3.10 high 2. learning mathematics is not frustrating. 2.61 high 3. the hours i spend doing mathematics are the ones i enjoy most. 2.92 high 4. i am highly motivated to learn mathematics. 3.05 high over-all weighted mean 2.92 high volume 10, no 2, september 2021, pp. 285-300 291 findings also elucidate that students highly perceive the subject matter content in mathematics as very significant to them. table 3. level of students’ motivation to learn mathematics in terms of relevance 3.1.2.3.level of students’ motivation to learn mathematics in terms of confidence table 4 displays the data which show students’ motivation level to learn mathematics in terms of confidence. among the given statements, statement 4, “learning mathematics gives me opportunities for personal advancement," ranks first as it has obtained the highest weighted mean of 3.33 which is interpreted as "very high." meanwhile, statement 9, "i expect to get high scores in mathematics tests," follows next as it has yielded the weighted mean of 3.05; statements 6, “i rarely expect to perform well in mathematics-related subjects” and 7, “i expect to be able to solve mathematical problems anywhere i come across them if they are of my level of education” which both earned the same weighted mean of 2.87, interpreted as "high." other statements have earned varied weighted mean but are all interpreted as “high.” analysis of the findings discloses that students' confidence to learn mathematics is "high" as strongly supported by the overall weighted mean of 2.67. the findings further reveal that students highly manifest their locus of control, personal causation, and learned helplessness in the different tasks and activities they always do when learning mathematics. items weighted mean interpretation 1. i aspire to study mathematics in college after graduating high school. 2.87 high 2. i am not sure whether there is a need for me to continue studying mathematics. 2.69 high 3. i find activities in mathematics lessons meaningful. 3.11 high 4. mathematics subject matter is related to my daily experiences. 3.18 high 5. mathematics is relevant to my needs and goals both in school and at home. 3.26 high 6. mathematics gives me opportunities for choice, responsibility, and interpersonal influence. 3.13 high 7. mathematics lessons give me opportunities for cooperative social interaction. 3.13 high 8. i would like a career that does not require mathematics. 2.32 low over-all weighted mean 2.96 high doño & mangila, mathematics teacher’s engagement and students’ motivation … 292 table 4. level of students’ motivation to learn mathematics in terms of confidence items weighted mean interpretation 1. i find it hard to work independently on mathematical problems. 2.23 low 2. i rarely expect to be able to apply mathematics in life situations. 2.08 low 3. i rarely expect to be successful in mathematical tasks given by teachers in mathematics classrooms. 1.95 low 4. learning mathematics gives me opportunities for personal advancement. 3.33 very high 5. i practice solving mathematical problems on my own during holidays. 2.72 high 6. i rarely expect to perform well in mathematics-related subjects. 2.87 high 7. i expect to solve mathematical problems anywhere i come across them if they are of my level of education. 2.87 high 8. i can work independently in mathematics exercises in and outside mathematics classrooms. 2.76 high 9. i expect to get high scores on mathematics athematics tests. 3.05 high 10. i expect to be able to apply mathematics easily to other situations in life. 2.82 high over-all weighted mean 2.67 high 3.1.2.4.level of students’ motivation to learn mathematics in terms of satisfaction table 5 presents the data that reflects students' motivation to learn mathematics in terms of satisfaction. of the given statements, statement 1, "learning mathematics is in itself rewarding," ranks first as it has garnered the highest weighted mean of 3.05; closely followed by statement 5, “i am satisfied with the way mathematics is taught in mathematics classrooms” 3.03; and statement 6, “i am satisfied with my performance in mathematics assignments, tests, and examinations” 3.00, which have received the same corresponding verbal interpretation of "high." other statements have received varied weighted mean but are also interpreted as “high.” analysis of the results denotes that students' satisfaction to learn mathematics is "high" as strongly evidenced by the overall weighted mean of 2.81. volume 10, no 2, september 2021, pp. 285-300 293 furthermore, the results reveal that the students are often motivated to perform their assigned tasks when they know they are appreciated by their teacher and are given or reinforced by certain rewards or recognition for a job or an output well done. table 5. level of students’ motivation to learn mathematics in terms of satisfaction items weighted mean interpretation 1. learning mathematics is in itself rewarding. 3.05 high 2. i am satisfied with the way i learn mathematics. 2.97 high 3. i feel uneasy during mathematics lessons. 2.27 low 4. i am dissatisfied with my participation in classroom mathematical activities. 2.53 high 5. i am satisfied with the way mathematics is taught in mathematics classrooms. 3.03 high 6. i am satisfied with my performance in mathematics assignments, tests, and examinations. 3.00 high over-all weighted mean 2.81 high 3.1.3. summary data on students’ motivation to learn mathematics table 6 displays the summary data on students’ motivation to learn mathematics in terms of interest, relevance, confidence, and satisfaction. it can be gleaned that relevance ranks first as it has yielded the highest over-all weighted mean of 2.96; closely followed by interest, 2.92; satisfaction, 2.81; and confidence with the lowest over-all weighted mean of 2.67. it can also be noticed that the said indicators only vary on their overall weighted mean but they all received the same verbal interpretation of "high." analysis of the findings elucidates students' motivation to learn mathematics as to the following indicators is "high" as strongly supported by the overall average weighted mean of 2.84. the findings further imply that students love to learn mathematics because they consider the subject not only fun and interesting but also an essential part of their lives which helps them grow personally, become highly confident, and be successful in their personal and professional lives. table 6. summary of data on students’ motivation to learn mathematics items weighted mean interpretation 1. interest 2.92 high 2. relevance 2.96 high 3. confidence 2.67 high 4. satisfaction 2.81 high over-all weighted mean 2.84 high doño & mangila, mathematics teacher’s engagement and students’ motivation … 294 3.1.4. testing of the hypothesis table 7 reveals the significance of the correlation between teacher's engagement and students’ motivation to learn mathematics. it can be gleaned that the teacher's engagement and the students’ motivation to learn mathematics registered a pearson "r" correlation coefficient value of 0.8051 with the probability value of 0.0108, which is less than the 0.05 level of significance. therefore, there is enough evidence to accept the alternative hypothesis and establish a significant correlation. the foregoing result tells that teacher's engagement is closely associated with the student's motivation to learn mathematics. furthermore, the result implies that a high level of teacher engagement in mathematics can be a vital contributing factor for students to be highly motivated in learning the subject and be academically successful in the future. table 7. the correlation between teacher’s engagement and students’ motivation parameters findings pearson “r” value 0.8051 probability 0.0108 decision of the hypothesis accept interpretation with significant relationship 3.2. discussion the data indicated in table 1 present the level of teacher’s engagement in teaching mathematics as a subject. from the given data, it can be inferred that the teacher is highly engaged as shown by her positive body language and behavior, consistent focus, verbal participation, confidence, interest and enthusiasm, individual attention, clarity of teaching, meaningfulness of work, rigorous thinking, and performance orientation. the foregoing result is supported by the interview data indicating that the teacher often pays attention to students’ needs by speaking in vernacular and repeating questions and answers to help her students comprehend. she often minimizes class disruptions by using the principle of withit-ness (having eyes at the back), as well as prohibiting them to do unnecessary things inside the classroom while she is teaching. she likewise often asks varied questions in order to help her students understand the lessons. the teacher, however, uses several strategies to help students think rigorously, to make their work meaningful, participate verbally, make teaching clear, become oriented with their performance, become interested and enthusiastic, as well as confident. these manifestations of engagement support mark’s (2000) finding that teachers’ engagement is essential to their professional motivation when their work is focused on making their students work well in the classroom as well as make them think and feel that teaching and learning are essential to their successes in life. furthermore, teachers who are highly engaged show interest in student performance and achievement outcomes as reflected in curriculum preparation, collaboration, quality of instruction, assessment modes, and student feedback (louis & smith, 1992). the data presented in tables 2, 3, 4, and 5 indicate students’ motivation to learn mathematics in terms of interest, relevance, confidence, and satisfaction. from the data presented, it can be deduced that students highly manifest curiosity to learn mathematical concepts and skills and perceive the subject matter content in their mathematics subject as very significant. they likewise demonstrate their locus of control, personal causation, and learned helplessness in different tasks and activities, as well as satisfaction in their volume 10, no 2, september 2021, pp. 285-300 295 performance of the assigned tasks, knowing that their teacher appreciates and reinforces rewards or recognitions for the jobs and outputs they have done well. in addition, the qualitative responses of students during the focus group discussion show that although learning mathematics is interesting and fun, they sometimes find it frustrating as some concepts and skills are difficult to understand. they also see mathematics as highly relevant as it really helps them in reaching their goals and dreams in life. they likewise perceive it very useful in solving their problems as learning mathematics allows them to confront and solve problems which relate to practical situations. they also view learning mathematics as a highly rewarding activity as it affords them learning that they can use in real life. the foregoing results affirm xiang, bruene, and chen’s (2005) finding that student’s individual interest plays an essential role in the learners’ preference to engage in classroom tasks and activities. they also support willms, friesen, and milton’s (2009) observation that students want their work to be intellectually engaging and relevant to their lives. they also prove willms, friesen, and milton’s (2009) finding that working with authentic problems engages students and builds a sense of purpose to the learning experiences. furthermore, the results highlight the importance of feedback and reinforcement as essential elements, as learners become motivated to learn when they appreciate the results, or their performances and outputs are appreciated by their teachers (wiers-jenssen et al., 2002). the summary data shown in table 6 reflect students’ motivation to learn mathematics as to interest, relevance, confidence, and satisfaction. from the said data, it can be inferred that students highly demonstrate curiosity to learn mathematical concepts and skills, view mathematics as a relevant subject, possess confidence to perform authentic classroom tasks and activities, as well as believe that learning mathematics is a rewarding and fulfilling activity. these results strongly support skinner and belmont’s (1993) claim that motivated students generally exhibit positive emotions like interest, curiosity, enthusiasm, and optimism. moreover, they select authentic tasks and initiate actions when given the opportunity, as well as exert intense effort and concentration in the implementation of learning tasks and activities. the data indicated in table 7 denote the significance of the correlation between teacher’s engagement and students’ motivation to learn mathematics. from the given data, the analysis shows that teacher’s engagement contributes positively to students’ motivation in learning mathematics as a subject. it entails that the more engaged the teacher is in teaching essential concepts and skills in mathematics, the more motivated the students will be in learning the same. the foregoing result affirms darling-hammond and snyder’s (2000) finding that teacher engagement significantly improves students’ gains not only in reading but also in mathematics. moreover, it strongly supports basikin’s (2007) claim that a high level of teacher engagement is an essential ingredient for the success of schools, and is an important predictor of academic achievement. 4. conclusion the findings of the study indicate that teacher’s engagement in teaching affects students’ motivation in learn mathematics as a subject, thereby establishing a close association between the two variables under investigation. furthermore, there is a corroboration between the quantitative data obtained from the survey and the qualitative data acquired during the interview and focus group discussion. however, the results obtained from this study could not be adequately accepted as the basis not to devise an intervention program. hence, the study then recommends that an action plan can be cooperatively formulated by teachers and students, through the guidance and assistance of school doño & 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(2009). the relations between student motivational beliefs and cognitive engagement in high school. the journal of educational research, 102(6), 463-472. https://doi.org/10.3200/joer.102.6.463-472 wiers-jenssen, j., stensaker, b. r., & grogaard, j. b. (2002). student satisfaction: towards an empirical deconstruction of the concept. quality in higher education, 8(2), 183195. https://doi.org/10.1080/1353832022000004377 willms, j. d., friesen, s., & milton, p. (2009). what did you do in school today? transforming classrooms through social, academic, and intellectual engagement. toronto: canadian education association. xiang, p., bruene, a., & chen, a. (2005). research. journal of teaching in physical education, 24(2), 179-197. https://doi.org/10.1123/jtpe.24.2.179 https://doi.org/10.3200/joer.102.6.463-472 https://doi.org/10.1080/1353832022000004377 https://doi.org/10.1123/jtpe.24.2.179 doño & mangila, mathematics teacher’s engagement and students’ motivation … 300 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p311-324 311 the role of equilibration in the formation of cognitive structures in mathematics learning imam kusmaryono1*, mohamad aminudin1, kartinah2 1universitas islam sultan agung, indonesia 2universitas pgri semarang, indonesia article info abstract article history: received jul 9, 2021 revised sep 22, 2022 accepted sep 25, 2022 the purpose of this study was to investigate the critical role of equilibration in organizing and building students' cognitive structure in the practice of learning mathematics. this qualitative research was designed using a phenomenological study approach. participants in this study were 30 high school students of xi level. the main research instrument was the interview guide format and the document of cognitive test questions with the circle equation material. research data obtained were tested for the validity of the data by triangulation. method triangulation was done by comparing student work data and interview data. theory triangulation is done by comparing the descriptions of the data obtained with the relevant theoretical sources. the results showed that in the context of knowledge construction (cognitive structure), there is a strong relationship between the process of assimilation and accommodation with equilibration. the role of balance in forming cognitive structures is to guide students to organize previous knowledge (schemas) and new knowledge and create thinking structures to become more complex by considering the socio-cultural dimension (environment) in mathematics learning. further studies suggest that when selecting topics for equilibration-based activities, teachers should provide an effective learning environment to support the formation of cognitive structures to become more prosperous and complex. keywords: cognitive conflict, cognitive structure, equilibration, piaget’s theory this is an open access article under the cc by-sa license. corresponding author: imam kusmaryono, department of mathematics education, universitas islam sultan agung jl. kaligawe raya km. 4, semarang, central java 50112, indonesia. email: kusmaryono@unissula.ac.id how to cite: kusmaryono, i., aminudin, m., & kartinah, k. (2022). the role of equilibration in the formation of cognitive structures in mathematics learning. infinity, 11(2), 311-324. 1. introduction process skills and strategies in solving problems are very important for students to master in mathematics learning. students are required to be able to find solutions in solving problems. this is because solving problems requires a new and different step or strategy compared to the steps or strategies in solving routine or usual problems. students will carry https://doi.org/10.22460/infinity.v11i2.p311-324 https://creativecommons.org/licenses/by-sa/4.0/ kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 312 out a thought process to find new solutions in problem-solving. in the process of thinking, there is processing between the incoming information and the schema (cognitive structure) that exists in the human brain (aloqaili, 2012; bormanaki & khoshhal, 2017; evtyugina et al., 2020). new experiences or information that come in will be processed with adaptation through the process of assimilation or accommodation (dorko, 2019). learning mathematics is not just transferring knowledge from teachers to students. however, there are many things that must be understood by the teacher, including how students construct the knowledge that goes into their cognitive structure (amineh & asl, 2015). when the teacher has presented a very well-planned lesson, the teacher wonders why some students "get it" and others don't. then the teacher tries to ask himself some tough questions: are they (students) unmotivated? are they unfocused, inattentive, lazy? are they just "slow"? meanwhile, there are students in the same class and there are those who excel. they thrive in well-prepared lessons, and they can learn from anyone, anytime by any method. they can do this because they know how to collect, process, and generate information. they have a well-developed cognitive structure (garner, 2007). cognitive structures are the basic mental processes that a person (individual) uses to understand information (garner, 2007). cognitive structures are also called mental structures or thought patterns. cognitive structures play a major role in students' information processing abilities as they serve as a frame of reference for understanding and working with several concepts. teachers have an important role in developing students' mental representations through graphic representation content, diagram visualization, and symbolic thinking and abstraction. the greater the number of illustrations, the greater the mental representation, and the faster students' ability to process information develops (navaneedhan & kamalanabhan, 2017). in problem-solving activities, students often experience uncertainty in determining whether the solution or reason stated/given is a right or wrong solution. giving answers or reasons to a question is certainly related to the cognitive abilities of the individual. in situations of conflict that occur in connection with individual cognitive abilities, where the individual is unable to adapt his cognitive structure to the situation at hand in learning, it is said that there is a cognitive conflict within the individual (maharani & subanji, 2018). cognitive conflict is a situation of cognitive imbalance caused by a person's awareness of information that is contrary to information stored in previous cognitive structures (lee & kwon, 2001). cognitive conflict can also arise when there is a conflict of opinion or thought between an individual and other individuals in the individual's environment (maharani & subanji, 2018). in fact, cognitive conflicts are formed and are related to the cognitive structure of the individual and their environment (chang et al., 2009). there are several opinions of several experts who reveal how cognitive conflict is built: (1) cognitive imbalance, namely; an imbalance between a person's cognitive structure and the information that comes from their environment, in other words, there is an imbalance between internal structures and external inputs (piaget, 1964); (2) cognitive imbalance or metacognitive conflict, namely: a conflict between schemata where there is a conflict between the old cognitive structure and the new cognitive structure (which is being studied or faced) (chang et al., 2009; kibalchenko & eksakusto, 2020) and (3) cognitive conflict, namely the conflict between new cognitive structures (involving newly studied material) and an environment that can be explained but the explanation refers to the initial cognitive structures possessed by individuals (lee & kwon, 2001). some of the opinions put forward by experts can be concluded that cognitive conflict is a condition in which the new information it receives does not match the existing cognitive structure. piaget's term cognitive conflict is called disequilibrium (aloqaili, 2012; bormanaki & khoshhal, 2017; piaget, 1964). volume 11, no 2, september 2022, pp. 311-324 313 piaget said that a cognitive structure is a well-organized knowledge structure in the brain that always integrates with the environment through a process of adaptation, namely assimilation and accommodation (piaget, 1964). if assimilation and accommodation occur in a conflict-free manner or there is compatibility with the environment, the cognitive structure is said to be in equilibrium with its environment. however, if this does not occur, it means that in the process there is a cognitive conflict so that a person is in a state of imbalance (disequilibrium). when a person experiences a disequilibrium, he will respond to this situation and seek a new balance (equilibrium) with his environment (bormanaki & khoshhal, 2017; piaget, 1964; rutherford, 2011; simatwa, 2010). for more details, how the performance of equilibration in the formation of thinking structures through adaptation is described in figure 1. figure 1. cycles of adaptation and equilibrium (bormanaki & khoshhal, 2017) figure 1 shows that the cognitive development process is a cycle of adaptation and equilibrium according to piaget (bormanaki & khoshhal, 2017). figure 1 illustrates a cycle diagram of the adaptation and equilibration process, to illustrate the process of cognitive development. when a person learns or receives a new stimulus, a disequilibration will occur which leads to a process of adaptation (assimilation and accommodation). through this process, the scheme will develop through the process of combining, changing or forming a new scheme until equilibrium occurs. in the adaptation and equilibration cycle, new experiences are assimilated for the first time into the existing schema. if it is not in accordance with the existing scheme, the result will be a cognitive imbalance (cognitive disequilibrium). then, accommodating (adjusting) the scheme brings into a state of cognitive equilibrium until a new scheme challenges again (assimilation). this process will continue when a person learns or receives a new stimulus so that one's thinking process will be increasingly complex (mature) (bormanaki & khoshhal, 2017). in the math forum at nctm 2018 it is emphasized that in mathematics learning, students build their own understanding of mathematical concepts. teachers do not lecture, or explain much, or try to "transfer" mathematical knowledge, but to create situations that will help them build the necessary understanding. thus, students must be actively able to construct mathematical knowledge, build mathematical connection processes and develop habits of thinking about solving mathematical problems (steffe, 2002; wang, 2011). meanwhile, the role of teachers in learning is as resource persons and facilitators who provide assistance as needed (scaffolding) to facilitate the knowledge construction process developed by students (chang et al., 2009; wang, 2011). equilibration cognitive equilibrium cognitive disequilibrium accommodation assimilation kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 314 the purpose of this study is to investigate the importance of the role of equilibration in organizing and building the structure of students' thinking in mathematics learning practice. this study focuses on students' thought processes when building a thinking structure by empowering a balance in the accommodation process (assimilation and accommodation). through the results of this study, it is hoped that educators can provide an effective learning environment to support the formation of students' cognitive structures to become richer and more complex an investigation was conducted to answer the researcher's curiosity about the question: a) what is the role of equilibrium in shaping students' thinking structures when solving math problems? b) how is the role of the teacher so that students can build their thinking structure in a more complex direction? 2. method 2.1. research design this study uses a qualitative method with a phenomenological approach to explore data and find the meaning of facts and experiences experienced by objects that are the focus of research (khan, 2014). researchers conducted many interviews to build a sufficient dataset to find symptoms that emerged from the object under study (creswell & creswell, 2017). the phenomenon under study focuses on the cognitive structure of the object (students) when solving the circle equation problem. 2.2. participants participants in this study were 30 students at level xi who took mathematics classes. after they have completed the test, a minimum of three students will be selected as representatives to be interviewed. the research was conducted at a high school, semarang city, indonesia. 2.3. instruments the main instruments in this study were the interview instruction format and the cognitive test question document with the circular equation material. the interview instrument contains a list of unstructured interview questions. interview questions can be developed according to need. cognitive test instrument consists of 1 item description of the equation of the circle which is presented below. 2.4. procedure mathematics learning is carried out for 4 learning meetings. at the end of the meeting, students completed a written test with one of the things being tested was the equation of a circle problem. the results of student work are analyzed and grouped into three categories of answer quality, namely: correct and perfect answers, correct and imperfect answers, and wrong answers. then each group is sampled to be interviewed to confirm its work. problem : a circle centered on the intersection of two lines g1 and g2. the equation for the line g1 is x + 2y + 6 = 0 and the equation for the line g2 is x 5y + 13 = 0. the circle is through a line equation 2x 3y + 6 = 0. question : determine the general form of the circular equation. volume 11, no 2, september 2022, pp. 311-324 315 determination of sample data sources for interviews was carried out purposively and snowballing, namely, the data sources chosen were the people who knew best about what was being asked, and if they did not have the desired information, the number of data sources could be increased or increased. overtime (miles & huberman, 2016). the results of the interviews were analyzed as a basis for describing the cognitive structure of students in solving circular equations. the research data obtained were tested for validity by triangulation. method triangulation is done by comparing student work data and interview data. theory triangulation is done by comparing the description of the data obtained with the relevant theoretical sources. to find out whether the findings are consistent or not with the findings of the previous theory. 2.5. data analysis, and validation data analysis is done by reviewing texts or documents and in-depth interviews. the results of the interviews were analyzed as a basis for describing the cognitive structures of students when solving circular equation problems. research data obtained were tested for the validity of the data by triangulation. method triangulation was done by comparing student work data and interview data. theory triangulation is done by comparing the description of the data obtained with the relevant theoretical sources. to find out whether the findings are consistent or not with the findings of the previous theory. 3. result and discussion 3.1. result test result data from 30 respondents that were collected were grouped according to the quality of the respondent's answers, namely correct (perfect) answers, correct but imperfect answers, and wrong answers. the following shows the results of grouping the quality of respondents' answers in table 1. table 1. quality of respondents' answers number of respondents quality of respondents' answers true and perfect answers true and imperfect answers wrong answers n = 30 12 10 8 percentage 40% 33% 27% based on the results of students' responses to the problems posed by the teacher (see table 1), three students were selected as representatives to be interviewed. the three students are students (s-r1) with correct (perfect) answers, students (s-m1) with correct but imperfect answers, and students (s-f1) with wrong answers. the purpose of the interview was to find out the students' thought processes in solving problems through the process of assimilation and accommodation. to find out the students' thought processes in solving problems, the researcher analyzed the text (documents) of students' responses and conducted a more in-depth interview with the subject (s-r1; s-m1; s-f1). so that later the subject's thought process can be explained when solving math problem. the following is the result of the subject's answer (s-f1) to the circle equation problem proposed in the assignment. kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 316 figure 2. the answer from the subject (s-f1) the answer response given by the subject (s-f1) in figure 2 shows that solving the circle equation problem is the wrong answer. for this reason, the researcher (r) confirms through interviews with the subject (s-f1) as a basis for carrying out a theoretical discussion. r : what are the ideas for solving this problem? s-f1 : at first, i thought of finding the center of the circle by finding the intersection of the two lines g1 and g2. r : are you sure of the benefits of step two? s-f1 : i realize that this second step does not support the solution r : are you sure of the action in step three? s-f1 : i am confused thinking about it, maybe i took the wrong step to determine the radius of the circle r : is the solution to the circular equation already? s-f1 : i have to double-check. r : did you understand that step three only applies to the circle centered on the point (0,0)? s-f1 : i understand there have been errors in several places and i will fix these errors. based on the results of the interview, the subject (s-f1) was given the opportunity to improve his work. not long after, the subject (s-f1) succeeded in making improvements as presented in the following interview. r : at which step do you begin to reflect? s-f1 : i calculated the radius of the circle 𝑟 = | 𝑎𝑥1 + 𝑏𝑦1 + 𝑐 √𝑎2 + 𝑏2 | = −13 √13 r : how to solve the circular equation? volume 11, no 2, september 2022, pp. 311-324 317 at the beginning of the problem-solving activity, the subject (s-f1) was still confused thinking about possible strategies or steps to be carried out, as shown in the interview excerpt. the mental condition of this confusion in the subject (s.f1) is called equilibration, which is the process of adjusting balance from disequilibrium to equilibrium for the purpose of increasing one's thinking and knowledge to a more complex or mature stage (bormanaki & khoshhal, 2017; zhiqing, 2015). because this problem is considered very complex, the subject (s.f1) is guided by the researcher to complete this task. vygotsky's theory states that educators (teachers) should help students engage in complex or higherlevel thinking through structured assistance (liu & matthews, 2005). in mentoring activities with the help of sufficient scaffolding, the subject (s-f1) realizes that there are an error in determining the radius of the circle r2 = x2 + y2 (see figure 2). this error is because in the subject's thinking process imperfect assimilation occurs so that when it is continued into the accommodation process it also experiences a cognitive disequilibrium. then after the subject reflects with the help of scaffolding, the subject can replace it with the appropriate formula. at the final completion stage of the subject (s-f1) succeeded in compiling the circle equation x2 + y2 + 16x – 2y + 52 = 0 as the correct solution. this means that the cognitive structure of the subject has reached cognitive equilibrium (bormanaki & khoshhal, 2017; zhiqing, 2015). the results of the response of the subject's answer (s-m1) to the circle equation problems proposed in the assignment. figure 3. the answer from the subject (s-m1) s-f1 : i will show you the equation of the circle (x + x1) 2 + (y – y1) 2 = r2  (x + 8)2 + (y – 1)2 = 13 r : is this the final solution? s-f1 : i still have one more step to complete. x2 + y2 + 16x – 2y + 65 = 13  x2 + y2 + 16x – 2y + 52 = 0 kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 318 the following is an excerpt from the researcher (r) interview with the subject (sm1) to confirm the process of solving the circle equation problem (figure 3) as a basis for theoretical discussion. r : what knowledge did you use to solve this problem? s-m1 : i started by determining the center of the circle through the intersection of two lines; determine the radius of the circle, and construct the equations of the circle. r : (x + 8)2 + (y – 1)2 = 13 is this a solution to the circular equation? s-m1 : it seems i am not sure and interested in finding a better solution to the circular equation. r : please reflect so that you can find the right answer. s-m1 : here is the solution to the circular equation in question. x2 + y2 + 16x – 2y + 65 = 13  x2 + y2 + 16x – 2y + 52 = 0 r : where did you get this solution idea? s-m1 : i feel that i have enough knowledge to better construct the equations of the circle. in the confirmation of the interview with the subject (s-m1), it turned out that the subject was not sure of the answers given. when an individual (subject s-m1) thinks about solving a problem, the adaptation process (assimilation and accommodation) will take place during the equilibration process until a condition of conformity (cognitive equilibrium) occurs. when someone has obtained a solution but is not satisfied, unsure, or unsure of the solution, there is still a cognitive disequilibrium (bormanaki & khoshhal, 2017; dorko, 2019). conversely, when someone is satisfied with the answer, the person's thought process has reached a state of equilibrium (bormanaki & khoshhal, 2017). this condition encourages the subject (s-m1) to reflect (check again) on the answers they get. when the reflection takes place the subject interacts with the environment through scaffolding given by peers. finally, the subject succeeded in finding a solution to the circle equation. this means that after the subject interacts with the environment (peers), equilibration has guided the subject's cognitive processes to improve their cognitive structure. in terms of cognitive development, when they (s-f1 and s-m1 subjects) can find themselves the intersection of the two lines as the center of the circle, thus they can be said to have been at the level of cpd (construction of proximal development) (kusmaryono et al., 2021), and when they receive assistance in the form of scaffolding in a social environment (from teachers or peers) the subject can find the radius of the circle and the solution of the circular equation, then the subject has reached the level of potential development (vygotsky, 1987). however, if it is understood more deeply, it is possible that even though in the end the subjects (s-f1 and s-m1) arrive at the cognitive equilibrium stage, but before getting scaffolding they cannot accommodate the scheme. theoretically, this can lead to disjunctive generalizations (dorko, 2019). the results of the response of the subject's answer (s-r1) to the circle equation problems proposed in the assignment. volume 11, no 2, september 2022, pp. 311-324 319 figure 4. the answer from the subject (s-r1) snippets of the researcher (r) interview to the subject (s-r1) to confirm the process of solving the circle equation problem (see figure 4) as a basis for theoretical discussion are presented as follows. r : are you sure this answer is correct? s-r1 : i really believe it is r : have you reflected? s-r1 : i have done my reflection by checking the completion steps r : try to describe the checking steps that you did. s-r1 : step 1: determine the intersection of the lines g1 and g2 as the center of the circle step 2: determine the values of the coefficients x, y, and constants or (a, b, c). step 3: determine the radius of the circle step 4: draw up the equations of the circle r : what do you think about the radius of this circle r2 = x2 + y2 s-f1 : the radius of the circle r2 = x2 + y2 only applies to the equation of the circle centered on the point (0,0) and does not apply to the circle centered at the point (a, b) in the interview with the subject (s-r1), the teacher deliberately creates a social environment in the form of cognitive conflict, which aims to build the subject's thinking structure (s-r1) to be more complex. the forms of cognitive conflict created by the teacher can be in the form of denial, proof, reasoning, search questions, and others. overall the subject (s-r1) can assimilate and accommodate information from outside (socio-cultural environment) that comes in contextually so as to allow cognitive balance. so this condition has facilitated the cognitive equilibrium process, avoiding cognitive disequilibrium. smoothness in the process of assimilation and accommodation allows the subject (s-r1) to carry out the organization of knowledge as a whole by rearranging the internal schemes that kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 320 they already have. subjects feel challenged to answer the problems posed. the subject can explain the steps for completion precisely. subjects can provide reasons for conclusions with logical reasoning. what happens to the subject (s-r1) is in line with the modern view of rutherford (2011) which has added a new dimension to the socio-cultural realm (including environmental factors) of the accommodation and assimilation processes needed in the equilibration process for one's cognitive development. according to piaget's view, children (individuals) also change their schema according to the organization as has been done by the subject (s-r1). the organization is a person's tendency to regulate mental processes (thinking) by rearranging internal schemas and exploring the relationships and associations between schemas (piaget, 1964; zhiqing, 2015). organizational processes aim at developing interconnected cognitive systems so that they become more effective. piaget calls organization a high-level cognitive system (aloqaili, 2012; bormanaki & khoshhal, 2017; joubish & khurram, 2011; zhiqing, 2015). to end cognitive conflict, it is necessary to have scaffolding both from the teacher and from peers who do not experience cognitive conflict, in addition to scaffolding the role of metacognition can also help to end cognitive conflict. with the existence of scaffolding and metacognition, there is a cognitive equilibrium (re-equilibrium) and reconceptualization of information so that there is a new balance of what was previously contradictory (cognitive conflict). cognitive balance occurs because of the intervention (scaffolding) that is carried out deliberately by the teacher or other sources and metacognition, so that the assimilation and accommodation processes take place smoothly. based on this, it can be said that cognitive disequilibrium or cognitive conflict needs to be conditioned so that an equilibrium occurs at a level higher than the previous equilibrium. 3.2. discussion through text review analysis (document) of the responses of the subject's answers, the results of interviews, and piaget's cognitive theory, the cognitive structure of the subject (s-r1; s-m1; s-f1) when solving the circle equation problem can be described in the scheme (see figure 5). figure 5. the cognitive structures in the realm of structural and cultural volume 11, no 2, september 2022, pp. 311-324 321 referring to figure 5, it can be explained how equilibration occurs. at level 1 (low level), cognitive balance (eq) occurs, so there is no cognitive conflict even though assimilation and accommodation occur. at this level (subject to s-m1) new information is assimilated and accommodated properly, in other words, incoming information can be captured, knowledge can be understood according to the schemata that have been in the child's mind. at level 2 (middle level) there is a cognitive imbalance (dis-eq) or cognitive conflict occurs due to a lack of data so that the information that is entered is incomplete and does not match the cognitive structure (schemata) that is owned so that when the information comes in, it is imperfect assimilation. at level 2, it appears that the role of the social environment (en) is in the form of scaffolding assistance from both the teacher and from peers who are free of cognitive conflicts. at level 2 conditions (experienced by the s-f1 subject), individuals reflect re-concept (re-conceptualize) the incoming information (knowledge) so that a new equilibrium that was previously conflicting (cognitive conflict) occurs. at level 3 (high level), cognitive balance occurs because of the ability of cognitive organization (subject s-r1) which is carried out on purpose and is realized by the subject so that the assimilation and accommodation processes take place perfectly. cognitive organization ability is owned (subject s-r1) because the subject's initial knowledge is complete. cognitive disequilibrium or cognitive conflict can be overcome and conditioned by the subject by regulating mental processes properly so that equilibrium occurs at a level higher than the previous equilibrium. according to a neo-piaget follower that assimilation and accommodation models in the cognitive (structural) and cultural (behavioral) realms (rutherford, 2011), we can understand from the diagram in figure 5 that: (1) cognitive accommodation requires a process in which individual cognitive constructs change through interaction with the environment (prior knowledge, interests, and motivation) for external suitability; (2) cognitive assimilation requires a process in which individual cognitive constructs grow from interaction with the environment (get scaffolding) for the purposes of internal conformity; and (3) cognitive organization requires a process whereby individual or group cognitive constructs adapt (intelligently) to the environment by rearranging internal schemes and exploring the relationships and associations between schemes for external conformity by involving socio-cultural environments, interests, and initial knowledge. in the equilibration process, the subject (student) will take advantage of their cognitive abilities in an effort to find the truth and justify their opinion. this means that their cognitive abilities have the opportunity to be empowered, or strengthened, especially if the student is still trying. for example, students will use their understanding of mathematical concepts or experiences to make the right decisions. in situations like this, students can get clarity from their environment (chang et al., 2009), scaffolding, among others, from teachers or students who are smarter (vygotsky, 1987). in other words, during equilibration where cognitive conflicts occur, a person must respond appropriately or positively by refreshing and empowering their cognitive abilities (bormanaki & khoshhal, 2017; zhiqing, 2015). piaget's learning theory provides an explanation that students (s-f1 subjects) are involved in reconstructive generalization because they try to assimilate experiences with their schemes (dorko, 2019), adapt new knowledge (s-m1 subjects) in case of disturbances and imbalances, and modify and rearrange schemes (subject s-r1) to balance back so that the development of cognitive structures occurs (bormanaki & khoshhal, 2017; dorko, 2019; grokholskyi et al., 2020). in the context of mathematics learning, it has been shown that the construction of cognitive structures (knowledge) in the construction of cognitive structures (knowledge) has a strong relationship between the process of assimilation and accommodation and the process of equilibrium (equilibration) in the development of students' cognitive structures. so the main key to intellectual development is the kusmaryono, aminudin, & kartinah, the role of equilibration in the formation of cognitive … 322 organizational process and the adaptation of schemes that children get (kholiq, 2020). thus, piaget's theory has been shown to have an influence in organizing and constructing the thinking structure of children (students) in the practice of learning mathematics. 4. conclusion based on the findings and discussion it can be concluded that the role of equilibration in the formation of cognitive structures is to guide students to compile previously and new knowledge (schemes) and to form or enrich thinking structures so that they become more complex. in schema formation (cognitive structure), equilibration has influenced the process of assimilation and accommodation by considering the socio-cultural (environmental) dimension in mathematics learning. the most important thing about equilibration is the resolution of a cognitive conflict and the formation of cognitive equilibrium. the role of the teacher when students experience cognitive conflicts in the structure of students' thinking is to provide a conducive learning environment by providing scaffolding to students so that cognitive equilibrium can occur at a higher level. further studies suggest that when selecting topics for equilibration-based activities, teachers should provide an effective learning environment to support the formation of cognitive structures to become richer. acknowledgements we would like to thank the university of pgri semarang for being willing to collaborate with universitas islam sultan agung in the implementation of this research. references aloqaili, a. s. 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(2015). assimilation, accommodation, and equilibration: a schema-based perspective on translation as process and as product. international forum of teaching & studies, 11(1/2), 84-89. https://www.marxists.org/archive/vygotsky/collected-works.htm https://doi.org/10.4304/tpls.1.3.273-277 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p223-236 223 the lateral thinking processes in solving mathematical word problems reviewed at adversity quotient and reflective cognitive style andi saparuddin nur1*, kartono2, zaenuri2, rochmad2 1universitas musamus, indonesia 2universitas negeri semarang, indonesia article info abstract article history: received jul 17, 2021 revised jun 20, 2022 accepted sep 6, 2022 solving word problems with a thought jump shows flexibility and the ability to use alternative procedures that are important for students to master. the thought jump is a feature of the lateral thinking process and is needed to overcome the various difficulties in solving mathematical word problems. however, lateral thinking has not been widely linked with adversity quotient and reflective cognitive style. this study aimed to describe students' lateral thinking processes in solving word problems in terms of adversity quotient and reflective cognitive style. this research is a qualitative descriptive study. the subjects in this study were junior high school students in gowa regency, south sulawesi province. the research instrument used the mfft diagnostic test, arp questionnaire sheet, word problem text, and interview guidelines. the results of this study indicate that climber-reflective subjects can think laterally and use them to solve the first and second-word problems well. camper-reflective subjects can only think laterally for situations that are still within reach, while for more complicated cases, camper subjects are easily distracted and even stop solving problems. quitter subjects solve word problems very procedurally, follow rigid algorithms, and cannot work backward when faced with difficulties. keywords: adversity quotient, cognitive style, lateral thinking, reflective, word problems this is an open access article under the cc by-sa license. corresponding author: andi saparuddin nur, department of mathematics education, universitas musamus jl. kamizaun mopah lama, rimba jaya, merauke, papua 99611, indonesia. email: andisaparuddin@students.unnes.ac.id how to cite: nur, a. s., kartono, k., zaenuri, z., & rochmad, r. (2022). the lateral thinking processes in solving mathematical word problems reviewed at adversity quotient and reflective cognitive style. infinity, 11(2), 223-236. 1. introduction thinking is the abstraction and ideas concepts developed through a series of systematic procedures needed to solve problems (de bono, 1970). learning activities in the classroom require thinking activities so that students understand the ideas structure from the material taught by the teacher. students are said to have carried out the thinking process if https://doi.org/10.22460/infinity.v11i2.p223-236 https://creativecommons.org/licenses/by-sa/4.0/ nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 224 they have analyzed a certain object or material only through comparing, observing, and abstracting the object from various viewpoints (wantika, 2019). thinking activities are directed at obtaining solutions to a given problem to activate each student's unique perspective. thinking ability related to creativity is lateral thinking (hadar & tirosh, 2019; nggaba et al., 2018). de bono suggests that lateral thinking be used to develop creativity by using a flexible way from one aspect to distinct (hadar & tirosh, 2019). lateral thinking is the processing of information that provides a means to rearrange the mindset to pave the way for the new ideas development that may be needed (mustofa & hidayah, 2020; nggaba et al., 2018). problem-solving ability is a necessary tool in lateral thinking (hadar & tirosh, 2019; julita et al., 2019). someone who thinks laterally will be able to develop a problem-solving perspective from various view points to come up with various ideas (nggaba et al., 2018). problem-solving is closely related to the think creatively ability. creativity supports the emergence of various ideas to obtain solutions problems (muliawati, 2016). students who think laterally have various viewpoints in understanding the problem and do not just rely on the same conclusion. the lateral thinking advantage is that it can help students develop creativity and foster an open-minded behaviour in dealing with problems. students can develop lateral thinking skills through word problems. in general, word problems are connected to students' daily life situations along with mathematical symbols that can be used in solving problems (sarjana et al., 2020). geometry is a concept that is often found in the everyday life context and has a relationship with word problems. in addition, geometry can be used to reveal students' lateral thinking skills because it is supported by visualization and imagination (susilawati et al., 2018). the curved side space concept is a special object in geometry that requires spatial thinking skills and creativity solving problems (johnston-wilder & mason, 2005). the stages of solving word problems geiger et al. (2018), namely; understanding, compiling, simplifying, interpreting context, making assumptions, formulating, mathematizing, working mathematically, interpreting the results of mathematization, comparing, criticizing, validating, communicating if the solution is considered to be following the model, and review if the solution is considered not according to the model. furthermore, according to toshio (tasni et al., 2020), there are four stages of problemsolving development, namely; students understand problems and think about problemsolving directions, students determine appropriate, logical, and representative information to plan problem solving, students verify problems and discover new knowledge through mathematical schemes, students evaluate previous processes, reconstruct all problemsolving processes, and generalize ideas to other domains. the steps taken in solving word problems can be exchanged sequences or can even be skipped. for example, a person can interpret the context of a word problem repeatedly while validating each stage and at the same time thinking about how to communicate the solution obtained. the problem-solving strategy can be used when a person has sufficient mental capacity. the problem-solving process carried out by each student can be different from each other due to differences in thinking, problem spaces, and learning struggles. a problem can be solved not only supported by intellectual and emotional intelligence but also adversity intelligence or adversity quotient (aq) (stoltz, 1997). aq contains four main dimensions, namely the control of adverse events, responsibility for bad results, the reach of difficult situations, and resilience to adversity (yakoh et al., 2015). aq is divided into three groups, namely quitters, campers, and climbers. the quitter group is students who have low struggle and tend not to make an effort to adversity. a quitter group is a group that easily hopeless and accepts difficult situations as something to be avoided. camper groups are students who volume 11, no 2, september 2022, pp. 223-236 225 have struggle at the comfort zone level. efforts are only made if the problem can still be solved. however, if the problem is more difficult then students tend to quit and immediately final solution. the last group is the climbers who are identified as mountaineers struggle with high spirits until they are at their peak. students with the climber type will try various ways so that the problems they face can be solved. climber groups always try to find solutions to the difficulties they face and turn them into profitable opportunities. the problem-solving process can also be determined by cognitive style, namely the response that a person gives when facing a problem in the viewpoint, the time required and response accuracy, or the dominant method used to respond (nur & nurvitasari, 2017). cognitive styles are divided into four cognitive styles (haghighi et al., 2015), namely; impulsive, slow inaccurate, fast accurate, and reflective. these cognitive styles are distinguished based on the tempo and accuracy response results. the faster response is given and the results obtained are not accurate is categorized as an impulsive cognitive style. conversely, the slower response with accurate results is called a reflective cognitive style. according to sa'dijah et al. (2020), there are four aspects of reflective thinking, namely; technique, monitoring, insight, and conceptualization. students who have a reflective cognitive style tend to think deeply, calculate all possibilities, and relatively accurately solve problems. students with reflective cognitive style are not much influenced by intuition in solving problems (qolfathiriyus et al., 2019), and more critical in using argumentation in analyzing each calculation result (masfingatin & suprapto, 2020). the students orientation with reflective cognitive style lies in the accuracy of problem-solving accompanied by logical thinking processes. the level of adversity quotient and students' cognitive style are important factors that support lateral thinking processes (oliveros, 2014). strength to overcome difficulties is a necessary strategy to build creative thinking processes and better problem-solving abilities (suryapuspitarini & adhi, 2018; wahyuningtyas et al., 2020). students who have resilience in dealing with problems always have a way to find solutions and are able to think reflectively at every stage. reflective thinking is a method that can be used in finding alternative procedures. based on this description, the research question posed is how is the lateral thinking process of students in solving word problems reviewed at adversity quotient and reflective cognitive style? 2. method this research is descriptive qualitative research that explores the lateral thinking process of students in solving word problems reviewed at adversity quotient and reflective cognitive style. the subjects of this study were ninth-grade students at the state junior high schools in gowa regency, amount 30 people. the process of selecting subjects used the purposive sampling technique is based on the determination of research criteria. the researcher acted as a key instrument and was supported by the matching familiar figure test (mfft) sheet, adversity response profile (arp) questionnaire sheet, word problem text, and interview guidelines. the student's aq type is identified through the adversity response profile (arp) score. arp is constructed by using several situational statements that stimulate students' responses to these conditions (stoltz, 1997). students who get an arp≤59 score are categorized as quitter, a score of 95≤arp≤134 is categorized as a camper, and a score of 166≤arp≤200 is categorized as a climber (pradika et al., 2019). in addition, students' reflective cognitive style was identified using mfft. this test consists of 14 items, each student is asked to find a model picture that matches the question among eight other identical pictures. students are given 15 minutes to complete the questions. warli (nur & nurvitasari, nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 226 2017) made the criteria for students' cognitive style after completing the mfft test as shown in table 1. table 1. criteria for mfft results time accurate ≤7 correct answers > 7 correct answers ≤ 7 minute 30 second impulsive fast accurate >7 minute 30 second slow inaccurate reflective the selection results of research subjects based on the mfft test and the aq questionnaire were obtained as shown in table 2. table 2. the selection results of research subjects cognitive style aq type total quitter camper climber impulsive 5 1 0 6 slow inaccurate 2 1 1 4 fast accurate 0 3 2 5 reflective 2 3 3 8 total 9 8 6 23 based on the selection results of research subjects in table 2. choosen one student with a reflective cognitive style was selected with the most quitter, camper, and climber types. students' lateral thinking processes were explored using two word problem related to curved side space (subchan et al., 2018), namely; (1) a tubular reservoir with a radius of 50 cm and a height of 2 m is used to irrigate the garden. currently, the reservoir contains water as much as 3/4 of the total volume, and there is a small hole in the bottom of the reservoir that causes water to flow out at a rate of 50 cm3/sec. determine how long the water in the reservoir will run out! (2) a cone measuring 36 cm in diameter and 24 cm in height is cut horizontally at the top with a height of 8 cm. what are the surface area and volume of the remaining cone? the data analysis technique used the miles and huberman model (sukestiyarno, 2020), namely; collect data, reduce data, verify data, and draw conclusions. students' lateral thinking processes are described through diagrams that describe the stages of solving word problems according to geiger et al. (2018). the results of solving word problems and lateral thinking processes were confirmed through interviews used the think-aloud technique to explore students' thoughts when solving word problems. interview transcripts were coded in three digits, each digit separated by a “–“. the first digit indicates the source of information, namely "r" for researchers, and "sq" for quitters, "sch" for campers, and "sc" for climbers. the second digit represents lateral thinking process data on first and second word problem by using the symbol "1" or "2". the third digit represents the sequence of questions in the interview process. for example, "sq-1-01" states the interview transcript of the quitter subject on the first word problem and the first question sequence. volume 11, no 2, september 2022, pp. 223-236 227 3. result and discussion 3.1. result the subject selection process was successfully identified students with reflective cognitive style characters for each aq type. the lateral thinking process is carried out by selected each reflective subject with the quitter, camper, and climber types and has nice communication skills. quitter subject understands the problem with a thought about the known information in the questions and components being asked. quitter subject identifies known and asked information in the word problem. quitter subject understands the problems encountered related to the cylinder volume and writes down how to find the volume. quitter subject try to compose and interpret the context of the word problems by writing the formula for the cylinder volume. quitter subject connects the reservoir shape with the volume building concept. quitter subjects can use the formula for the cylinder volume and obtain a mathematical solution. quitter subject thinks about 3/4 of the reservoir volume filled with water used appropriate computations. after the water volume in the reservoir is known, quitter subject then relates the information on water leakage at a rate of 50 cm3/second with the water volume in the reservoir. quitter subject was initially confused when determining the time required for the water to run out in the reservoir. quitter subject used analogy and finally divided the water volume by the rate of water leakage in the reservoir. quitter subject interprets the results she gets and communicates them in the initial context. quitter subject does not validate the solution that has been obtained even thought there is a notation error (see figure 1). figure 1. the solution quitter subject in solving first word problem camper subjects understood the problem by compiled the information that is known and asked in the question. camper subjects identified the reservoir shape as a tube model so write down the formula for the cylinder volume. camper subjects can relate the relation of reservoir volume in mathematical form. in addition, the camper subject did the mathematization process to determine the water volume in the reservoir. camper subjects try to interpret the meaning of 3/4 reservoir volume and relate to the entire reservoir volume. camper subjects could identified the elements in the tube and see their relationship in problem-solving (see figure 2). nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 228 figure 2. the solution camper subject in solving first word problem camper subjects interpret the mathematization results as well as validate the solutions they get. the camper subject tries to relate the water volume and the leakage rate to determine time it takes for the water to run out in the reservoir. the camper subject tries to understood the context and then illustrates if every second the water leaves the reservoir as much as 50 cm3/second and used the division operation to determine time it takes for the water to reservoir run out. camper subjects work mathematically while compared the solutions him get. the camper subject tries to validate the solution by observing the accuracy of the computational process used. when the camper subject considers the solution obtained is irrelevant, the process is repeated until a convincing solution is obtained. camper subjects communicate the solutions obtained after validating each step. the camper subjects use leaps of thought to obtain logical relationships in solving word problems. camper subjects did not completely follow sequential steps because they sometimes have to work backward, or if the problem seems easy can move directly to the next step and think about its relevance to the word problem context. when faced with obstacles, camper subjects think of effective ways to found solutions and try rational solutions. furthermore, the climber subject began to understood the problem by written down each information piece and making logical connections in each statement. the climber subject understood the tube concept as an reservoir abstraction shape to analyze the water volume. at the same time, the climber subject works mathematically used the tube volume concept. the climber subject then determined water volume in the reservoir but had difficulty assumed 1/4 as a subtracting quantity from the total volume. the climber subject realized that the process he was doing did not make sense, so he started thinking about other ways. the climber subject tries to think backward by interpreting the context of 3/4 of the volume. the climber subject makes a model and works mathematically after understood that what is meant by the word problem is 3/4 part of the reservoir volume. the climber subject determines the time it takes for the water to run out in the reservoir by dividing the water volume by the water velocity. the climber subject tries to validate the solution by checking every step that has been passed. after the examination is carried out, the climber subject gains confidence in the solution obtained and tries to communicate it according to the initial context (see figure 3). volume 11, no 2, september 2022, pp. 223-236 229 figure 3. the solution climber subject in solving first word problem different aq types show different thinking processes in solving word problems even though all reflective subjects get the same solution. this shows that aq has a role in designing strategies, choosing procedures, and increasing efforts to overcome difficulties. furthermore, the second word problem was used to verify the lateral thinking process of each reflective subject with a higher adversity level. the second word problem has a more complex challenge and requires the subject to make mathematical manipulations and decompositions to simplify the problem. the quitter subject began to writen down the information that is known on the question and connects the cone shape. quitter subjects use the surface area and volume of the cone as a mathematical form to solve the problem but have difficulty connected various information on the problem. quitter subjects try to remember the relationship between the diameter of the cone that has been cut by used a similarity ratio. however, the quitter subject has difficulty found the relationship between the two concepts. many times, the quitter subject tries to understand the problem context, but is unable to found the right relationship and ends up rushing to use the formula for the cone volume. in addition, the quitter subject gave up and tried to communicate the solution obtained even though it was irrational. the quitter subject is unsure of the solution obtained, but cannot use other methods and chooses to quit (see figure 4). figure 4. the solution quitter subject in solving second word problem nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 230 camper subjects showed a more flexible thought process to solve the second word problem. the camper subject begins to think about the problem context and written down the surface area and volume as the elements being asked. however, camper subjects had difficulty in determining the diameter and volume of the cut cone. therefore, the camper subject tried to use the similarity concept to obtain the cut cone diameter. furthermore, the camper subject used this diameter to obtain the volume of each cone before and after being cut. the camper subject was could distinguish between diameters and radius so that they are not mistakenly applied to the formula. however, the camper subject made an error in applying the cone height measurement to the complete cone size. the subject camper did not verify the error has been made and continues with troubleshooting stages. after the cone volume is obtained, the camper subject tries to found the relationship between the two volumes. the camper subject had difficulty understanding the relationship between the two cone volumes he had obtained and decided shift to the next problem, which was to determine the cone surface area (see figure 5). figure 5. the solution camper subject in solving second word problem camper subjects understand well that determining the cone surface area will require the painter's line size. camper subjects determine the painter's line from each cone by using the pythagorean theorem to then be applied to the formula for the cone surface area. after obtained the cone surface area before and after being cut, the camper subject thought that the solution was similar to determining the volume. the camper subject understands that the solution has not been solved, but the process to determine the relationship of the known components is so difficult to solve. the camper subject chooses to stick with the results got him. when faced with a problem that is quite difficult to solve, the camper subject chooses to turn his attention to another problem or stop at the solution that has been obtained. camper subjects did not have alternative procedures to be sure of the solution. this condition shows that the camper subject's thought process was largely determined by the him situation. when the situation was most difficult peak, the camper subject tries to avoid it and chooses to stop. in addition, the climber subject began to understood the cone surface area and volume as the difference between the entire cone and the cut cone. the climber subject did not focus on the relationship between the cone concept, but instead thinks mathematically to found the relationship between the cut cone surface area and volume. the climber subject seemed more flexible in using mathematical procedures and identifying cone model. volume 11, no 2, september 2022, pp. 223-236 231 however, the climber subject has difficulty in determining the cut cone radius and the painter's line length. the climber subject tries to found the painter's line length with using the pythagorean theorem and draws the relationship of the radius, height, and painter's line like a right triangle. in addition, the cut cone radius was determined by the congruence concept. climber subjects used cone radius to determine the painter's line length that has been cut. the elements that have been found become an important tool for the climber subject to solve problems (see figure 6). figure 6. the solution climber subject in solving second word problem the climber subject tries to found a logical relation in determining the cut cone radius length. the climber subjects used the similarity concept to determine missing information. the climber subjects can understand the interrelationships between concepts and used him for solving word problems. after all the necessary elements have been obtained, the climber subject can apply them to the right formula. the climber subject interprets each result obtained and relates it to the context. the climber subjects always tried to validate the solutions obtained by ensuring the calculations accuracy at each stage. after gaining confidence in the solution obtained, the climber subject communicates and drew conclusions to answer the problem. 3.2. discussion the climber subject use think laterally to solve the first and second word problems well. the camper subjects can only think laterally for situations that are still within reach, while for more difficult situations. the camper subject was easily distracted and even stop solving problems. quitter subjects solve word problems very procedurally, follow rigid algorithms, and could not work backward when faced with difficulties. the differences description in the lateral thinking processes of each subject in solving word problems is shown in figure 7. nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 232 figure 7. the lateral thinking process of each subject in solving word problems the climber subject shows lateral thinking processes in solving the first and second word problems. the climber subjects can identify information on the problem, make relationships and express the conditions needed to solve problems. climber subjects can make mathematical assumptions, use relevant concepts, and develop various logical problem-solving strategies. the ability to overcome difficulties and struggle high level has a better problem-solving process because him always think of various alternative ways to get solutions. the results of this study are in line with the opinion (oliveros, 2014) which shows that the higher level of overcoming a person's difficulties, the better his mathematical problem-solving abilities. the climber subject shows an effort to obtain alternative solutions when facing problems, while the camper subject is only able to think of solutions that are still within his cognitive range and the quitter subject chooses to use more intuition and is unable to find alternative solutions when facing problems. this is in line with the opinion (malik et al., 2019) which states that quitter type students have not been able to carry out all stages of problem-solving, camper type students have been able to carry out problem-solving stages but have not been able to check the validity of each stage taken, while climber type students can carry out all stages of problem-solving properly. this finding supports the opinion (tasni et al., 2020) which states that students' difficulties in solving problems are related to the inability to collect representative data, plan effective strategies, understand mathematical concepts and evaluate problem-solving processes. based on the stages of problem-solving carried out by the three reflective subjects, the stages of identifying and analyzing problems can be understood well. this can be observed in the ability of quitter, camper, and climber reflective subjects who can write down information on questions and use it in solving problems. at the stage of understanding the problem, the reflective subject writes down information that is known and asked, presents graphs, pictures, or other mental representations to describe the problem, states the validity of the arguments expressed, and can conclude the truth of what is understood (sudia & lambertus, 2017). word problem allow students to make analogies between problems and their solution models so that them require a good understanding (maulyda et al., 2020). reflective subjects generally tend to think partially which allows the problemsolving process to be carried out laterally (qolfathiriyus et al., 2019). in addition, students who think reflectively can solve problems creatively using visual and symbolic representations (nugroho et al., 2020). this is indicated by the solutions that the subject can give to each problem. this finding is in line with the opinion sarjana et al. (2020) which states that solving word problems can be optimized by practicing verbal skills, and the ability to create models simultaneously. however, the difficulty level causes different attitudes towards problems where the quitter subject is unable to alternative think solutions and only uses intuition. the camper subject is only able to provide alternative solutions in situations that are still understandable and the climber subject tries to use alternative solutions until the volume 11, no 2, september 2022, pp. 223-236 233 problem can be solved. climber students have better problem-solving skills than a quitter and camper students (suryapuspitarini & adhi, 2018; wahyuningtyas et al., 2020). aq has a significant role in students' lateral thinking processes. the results of this study are in line with the findings hulaikah et al. (2020), which show that there is a link between the learning experience and struggle to adversity with students' problem-solving abilities. this strengthens the opinion gusau et al. (2018), that cognitive style is not the main factor in determining problem-solving ability. lateral thinking skills can be grown through learning that provides students with challenges in the form of problem-solving so as to facilitate the discovery process, social interaction, and reflective thinking for students (susilawati et al., 2018). lateral thinking skills can emerge if begin with challenging problems and allow lots of creative ideas. students who have been proficient in solving problems will be able to think laterally and show different behavior from novice students. this finding is supported harisman et al. (2020), which states that someone who is advanced will more easily recognize patterns, change models, and be able to find unique strategies in solving problems, while beginners are only able to recognize problems directly, manipulate numbers, and solve problems. unable to change the problem-solving model. this is in line with the opinion julita et al. (2019), which suggests that students are accustomed to using creative problem-solving types so that they are able to think mathematically laterally better. 4. conclusion students with reflective cognitive style have problem-solving abilities, but differences in efforts to adversity result in differences in skills at a higher stage. climber subjects are able to think laterally and find alternative ways that allow logical solutions to be obtained. camper subjects are able to show lateral thinking processes, but when faced with more difficult situations then him distract more. the quitter subject is able to solve word problems using a coherent procedure, but when facing difficulties the quitter subject tends to be in a hurry and does not show lateral thinking processes. the quitter subject chooses to stop at the solution obtained. the adversity quotient needs to get the teacher's attention in order to understand students' difficulties in solving word problems. teachers should provide opportunities for students with reflective cognitive styles to develop lateral thinking processes so as to bring up various alternative solutions in developing word problem-solving skills. lateral thinking processes are needed so that students are able to find new ideas, especially in solving word problems. references de bono, e. 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(2019). kemampuan berpikir lateral siswa smp pada pemecahan masalah geometri. in prisma, prosiding seminar nasional matematika. https://doi.org/10.33122/ijtmer.v2i4.61 https://doi.org/10.1088/1742-6596/1306/1/012016 https://doi.org/10.1088/1742-6596/1306/1/012016 https://doi.org/10.22460/infinity.v9i2.p159-172 https://doi.org/10.20414/betajtm.v13i2.390 https://doi.org/10.2991/icie-18.2018.17 https://doi.org/10.20414/betajtm.v13i1.371 nur, kartono, zaenuri, rochmad, the lateral thinking processes in solving mathematical … 236 yakoh, m., chongrukasa, d., & prinyapol, p. (2015). parenting styles and adversity quotient of youth at pattani foster home. procedia social and behavioral sciences, 205, 282286. https://doi.org/10.1016/j.sbspro.2015.09.078 https://doi.org/10.1016/j.sbspro.2015.09.078 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.233 29 analysis of students' mathematical power in terms of stifin test isna rafianti 1 , heni pujiastuti 2 1,2 mathematics education sultan ageng tirtayasa university, serang, indonesia 1 isnarafianti@untirta.ac.id, 2 henipujiastuti@untirta.ac.id received: december 18, 2016; accepted: january 2, 2017 abstract this research is motivated by the poor performance of students in particular mathematical power. one reason is that learning tends to be centered on teachers who emphasize the procedural process, mechanistic task and less provide opportunities for students to develop the ability to think mathematically. in terms of learning, each individual has their advantages and disadvantages to absorb the lessons given. but in the world of education is now known various methods in order to meet the demands of individual differences, one of which is stifin to determine the dominance of machine intelligence to make students more comfortable in the learning process that is expected to enhance the mathematical power. this research approach is qualitative descriptive that seeks to analyze the characteristics of students’ mathematical power in terms of stifin. this research was conducted in sman 2 kota serang, with research subjects are five students of eleven grade derived from each type stifin different categories according to stifin. instruments in this study are mathematical power test and interview. the results showed that mathematical power of thinking and sensing type has higher than the other type, followed by the type feeling, intuiting and the last is the type of instinct. keywords: mathematical power, stifin abstrak penelitian ini dilatarbelakangi oleh rendahnya prestasi siswa terutama dalam penguasaan matematika. salah satu alasannya adalah bahwa pembelajaran cenderung berpusat pada guru yang menekankan proses prosedural, tugas mekanistik dan kurang memberikan kesempatan bagi siswa untuk mengembangkan kemampuan berpikir matematis. dalam hal belajar, setiap individu memiliki kelebihan dan kekurangan mereka untuk menyerap pelajaran yang diberikan. namun, dalam dunia pendidikan saat ini sudah dikenal berbagai metode untuk memenuhi tuntutan perbedaan individu, salah satunya adalah stifin yaitu metode untuk menentukan dominasi kecerdasan mesin untuk membuat siswa lebih nyaman dalam proses pembelajaran sehingga diharapkan dapat meningkatkan daya matematis siswa. pendekatan penelitian ini adalah deskriptif kualitatif yang bertujuan untuk menganalisis karakteristik daya matematis siswa ditinjau dari tes stifin. penelitian ini dilakukan di sman 2 kota serang, dengan subjek penelitian adalah lima siswa dari sebelas kelas yang berasal dari masing-masing jenis stifin kategori yang berbeda sesuai dengan stifin. instrumen dalam penelitian ini adalah tes daya matematis siswa dan wawancara. hasil penelitian menunjukkan bahwa daya matematis tipe thinking dan sensing memiliki skor yang lebih tinggi daripada tipe lainnya, kemudian diikuti oleh tipe feeling, intuiting dan yang terakhir adalah tipe instinct. kata kunci: daya matematis, stifin how to cite: rafianti, i. & pujiastuti, h. (2017). analysis of students' mathematical power in terms of stifin test. infinity, 6 (1), 29-36. mailto:1isnarafianti@untirta.ac.id mailto:2henipujiastuti@untirta.ac.id rafianti & pujiastuti, analysis of students' mathematical power in terms … 30 introduction national council of teachers of mathematics (2000) stated, the purpose of learning mathematics is to develop: the ability to explore, construct a conjecture; and arrange reason logically, non-routine problem solving skills; the ability to communicate mathematically and use mathematics as a tool of communication, the ability to connect between mathematical ideas and between mathematics and other intellectual activity. abilities in the learning objectives referred to mathematical power (mathematical power) or math skills (doing math). japar (2015) stated that the results of previous studies show low achievement of students for subjects, especially mathematics students' mathematical power, namely: (1) understanding the concept 41.73; (2) the mathematical reasoning 40.79; (3) troubleshooting 17.31; (4) the mathematical connection 26.35; and (5) mathematical communication 40.32. at this time, students' mathematical power has not been fully achieved. according suwarsono (suradi, 2006: 9) difficulties experienced by students in studying mathematics is inseparable from teaching strategies that have been used in schools, the teaching strategies classical with the lecture method as the primary method. the learning which tends to be centered on teachers who emphasize the procedural process, the task of training the mechanistic and less provide opportunities for students to develop the ability to think mathematically (djohar, 2003, imstep-jica, 1999, and marpaung, 2003). in fact, the importance of developing the ability to think and the role of the teacher has long been proposed by polya that to teach you how to think, teachers not only provide information but also putting themselves according to the conditions of students, and understand what is happening in the minds of students. in line with this, in terms of learning each individual has advantages and disadvantages to absorb the lessons given. each has a more convenient way of learning and learning is strongly influenced by the tendency of the workings of the brain is dominant. this is because that each individual is unique, meaning that every individual has the difference between the one with the other. the differences are manifold, ranging from physical differences, thought patterns, and ways to respond to or learn new things. for education, the failure to know how to learn will result in a learning process that saturate, it is difficult to get maximum results in the end achievement would be decreased. in the world of education is now known various methods in order to meet the demands of the individual differences. in the developed countries and even the education system is created so that individuals can freely choose the pattern of education in accordance with his characteristics. various measures can be undertaken by the school, such as by giving a good learning media, or by providing appropriate teaching methods for students. stifin is one answer to it. stifin tests initiated by poniman (2009) relies more scientifically the psychological analytical approach pioneered by carl gustav jung, compiled with the theory of the whole brain concept of ned herrmann and theory tiune brain. stifin test is a test that is done by scanning the tenth fingertips (take no more than one minute). fingerprint carries information about the composition of the nervous system are then analyzed and linked to specific parts of the brain that acts as a dominant operating system and as well as a machine intelligence. in stifin, learned patterns of each machine intelligence modeled as follows: sensing (s) good in memorizing, thinking (t) is great at calculating, intuiting (i) champion in creativity, feeling (f) happy if discussion and instinct (inclusive) learners versatile but need the peace to optimize brain function center (instinct) or better known as stifin through the fingerprint test. volume 6, no. 1, february 2017 pp 29-36 31 it is important for students to know that there is a potential or force on them in optimizing capabilities. so by knowing the dominant intelligence engine, students are more comfortable in the learning process. they can customize the learning patterns that they have learned through the results of these tests. students who know the types of learning patterns they will adjust to learning in the classroom in order to be successful in learning and help students to become effective problem solver. pattern or way of learning itself is one of the factors that affect how students learn mathematics. in addition, teachers will be more tolerate and put the maximum attention to the plurality of machine intelligence of each student. this then becomes very important for teachers to analyze and determine the pattern of student learning that led to a lack of students' mathematical power. because the type of machine intelligence stifin different causes in different ways of learning so that the mathematical power is also different. by directing students by machine intelligence, mathematical power is expected that students can be better. in addition, teachers can also find out students' mathematical power is lacking if each student has an intelligence engine stifin different so as to help teachers to meet the demands of a difference in the classroom and be able to carry out meaningful learning. based on the above, the question in this research is is "how can students' mathematical power if the terms of the test stifin?". so the purpose of this study is to analyze the students' mathematical power in terms of test stifin (sensing (s), thinking (t), intuiting (i), feeling (f), and instinct (in). method this research has the characteristics of qualitative research that the researcher as a lead instrument, using qualitative methods, has a natural setting, descriptive, inductive data analysis, and more concerned with process than results. therefore, the approach of this study is a qualitative approach. as the definition of bogdan and taylor in moleong (2000: 3), a qualitative approach is a research procedure that produces data in the form of words written or spoken of people and behaviors that can be observed. the characteristics of qualitative research by moleong (2000: 121) are: (1) has the natural background, (2) researchers as the main instrument, (3) using qualitative methods, (4) inductive data analysis, (5) the theory of basic , (6) is descriptive, (7) is more concerned with process than results, (8) the limits specified by the focus, (9) the specific criteria for the validity of the data, (10) the design of which is temporary, and (11) the results of research negotiated and agreed. the research was conducted at sman 2 serang, the reason why the researchers chose sman 2 serang, because in this study one of the variables discussed is the test stifin, sman 2 serang seemed appropriate, because it is one of the schools that do test stifin to students, for that researchers no longer need to perform tests stifin research on the subject because it is less effective in terms of cost given for tests stifin requires no small cost. when the study was conducted in the first semester of the academic year 2016/2017. to determine the sample in this study the researchers used a sampling technique by purposive sampling were selected based on the goal to be achieved is to know the students' mathematical power in terms of test stifin. subjects in this study were selected by considering the teacher's explanation about the type of dominant stifin owned by the students. the research subjects selected were three eleventh-grade students representing every type of stifin so that in total there were 15 students, then after being given a power of rafianti & pujiastuti, analysis of students' mathematical power in terms … 32 mathematical tests, chosen by each of the students who test scores of each type stifin dominant. aspects of the mathematical ability of students include; mathematical problem solving ability by using indicators identify the elements that are known, asked, and the adequacy of the required elements; mathematical communication skills by using indicators explaining ideas, situations, and mathematical relationships in writing; mathematical connection capabilities with the indicators using mathematical mathematics in other areas of study or daily life (sumarmo, 2010). then to aspects of mathematical reasoning ability and mathematical representation of each used indicator of ratnaningsih (2008), namely checking the validity of the argument and choose, apply and; changing representation to solve the problem. to be able to give an objective assessment, the criteria scoring answers to questions test students' mathematical power ability by using the guidelines on holistic scoring rubrics proposed by cai, lane & jakabcsin (1996) which was adapted. table 1: criteria of student’s answer score score students response 0 1 2 3 4 no answer / reply does not correspond to the question / no correct only some aspects of the statement is answered correctly answer incomplete (partial instructions followed) but contains incorrect calculations almost all aspects of the question is answered correctly all aspects of the questions were answered with complete / clear and true results and discussion results mathematical power test given to students in grade eleven sman 2 serang is a quadrilateral material, consisting of five questions representing each indicator of the ability of problem solving, reasoning, communication, connections and representation. the research subject was taken as a student selected by teachers to consider the dominant stifin intelligence engine owned. then, after being given the power of mathematical tests, chosen by each of the students who score most dominant mathematical power. based on the test results of mathematical power capabilities, the following list of names and scores of research subjects test results mathematical ability of students. table 2: score of mathematical power ability test results no subjek stifin score 1. dns thinking 16 2. ra sensing 16 3. mdp feeling 11 4. ln intuiting 9 5. sns insting 8 volume 6, no. 1, february 2017 pp 29-36 33 based on table 2, it can be seen that subjects with type thinking and sensing has the highest score is 16. subsequently followed by subjects with type feeling with a score of 11, the type intuiting with a score of 9 and type of instinct with a score of 8. the score for each ability in mathematical power capability: problem solving ability (ps), reasoning (r), communication (cm), connections (cn), representation (rp) may be seen in the following table. tabel 3: score of mathematical power ability per item no subject stifin mathematical power ability score total ps r cm cn rp 1 dns thinking 2 4 4 4 2 16 2 ra sensing 2 4 4 4 2 16 3 mdp feeling 1 3 2 4 1 11 4 ln intuiting 2 2 2 2 1 9 5 sns insting 1 2 2 3 0 8 based on table 3. it can be seen that students with thinking and sensing types have the same score and the highest. both do the calculations correctly and the steps are clear. it's just the matter of solving problems and questions about representation, the answer is incomplete (partial instructions followed) but contains incorrect calculations. the observation of the researcher at the time of the test, that students with types thinking and sensing very serious in doing the given problem, and to maximize the allotted time. so that both students are students who most recently collecting test questions and answers. when interviewed about the answers that are not answered correctly, the student with the type sensing replied that he forgot the formula, thus making its calculations wrong. while type thinking replied that she remembered with the formula, but she did not understand about the comparison contained in the problem solving and representation task. discussion it was mentioned earlier that the mathematical aspects of the capability of the students in this study include the ability of mathematical problem solving, mathematical reasoning, mathematical communication, mathematical connection and mathematical representation. the items used in this study are as follows: items 1. (mathematical problem solving; identify the elements that are known, asked, and the adequacy of the required elements) “ibu mempunyai selembar kain berbentuk persegi panjang dengan keliling 100m. perbandingan ukuran panjang dan lebar kain tersebut adalah 3 : 2. a. unsur apa saja yang diperlukan untuk mencari luas kain ibu? b. bagaimana cara menentukan luas kain ibu?” items 2. (mathematical reasoning; checking the validity of the argument) “jika sebuah jajargenjang memiliki luas 126 cm² dan memiliki tinggi 9 cm. benarkah jajar genjang tersebut memiliki alas 14 cm? buktikanlah jawabanmu.” rafianti & pujiastuti, analysis of students' mathematical power in terms … 34 items 3. (mathematical communication; explaining ideas, situations, and mathematical relationships in writing) “luas kebun berbentuk persegi panjang sama dengan luas kebun berbentuk persegi yang panjang sisinya 8m. jika lebar kebun yang berbentuk persegi panjang adalah 4 m. a. nyatakanlah dalam bentuk gambar b. bagaimanakah cara menentukan panjang kebun yang berbentuk persegi panjang tersebut? items 4. (mathematical connection; using mathematical mathematics in other areas of study or daily life) “seorang siswa memiliki buku gambar berukuran 0,4m x 0,6m. buku gambar itu akan digunakan untuk menggambar bingkai foto berukuran 5cm x 5cm. bingkai foto tersebut tidak boleh saling berpotongan. a. bentuk bangun apakah buku gambar yang dimiliki oleh siswa? b. berbentuk bangun apakah bingkai foto yang akan digambar oleh siswa? c. berapa banyak bingkai foto yang akan digambar oleh siswa pada buku gambar?” items 5. (mathematical representation; changing representation to solve the problem) “bagian atas ∆𝐴𝐵𝐶 dilipat ke arah atas pada bagian de seperti tampak pada gambar di atas. 𝐴𝐵 dan 𝐷𝐸 sejajar dengan panjang berturut – turut 10 𝑐𝑚 dan 8 𝑐𝑚. tinggi ∆𝐴𝐵𝐶 adalah 15 𝑐𝑚 perbandingan ukuran panjang 𝐹𝐺: 𝐴𝐵 = 3 : 5 dan perbandingan tinggi 𝑡𝐴𝐵𝐶:𝑡𝐷𝐴𝐹 = 4:1. bagaimanakah cara menentukan luas daerah yang diarsir?” based on research results, the student with the type feeling, has a score under the type of thinking and sensing. most of the students' answers feeling types are only some aspects were answered correctly that in a matter of problem-solving abilities and representation, while the answer to the problem of communication incomplete (partial instructions followed) but contains incorrect calculations. the highest score is on a matter of connections, the answer is right but in a different way, he described the sketch book and a picture frame with a small scale so that it can find the number of picture frames that will be in the image of students in accordance with which he describes. when interviewed about the answer to the question, the student types feeling said that the question of a matter can he understood, but he had difficulties in answering especially what steps should be done first, and he said most forgotten by the formula. then to students intuting type and instinct to score each of 9 and 8. almost all the answers are less complete (some instructions followed) but contains incorrect calculations. most of the responses of the type intuiting and instinct is the answer to the point, so there is no systematic steps, although some of them the correct answers. when interviewed about the answer, the student with the type intuiting replied that in general of all the questions he knew what was being asked, just confused answer because it already forget the material and formula, so some volume 6, no. 1, february 2017 pp 29-36 35 answers in the answer by guessing only. as for the students' instinct, when interviewed about the results of the answer is approximately the same as the type intuiting answer, answered with a guess as to forget the formula and how. but for students with the type of instinct, the answer is almost every answer sheet made image first and then answer directly without using the formula. furthermore, researchers also interviewed teachers guidance and counseling (bk) in the school about the test results were reviewed based on the mathematical power stifin types of students. based on information from the teacher bk, that's the type of sensing and thinking is consistent with any problems or issues related to the calculation, since both types of work by using the left brain. this is consistent with the theory of explanation poniman (2012) who mapped the cerebral hemispheres, where each part of the brain has a special function that no other part. for the type of thinking located in the left brain, and sensing types are on the left limbic. similarly, research conducted by lutfiananda & rosyidi (2014) states that the type sensing and thinking more detail and sequentially in performing the troubleshooting steps and more planned in answering questions and its reasons are logical as for the type of feeling more students to use their feelings, according to bk teacher who is a trainer stifin, students with feeling types are actually able to solve the problem but they are often affected by mood or feelings, so that test scores can change. then for intuiting type is the type of people who are creative, but sometimes creativity can come out of academia, so it's natural that mathematics scores obtained are smaller than the type in the left hemisphere. in accordance with the explanation poniman (2012) that for the type of intuiting are on the right brain and feeling types are on the right limbic. then for the last type is instinct which is at the midbrain. according to the results of interviews with teachers bk, the type of instinct is the type of person who versatile, in fact he was able to solve various problems, but the type of instinct can be said to be a follower, it will be a positive person if it follows a positive person, and that being negative if followed negative person. in accordance with the explanation poniman (2012) that for the type of instinct to be in the midbrain. for students with the type of instinct selected as research subjects seem to be affected by the surrounding environment is negative, so that the results of tests done not in accordance with the expected potential of the type he had. conclusion from the research and discussion above, it can be concluded that students with stifin types that are in the left hemisphere (sensing and thinking) higher than the power of mathematical stifin types that are in the right hemisphere (intuiting and feeling). as for the students' mathematical power capability with stifin types that are in the midbrain (instinct) were at the bottom among the other types. stifin can serve as guidelines for teachers to understand the behavior of the students that although essentially as an individual, but have differences in ability, personality, and experience the environment (rosita, 2013). teachers are also expected to determine the learning method in accordance with the type of the students stifin. in addition, students who already know each machine intelligence can further optimize their ability and receive more focus in learning, especially mathematics. rafianti & pujiastuti, analysis of students' mathematical power in terms … 36 references cai, j., lane, s., & jakabcsin, m. s. 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(2006). interaksi siswa smp dalam pembelajaran matematika secara kooperatif. disertasi program pascasarjana universitas negeri surabaya. surabaya: tidak diterbitkan. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p95-110 95 mathematical literacy’s vocational students based on logical and numerical reasoning lathifatun ni’mah 1 , iwan junaedi 2 , scolastika mariani 3 1 smk nu lasem, jl. sunan bonang km 01 lasem rembang, jawa tengah, indonesia 2,3 semarang state university, jl. sekaran, gunung pati, kota semarang, jawa tengah, indonesia 1 nlathifatun@yahoo.co.id received: march 03, 2017 ; accepted: may 01, 2017 abstract the research aims is (1) to obtain learning quality of ctl model to students mathematical literacy, (2) to obtain mathematical literacy description based on logical reasoning, (3) to obtain mathematical literacy description based on numerical reasoning and (4) to obtain mathematical literacy description based on logical and numerical reasoning. the research type is descriptive study. the subject is xi ak smk nu lasem were taken 6 students high, medium and low logical reasoning, 6 students high, medium, and low numerical reasoning, 6 students high, medium, and low logical and numerical reasoning, 2 students high logical and medium numerical reasoning, 2 students medium logical and high numerical reasoning. the research result is (1) ctl models learning quality for mathematical literacy is good, (2) student mathematical literacy based on high logical reasoning level 4 and 5, medium level 3, low level 1 and 2, (3) student mathematical literacy based on high numerical reasoning level 5, medium level 4, low level 2 and 3, (4) student mathematical literacy based on high logical and numerical reasoning level 5, medium level 3 and 4, high logical and medium numerical reasoning or medium logical and high numerical reasoning level 4 and 5, low level 1 and 2. keywords: ctl, logical reasoning, mathematical literacy ability, numerical reasoning. abstrak penelitian bertujuan untuk (1) memperoleh gambaran kualitas pembelajaran model ctl terhadap kemampuan literasi matematika, (2) memperoleh gambaran literasi matematika ditinjau dari penalaran logis, (3) memperoleh gambaran literasi matematika ditinjau dari penalaran numerik dan (4) memperoleh gambaran literasi matematika ditinjau dari penalaran logis dan numerik. jenis penelitian ini adalah penelitian deskriptif. subjek penelitian adalah siswa xi ak smk nu lasem diambil 6 orang kategori penalaran logis tinggi, sedang, dan rendah, 6 orang penalaran numerik tinggi, sedang, dan rendah. 6 orang penalaran logis dan numerik rendah, sedang, tinggi, 2 orang penalaran logis tinggi dan numerik sedang, serta 2 orang penalaran logis sedang dan numerik tinggi. hasil penelitian menunjukkan bahwa (1) kualitas pembelajaran model ctl terhadap kemampuan literasi matematika berkategori baik, (2) literasi matematika siswa penalaran logis tinggi mencapai level 4 dan 5, sedang level 3, dan rendah level 1 dan 2, (3) literasi matematika siswa penalaran numerik tinggi pada level 5, sedang level 4, serta rendah level 2 dan 3, (4) literasi matematika siswa penalaran logis dan numerik tinggi level 5, sedang level 3 dan 4, logis tinggi dan numerik sedang maupun logis sedang dan numerik tinggi level 4 dan 5, serta rendah level 1 dan 2. kata kunci: ctl, kemampuan literasi matematika, penalaran logis, penalaran numerik. how to cite: ni’mah, l., junaedi, i. & mariani, s. (2017). mathematical literacy’s vocational students based on logical and numerical reasoning. infinity, 6 (2), 95-110. doi:10.22460/infinity.v6i2.p95-110 mailto:nlathifatun@yahoo.co.id ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 96 introduction quality of education in indonesia is currently low. this is consistent with the results of research rusmining, waluya, & lewis (2014) that indonesia has a low education quality based on the results of the acquisition of indonesian students at the international ratings (international assessment). one of the factors indonesian students score low are indonesian students trained solve the problems that substance pisa and timss contextual, demanding reasoning, argumentation and creativity to solve them poorly (wardhani and rumiati, 2011). mathematics and mathematical reasoning are two things that can not be separated (ministry of education in shadiq, 2004). math is believed can increase the power of reason (kariadinata, 2012). human reasoning is also needed when troubleshooting problems or when the decision making process (the us president thomas jefferson in shadiq, 2004). reasoning is one indicator of literacy (turner, 2011). "problem solving, reasoning, and numeracy is one of the eyfs areas of learning and development" (adonis, 2006). studying mathematics literacy is one of the prerequisites for someone to be successful in the 21st century (murnane, sawhill & snow, 2012). some activities that can encourage literacy math is (1) reasoning mathematically and mathematical concepts, (2) recognizing the role that mathematics plays in the world, (3) making well-founded judgments and decisions, (4) solving problems set in the pupil's life world context (sandstorm, nilsson & lilja, 2013). "four interrelated thinking processes items, namely problem solving, representating, manipulating and reasoning underpin mathematical literacy" (pugalee in diezmann, watters & english, 2001). this means that the reasoning underlying the mathematical literacy skills. the results of the diezmann, watters & english (2001) also mentions that four thought process on equal influence on mathematics literacy. bokar (2013) argues that today's students should be given the problems associated with the real world to prepare students to be able to resolve the issue properly in accordance with logical and mathematical reasoning. durrant-law (2013) argued that logic is the philosophical study of valid reasoning. by the time students complete the real problem is given, there is a process that includes phases employing mathematical concepts, facts, prosedures, and reasoning (stacey, 2012). according to venkat, graven, lampen & nalube (2009), two factors suggested as the central development of the literacy skills are mathematical reasoning (reasoning) and problem solving (problem solving). for more venkat, graven, lampen & nalube (2009) also stated that the needed in the reasoning is numerical and spatial thinking. therefore, the point of logical and numerical reasoning is needed to be studied. there are several learning models are suitable for reasoning. based on shadiq (2004) learning theory that fits with reasoning are rme, pbl, and ctl. ctl subject matter associated with real life / simulation (mulyatiningsih, 2010). ctl characteristics include, among others relating, experiencing, applying, cooperating, and transferring (cor in kasihani, 2002). volume 6, no. 2, september 2017 pp 95-110 97 based on background that has been described, there are several point of this research: (1) how is the quality of the ctl model learning mathematics literacy class xi student of smk? (2) how does the literacy skills math class xi student of smk based on logical reasoning? (3) how does the literacy skills math class xi student of smk based on numerical reasoning? and (4) how does the literacy skills math class xi student of smk based on logical and numerical reasoning? method this type of research is qualitative descriptive study. the research was conducted at smk nu lasem ak xi classes in the second semester of the academic year 2014/2015. results of tests of mathematical literacy skills of students were analyzed and deepened by interviewing the subject of research as triangulation. for mathematical literacy test using adoption mathematical literacy test from south africa 2014. in this study, reasoning (reasoning) is the activity of thinking to draw conclusions or make new statements and was based on some statements whose truth has been proven or assumed before (shadiq, 2004). definition of logical reasoning in this study is an activity for the reasons, decide to accept and reject information, and explain the idea (dowden, 2011). the principles of logical reasoning has been put forward by dowden (2011) are (1) to find out the underlying reasons before accepting a conclusion, (2) provide arguments supporting the conclusion, (3) revealed the reasons underlying the decision making, (4) design reasons which implies the conclusion, (5) introduce the importance of relevant information, (6) the pros and cons, (7) to consider possible actions, (8) to see the consequences of various actions which do, (9) evaluate the consequences, (10) consider if the consequences actually occurred, (11) delaying decision-making in the state of practice, (12) to assess what was said in the actual situation, (13) to avoid judging someone literally, (14) using background knowledge and sense healthy to draw conclusions, (15) given that remarkable statement requires extraordinary evidence, (16) put off asking an expert, (17) given that firm conclusions require strong reasons, (18) consistent reasoning oneself, (19) looking for inconsistencies reasoning oneself and others, (20) check some explanation that fits all the facts, (21) to make explanations others less so calculated by showing alternative explanations are taken into account, (22) reasoning adapted to the subject, and (23) draw conclusions if've got enough evidence. logical reasoning ability is measured using kenexa logical reasoning test (lrt). measuring numerical reasoning skills students used numerical reasoning by paul newton and helen bristoll. reasoning can be classified into 3 groups: (1) a group of high reasoning: the value ≥ , (2) a group of medium reasoning , (3) groups of low reasoning the (suherman and sukjaya in riyanto & siroj, 2011). logical reasoning can be classified into 3 groups: (1) a group of logical reasoning high the value sdx 1 , (2) groups of logical medium reasoning sdxsdx 1value1  :, (3) a group of low logical reasoning the value sdx 1 : as well as numerical reasoning can ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 98 be classified into 3 groups: (1) a group of high numerical reasoning the value sdx 1 , (2) groups of medium numerical reasoning sdxsdx 1value1  , (3) groups of low numerical reasoning the value sdx 1 . according ojose (2011) mathematical literacy is the ability of students to be able to understand and apply some math applications such as facts, principles, operations, and problem solving in everyday life in the past and also the present. mathematical literacy skills in this study is the individual's ability to formulate, employ, and interpret mathematics in various contexts (oecd, 2013). some aspects related to mathematical literacy based on the oecd (2013) are as follows. (1) the mathematical processes that describe what individuals do to connect the context of the problem with mathematics and thus solve the problem, and the capabilities that underlie those processes. (2) the mathematical content that is targeted for use in the assessment items. (3) the context in which the assessment items are located. table 1. proportion score sub-sub process components tested in the pisa study (oecd, 2013) component ability tested score (%) process formulating situations mathematically 25 employing mathematical concept, facts, pocedures and reasoning 50 interpreting, applying, and evaluating mathematical outcomes 25 there are fundamental mathematical capabilities that include in mathematical processes. the fundamental mathematical capabilities are communicating, mathematising, representation, reasoning and argument, devising strategies for solving problems, using symbolic, formal and technical language and operations, and using mathematical tools. fomulating include it, employing and interpreting too. table 2. proportion score sub-sub process content components tested in the pisa study (oecd, 2013) component ability tested score (%) content space and shape 25 change and relationship 25 quantity 25 uncertainty and data 25 volume 6, no. 2, september 2017 pp 95-110 99 table 3. proportion score sub-sub process context components tested in the pisa study (oecd, 2013) component context understanding score (%) context personal 25 occupational 25 societal 25 scientific 25 table 4. mathematics literacy level based on pisa (oecd, 2013) level student activity level 6 (≥669,3) students can conceptualise, generalise and utilise information based on their investigations and modelling of complex poblem situations. they can link different information sources and representations and flexibility translate among them. students capable of advanced mathematics thinking and reasoning. these students can apply their insight and understandings along with a mastery of symbolic and formal mathematical operations and relationship to develop new appoaches and strategies for attacking novel situations. students at this level can formulate an precisely communicate their actions and reflections regarding their findings, interpretations, arguments and the appropriateness of these to the original situations. level 5 (≥607,0) students can develop and wok with models for complex situation, identifying constraints and specifying assumptions. they can select, compare and evaluate appopriate poblemsolving strategies for dealing with complex problems related to these models. students at this level can work strategically using broad, well-developed thinking and reasoning skills, appropriate linked representations, symbolic and fomal characterisations and insight pertaining to these situations. they can reflect on their actions and formulate and communicate their intepretations and reasoning. level 4 (≥544,7) students can work effectively with explicit models for complex concrete situations that may involve constraints or call for making assumptions. they can select and intergrate different representations, including symbolic, linking them directly to aspects of real wold situations. students at this level can utilities well-developed skills and reason flexibly, with some insight, in these context. they can construct and communicate explanations and arguments based on their interpretations, arguments and actions. ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 100 level student activity level 3 (≥482,7) students can execute clearly described pocedures, including those that require sequential decisions. they can select and apply simple problem-solving strategies. students at this level can interpret and use representations based on different infomation sources and reason directly from them. they can develop short communications when reporting their intepretations, results and reasoning. level 2 (≥420,1) students can interpret and recognise situations in context that require no more than direct inference. they can extract relevant information from a single source and make use of a single expresentational mode. students at this level can employ basic logaithms, formulae, pocedures, or conventions. they are capable of direct reasoning and making literal interpretations of the results. level 1 (≥357,8) students can answer questions involving familiar context where all relevant infomation is present and the questions are clearly defined. they are able to identify information and to carry out routine procedures according to direct instructions in explicit situations. they can perform actions that are obvious and follow immediately from the given stimuli. determination of the subjects in this study based on the result of logical and numerical reasoning tests using kenexa test and numerical reasoning tests from paul newton and helen bristoll. this study took two students from each of the test results of logical and numerical reasoning. students as research subjects are 2 students with low logical reasoning scores, 2 students with medium logical reasoning scores, 2 students with high logical reasoning scores, 2 students with low numerical reasoning scores, 2 students with medium numerical reasoning scores, 2 students with high numerical reasoning scores, and if there are students have same level of logical and numerical reasoning will be examined too. 2 students with low logical and numerical reasoning scores, 2 students with medium logical and numerical reasoning scores, 2 students with high logical and medium numerical reasoning scores, 2 students with medium logical and high numerical reasoning scores, as well as 2 students with high logical reasoning and numerical scores. results and discussion some domains from charlotte danielson's framework for measuring the success of learning, those are (1) planning and preparation, (2) classroom environment, (3) instruction, dan (4) professional responsibilities (macgregor, 2007). krause, dias, & schedler (2015) said that there are 7 aspect fo measuring leaning quality (1) competencies and learning activities, (2) assessment and evaluation, (3) learning sources, (4) technology aand navigation, (5) learner support, (6) accessibility, dan (7) policy compliance. from them, the aspects are assessed on the quality of learning include preparation, process, and evaluation. during the preparation stage includes three dimensions: the device (syllabus, lesson plans, worksheets, lts, volume 6, no. 2, september 2017 pp 95-110 101 supplement teaching materials, materials (uncertainty), and assessment (assessment tools). stage of the process also consists of three dimensions: competencies and learning activities, learning resources, using technology (related to learning media). evaluation stage consists of tests of mathematical literacy (tklm). the preparation phase on the quality of learning consulted to the lecturers through several revisions. learning device that be prepared covering are syllabus, lesson plans, worksheets, lts, supplement teaching materials, and assessment tools. table 5. result preparation learning tools no. aspect criteria 1. syllabus good and can be used 2. lesson plan good and can be used 3. worksheet good and can be used 4. lts good and can be used 5. supplement teaching materials good and can be used 6. assesment tools good and can be used at the process stage including the competence and learning activities, learning resources, using technology (related to learning media) that used to help math teacher smk nu lasem. there are three teachers who observed eny handayani, s.pd., sunawan, s.pd., and sri winarti, s.pd. the results of the observations made by three teachers mentioned that the learning process with ctl model categorized good (average score of quationare is 3). in accordance with the results of melville & yaxley (2009) which states that one of the model that can make pofessional learning is contextual learning. the following are observations about the learning process. table 6. result process learning no. observer score (criteria) 1. observer 1 3,32 (good) 2. observer 2 3,59 (good) 3. observer 3 3,41 (good) based on the results of research conducted by suyono (2009) indicating that literacy based on effective and productive learning can improve the quality of learning and graduate school. tklm results indicate that there are 2 students who achieve level 1, 2 students at level 2, 5 students level 3, 7 students at level 4, 2 students have reached level 5, and no student who reaches level 6. table 7. list of subjects research by category logical reasoning logical reasoning category student (tklm score) high e1 (639,8), e21 (579,3) medium e15 (523,9), e16 (518,9) low e3 (370,3), e8 (435,8) ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 102 the results from the study showed that the literacy skills of mathematics students with high category logical reasoning that the subject e1 (639,8) and e21 (579,3) which have respectively reached level 5 and level 4. this is consistent with the results of research bokar (2013) indicating that the student should be given the problems associated with the real world to prepare students to be able to solve the problem well in accordance with the logical and numerical reasoning. subject e1 and e21 able to do the formulating problem with basic capabilities of communicating, representation, and devising strategies for problem solving. in general, e1 and e21 can do the employing problem with basic capabilities of mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. for the process of interpreting the subject e1 and e21 has been completed perfectly with basic capabilities of communicating, representation, and devising strategies for problem solving. on the subject matter 5.2.2.d e1 and e21 have been able to solve employing problems with basic capabilities category mathematising. this suitable with interview result that both of them understand and can solve the problem: 5.2.2 pak rudi memberi tugas matematika pada siswanya untuk melakukan survei banyaknya uang saku siswa laki-laki dan perempuan kelas xi ak yang dihabiskan selama istirahat makan siang di sekolah pada hari tertentu. banyaknya uang saku siswa laki-laki yang telah di survei 9.000 10.000 10.000 12.000 12.000 12.000 12.000 12.000 14.000 15.000 15.000 16.000 18.000 20.000 25.000 banyaknya uang saku siswa perempuan yang telah di survei 0 6.000 6.000 9.000 9.000 10.000 10.000 10.000 11.000 11.000 11.000 11.000 12.000 20.000 25.000 30.000 5.2.2.d hitunglah median uang yang dihabiskan oleh siswa perempuan e1: e22: figure 1. sample results subject e1 and e21 for employing problem with category mathematizing number 5.2.2.d subject e15 (523,9, level 3) and e16 (518,9, level 3) is already able to do formulating problems with basic of capabilities communicating, representation, and devising strategies for problem solving. in the process of the subject employing e15 and e16 have been able to solve problems with basic capabilities of communicating, representation, devising strategies for solving problems, and using mathematics tools. if viewed from the process of interpreting the subject of e15 and e16 already completed perfectly communicating problems with basic capabilities of representation and reasoning and argument. in interview both of them said that they can not operate mathematical model in 3.2.4. volume 6, no. 2, september 2017 pp 95-110 103 on 3.2.4 e15 and e16 have not been able to solve employing problems with basic capabilities category mathematising. 3.2.4 pada tahun 2010, laju pertumbuhan penduduk indonesia adalah 1.49 %. tentukan populasi indonesia pada tahun 2000 jika populasi pada tahun 2010 adalah 237.641.326. presentase pertumbuhan ( ) e15: e16: figure 2. sample results subject e15 and e16 for employing problems with basic capabilities category mathematizing number 5.2.2.d mathematics literacy skills of students with lower category logical reasoning indicates that the subject chosen e3 (370,3) and e8 (435,8) respectively reached level 1 and level 2. when viewed from category formulating, employing, and interpreting the subject of e3 and e8 are only able to work on the problems with basic capabilities communicating. on 3.1.7 e3 and e28 have not been able to solve employing problems with basic capabilities category mathematizing. interview result write that for e3 can not rounding number but e8 do not understand to calculate probability. 3.1 tabel di bawah ini menunjukkan informasi dari profil 2014 penduduk indonesia. simbol persentase islam islam i 87,2 non islam kristen kr 6,9 katolik kt 2,9 hindu h 1,7 buddha b 0,7 konghucu kc 0,05 3.1.7. jika dipilih secara acak penduduk indonesia hitunglah peluang terpilihnya orang beragama hindu? e3: e8: no answer figure 3. sample results subject e3 and e8 for employing problems with basic capabilities category mathematizing number 3.1.7 ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 104 table 8. list of subjects research by category logical reasoning numerical reasoning category student (tklm score) high e1 (639,8) medium e4 (566,7), e13 (569,2) low e20 (478,6), e22 (493,7) mathematics literacy skills of students with high numerical reasoning category indicates that the subject e1 been reached level 5. subject e1 has been able to do formulating problem with the basic capabilities of communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. when viewed from employing problem e1 has been able to solve problems with basic capabilities of communicating, representation, reasoning and argument, devising strategies for problem solving, and using symbolic, formal and technical language and operation. in the process of interpreting the subject e1 has been completed perfectly with basic capabilities of communicating, mathematising, representation, using symbolic, formal and technical language and operations, and using mathematics tools. the results showed that mathematics literacy skills of students with medium numerical reasoning the subject e4 and e13 which have both reached level 4. subject e4 and e13 is already able to do the problems formulating with basic capabilities of communicating, mathematising, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. based on a review of the employing problem e4 and e13 have been able to solve problems with basic capabilities of communicating. in the process of interpreting the subject of e4 and e13 has been completed perfectly problem with basic capabilities of mathematising, representation, using symbolic, formal and technical language and operations, and using mathematics tools. on 3.1.7 e4 and e13 have not been able to solve employing problems with basic capabilities category mathematizing. e4 clarify that she does not read problem correctly but for e13 still difficult in rounding. 3.1.7 jika dipilih secara acak penduduk indonesia hitunglah peluang terpilihnya orang beragama hindu? e4: e13: figure 4. sample results subject e4 and e13 for employing problems with basic capabilities category mathematizing number 3.1.7 subject e20 (478,6, level 2) and e22 (493,7, level 3) is already able to do formulating problems with basic capabilities of mathematising, representation, reasoning and argument, using symbolic, formal and technical language and operations, and using mathematics tools. based on a review of the employing problem e20 and e22 have been able to solve problems volume 6, no. 2, september 2017 pp 95-110 105 with basic capabilities of communicating, and representation. in the process of interpreting the subject of e20 and e22 already completed perfectly with basic capabilities of mathematising, using symbolic, formal and technical language and operations, and using mathematics tools. for clarification both of them said that they can not operate mathematical model in 3.2.4. examples of the work of a low numerical reasoning subject matter of numbers 3.2.4. e20: e22: figure 5. sample results subject e20 and e22 for employing problems with basic capabilities category mathematizing number 3.2.4 table 9. list of subjects research by category logical and numerical reasoning reasoning category high numeric (tklm score) medium numeric (tklm score) low numeric (tklm score) high logic e1 (639,8) e21 (579,3) medium logic e14 (619,6) e16 (518,9),e19 (604,5) low logic e3 (370,3), e20(478,6) subject e1 (level 5) has been able to do about formulating problem with basic capabilities of communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. based on a review of the subject e1 employing been able to solve problems with basic capabilities of communicating, representation, reasoning and argument, devising strategies for problem solving, and using symbolic, formal and technical language and operation. in the process of interpreting the subject e1 has been completed perfectly with basic capabilities communicating, mathematising, representation, using symbolic, formal and technical language and operations, and using mathematics tools. subject e16 (level 3) and e19 (level 4) is already able to do formulating problems with basic capabilities of communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. employing the process of the subject on the e16 and e19 have been able to solve problems with basic capabilities of communicating, and representation. in the process of interpreting the subject of e16 and e19 are already solving problems with basic capabilities of communicating, mathematising, representation, using symbolic, formal and technical language and operations, and using mathematics tools. for number 3.1.7 e16 and e19 have not been able to solve employing problems with basic capabilities category mathematizing. in interview e16 said that she does not read problem correctly but for e19 still difficult in rounding. ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 106 e16: e19: figure 6. sample results subject e16 and e19 for employing problems with basic capabilities category mathematizing number 3.1.7 subject e14 (level 5) and e21 (level 4) capable of doing a matter of formulating with basic ability of communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal and technical language and operations, and using mathematics tools. based on a review of the subject employing e14 and e21 have been able to solve problems with basic capabilities of communicating, and representation. in the process of interpreting e14 and e21 are already solving problems with basic capabilities of communicating, mathematising, representation, using symbolic, formal and technical language and operations, and using mathematics tools. examples of the work of e14 and e21 for number 3.2.4. for clarification they said that can not operate mathematical model in 3.2.4. e14: e21: figure 7. sample results subject e14 and e21 for employing problems with basic capabilities category mathematizing number 3.2.4 subject e3 (level 1) and e20 (level 2) is already able to do the problems formulating with basic capabilities of mathematising, representation, reasoning and argument, using symbolic, formal and technical language and operations, and using mathematics tools. e3 and e20 able to solve employing problem only with basic capabilities of communicating. based on a review of the process of interpreting the subject of e3 and e20 already solving the basic capabilities of mathematising, using symbolic, formal and technical language and operations, and using mathematics tools. for number 3.1.7 e3 and e20 have not been able to solve employing problem with basic capabilities category mathematizing. in interview process e3 said that still difficult with rounding but e20 not read corectly in this poblem. volume 6, no. 2, september 2017 pp 95-110 107 problem : number 3.1.7 answer e3: answer e20: figure 8. sample results subject e3 and e20 for employing problems with basic capabilities category mathematizing number 3.1.7 conclusion based on the result of data analysis, the conclusions are: 1. the quality of learning with ctl model towards mathematics literacy for student grade xi of smk categorized well. this is indicated in the preparatory phase learning device consultated to the supervisor through several revisions categorized good and can be used. at this stage of the process based on the observation of math teacher smk nu lasem good category. in the evaluation phase are two students reached level 1, 2 students at level 2, 5 students at level 3, 7 students on level 4, and 2 students at level 5. 2. literacy mathematical vocational students of class xi based on logical reasoning is: a) students with high logical reasoning achieving level 4 and 5, good in formulating with communicating, representation, and devising strategies for problem solving, for employing with communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal, and technical language and operation, and using mathematics tools and fo interpreting with communicating, representation, and devising strategies for problem solving; b) students with medium logical reasoning reached level 3, good in fomulating with communicating, representation, and devising strategies for problem solving, employing with communicating, representation, devising strategies for solving problems, and using mathematics tools, and intepreting with communicating, representation, dan reasoning and argument; c) students with low logic at level 1 and 2, good in formulating, employing and interpreting with communicating. 3. the ability of the mathematical literacy of students based on numerical reasoning is: a) students with high numerical reasoning reaches level 5, good in formulating with communicating, representation, and devising strategies for problem solving, employing with communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal, and technical language and operation, and using mathematics tools, and interpreting with communicating, representation, and devising strategies for problem solving; b) students with medium numerical reasoning was reached level 4, good in fomulating with communicating, representation, and devising strategies for problem solving, employing with communicating and using symbolic, formal and technical language and operations and intepreting with communicating, representation, dan reasoning and argument; c) students with low numerical reasoning at the level of 2 and 3, good in formulating with dasar representation dan devising strategies for problem solving, and employing with communicating and representation. 4. ability mathematical literacy of students based on logical and numerical reasoning is: a) students with high logical and numerical reasoning reached level 5, good in formulating with communicating, representation, and devising strategies for problem solving, ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 108 employing with communicating, mathematising, representation, reasoning and argument, devising strategies for problem solving, using symbolic, formal, and technical language and operation, and using mathematics tools, and intepreting with communicating, representation, dan devising strategies for problem solving; b) students with logical and numerical reasoning was reached level 3 and 4, good in formulating with communicating, representation, reasoning and argument, and devising strategies for problem solving, employing with communicating, and reasoning and argument, and intepreting with communicating, representation, dan reasoning and argument; c) students with high logical and medium numerical reasoning and also medium logical and high numerical reasoning reached level 4 and 5, good in formulating with communicating, representation, and devising strategies for problem solving, employing with communicating, devising strategies for problem solving, using symbolic, formal, and technical language and operation, and using mathematics tool and interpreting with communicating, representation, reasoning and argument, dan devising strategies for problem solving; d) students with low logical and numerical reasoning only in level 1 and 2, good in formulating with devising strategies for problem solving, employing with communicating, and interpreting with communicating, representation, and reasoning and argument. logic and numerical reasoning affects student ability to think, calculate and understand the problem so that it also affects the ability of students' mathematical literacy. the ability of the mathematical literacy should be owned by all students in order to understand and solve problems appropriately in order to future challenges. teachers should help strive for the achievement of these abilities. teachers can use different ways to know and analyze students' literacy skills. one can develop instruments to analyze, measure, or identify students' mathematical literacy skills. in addition, teachers need to pay attention to these conditions in classroom learning activities. different logical and numerical reasoning abilities allow different abilities of students' mathematical literacy. the ability of logical and numerical reasoning can be honed as well as the ability of mathematical literacy. teachers attention to differences in students' logical and numerical reasoning abilities will have implications for the selection of appropriate learning models. model selection should be tailored to the characteristics of students' abilities. learning with the right model and quality is expected to provide increased mathematical literacy skills. references adonis, a. (2006). primary framework for literacy and mathematics. united kingdom: department education and skills. bokar, a. j. (2013). solving ang reflecting on real-world problems: their influences on mathematical literacy and engagement in the eight mathematical practises. thesis: ohio university. diezmann, c. m., watters, j. j., & english, l. d. (2001). implementing mathematical investigation with young children. proceedings 24th annual conference of the mathematics education research. sydney: group of australia. dowden, b. h. (2011). logical reasoning. california: california state university sacramento. volume 6, no. 2, september 2017 pp 95-110 109 durrant-law, g. (2013). logical thinking. canberra: university of canberra. kariadinata, r. (2012). menumbuhkan daya nalar (power of reason) siswa melalui pembelajaran analogi matematik. infinity, 1(1), 10-18. doi:10.22460/infinity.v1i1.3. kasihani, e. s. (2002). contextual learning and teaching (ctl) (pengajaran dan pembelajaran kontekstual). prosiding seminar akademik, 2, 1-6. krause, j., dias, l. p., & schedler, c. (2015). krause, j., dias, l.p., dan schedler, c. competency-based education: a framework for measuring quality courses. online journal of distance learning administration spring, 18(1), 1-9. macgregor, r. r. (2007). the essential practices of high quality teaching and learning. the center for educational effectiveness, inc. melville, w., & yaxley, b. (2009). contextual opportunities for teacher professional learning: the experience of one science department. eurasia journal of mathematics, science & technology education, 5(4), 357-368. mulyatiningsih, e. (2010). pembelajaran aktif, kreatif, inovatif dan menyenangkan (paikem). jakarta: direktorat jendral peningkatan mutu pendidik dan tenaga kependidikan. murnane, r., sawhill, i., & snow, c. (2012). literacy challenges for the twenty-first century: introducing the issue. the future of children, 22(2), 3-15. oecd. (2013). pisa 2012 assessment and analytical framework. oecd publishing. ojose, b. (2011). mathematics literacy: are we able to put the mathematics we learn into everyday use? journal of mathematics education, 4(1), 89-100. riyanto, b., & siroj, r. a. (2011). meningkatkan kemampuan penalaran dan prestasi matematika dengan pendekatan konstruktivisme pada siswa sekolah menengah atas. jurnal pendidikan matematika, 5(2), 111-127. rusmining, waluya, s. b., & sugianto. (2014). analysis of mathematics literacy, learning constructivism and character education (case studies on xi class of smk roudlotus saidiyyah semarang, indonesia). international journal of education and research, 2(8), 331 – 340. sandstorm, m., nilsson, l., & lilja, j. (2013). displaying mathematical literacy-pupils’ talk about mathematical activities. journal of curriculum and teaching, 2(2), 55 – 61. shadiq, f. (2004). pemecahan masalah, penalaran dan komunikasi. yogyakarta: pppg matematika yogyakarta. stacey, k. (2012). the international assessment of mathematical literacy: pisa 20112 framework and items. 12th international congress on mathematical education. seoul, korea: coex. suyono. (2009). pembelajaran efektif dan produktif berbasis literasi: analisis konteks, prinsip, dan wujud alternatif sebagai implementasinya di sekolah. bahasa dan seni, 37(2), 203-217. turner, r. (2011). exploring mathematical competencies. research development, 24(5), 1-6. ni’mah, junaedi & mariani, mathematical literacy’s vocational students … 110 venkat, h., graven, m., lampen, e., & nalube, p. (2009). critiquing the mathematical literacy assessment taxonomy: where is the reasoning and the problem solving? pythagoras, 0(70), 43-56. doi:10.4102/pythagoras.v0i70.38. wardhani, s., & rumiati. (2011). modul matematika smp program bermutu instrumen penilaian hasil belajar matematika smp: belajar dari pisa dan timss. jakarta: kemendiknas dan pppptk. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p17-32 17 alternative learning during a pandemic: use of the website as a mathematics learning media for student motivation suripah, weni dwi susanti* universitas islam riau, indonesia article info abstract article history: received feb 15, 2021 revised july 24, 2021 accepted july 26, 2021 in march 2020, direct learning activities were transferred online due to the covid-19 pandemic. there are many alternative technology-based learning media used for online learning systems. however, the use of this media has not been able to increase students' motivation to learn mathematics. one of the media that can be used as an alternative learning media during this pandemic is a website. the purpose of this study was to find out how to use the website as a learning media in the midst of this pandemic, and to determine student learning motivation. this research is a descriptive study with a quantitative approach, the subjects in this study were 25 students of grade 8 and 9 junior high school. the results of this study were that many students agreed with the use of websites as alternative learning media during this pandemic and students' high motivation to learn mathematics when using the website with an average percentage of 66.3%. from this research, it can be concluded that the website can be an alternative media for learning mathematics in the midst of this pandemic because the website has various advantages that can also increase student learning motivation. keywords: alternative learning, mathematics, motivation, pandemic, website this is an open access article under the cc by-sa license. corresponding author: weni dwi susanti, departement of mathematics education, universitas islam riau jl. kaharuddin nasution no.113, bukit raya, pekanbaru, riau 28284, indonesia email: wenidwisusanti28@gmail.com how to cite: suripah, s., & susanti, w. d. (2022). alternative learning during a pandemic: use of the website as a mathematics learning media for student motivation. infinity, 11(1), 17-32. 1. introduction the covid-19 pandemic has hit all countries in the world, including indonesia. the covid-19 pandemic has an impact on all sectors of life, such as the economy, health, education and others (damanhuri, 2020; susilawati et al., 2020; zhang & ma, 2020). in the education sector, the covid-19 virus pandemic has stopped educational activities from the lowest levels of education (pre-primary school) to higher education. in connection with this, the minister of education and culture of the republic of indonesia issued circular number: 36962 / mpk.a / hk / 2020, march 17, 2020 regarding online learning and working from home in the context of preventing the spread of corona virus disease (covid-19). this https://doi.org/10.22460/infinity.v11i1.p17-32 https://creativecommons.org/licenses/by-sa/4.0/ suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 18 circular contains 4 important things, (1) study at home through daring or distance learning implement to experience without being burdened to complete the curriculum target for grade promotion or graduation, (2) distance learning can be focused on life skills education example covid-19, (3) students activities and task could be varied among students in accordance with their interest and condition including learning gap and facilities at home, (4) proof or product activity should be given feedback which is qualitative and useful for teachers without giving a score or quantitative grade. it is hoped that the sudden change in conditions in the world of education will not hinder the learning process. the government policy regarding learning from home is one of the efforts to prevent the spread of the covid-19 outbreak widely and to follow up to comply with social distancing and physical distancing rules (pratomo, 2020; yanti et al., 2020). mathematics learning which is part of the school curriculum is also affected by learning from home policy. this needs to be a concern because it is related to the characteristics of mathematics. learning mathematics is still considered difficult by many students, this is caused by several factors. first, many students think that the concept of learning mathematics is abstract and difficult to understand (putra et al., 2018). because the concept of learning mathematics is considered abstract, there are still many students who have difficulty in solving mathematical problems. solving these math problems requires problem-solving skills and creativity, because math problems sometimes require complex solutions that require students' creative thinking (maulidia et al., 2019). in addition to creative thinking, in solving mathematical problems, students’ critical thinking skills are also needed to be responsive in solving given problems. but unfortunately, students’ critical thinking skills are still low in solving math problems (rachmawati et al., 2021; zetriuslita et al., 2017). due to some of these factors, making mathematics as a difficult subject to learn so it makes teaching mathematics, especially during this pandemic teachers must be able to prepare learning media to help students understand the concept of learning mathematics. therefore, there must be a new breakthrough in the online mathematics learning process, namely by utilizing technology so that mathematics learning continues to run optimally (susanti & suripah, 2021). in the use of technology as a media for learning mathematics, teachers are required to be more creative and should take advantage of the facilities offered by information technology (suripah, 2017). it is intended that the learning media designed or created by the teacher can be easily used by students to understand the mathematics learning material provided. when a pandemic occurs, many learning media can be used by utilizing information technology. many teachers have started using the learning management system (lms) platform such as google classroom, edmodo, schoology and others (okmawati, 2020; purnawarman et al., 2016; roqobih & ambarwati, 2020). in addition, teachers also use video conference technology facilities such as zoom, google meet (fakhruddin, 2018; sajaril et al., 2020) and create interesting learning videos to upload on youtube as an alternative learning media for students (samosir et al., 2018). another learning alternative that can be used is software such as geogebra that can visualize mathematical concepts to students (zetriuslita et al., 2020). even though there are many technological facilities that can be used as alternative media during this pandemic, students' motivation to learn mathematics has actually decreased. several previous studies have shown that student motivation decreases during the online learning process (gustiani, 2020; muslimin & harintama, 2020; subakthiasih & putri, 2020). the decrease in motivation is caused by several factors, including bad internet network connection, improper media selection, and less conducive learning conditions. whereas learning motivation is important for every student to have, with learning motivation volume 11, no 1, february 2022, pp. 17-32 19 will bring out the intention from within students to continue to carry out learning activities so that the desired goals can be achieved (cahyani et al., 2020). therefore, interactive media is needed and can be the right alternative for use in online learning activities. one of the learning media that can be used in online learning activities is a website. website is a collection of pages summarized in a domain or subdomain that contains multimedia in the form of audio, text, images and video and can be accessed via a web browser (destiningrum & adrian, 2017; marisa, 2017). one of the advantages of the website as a learning media is that there is interactive multimedia that can be used in the learning process so that it can encourage student motivation to learn independently (danaswari & gafur, 2018). based on this, the website becomes an efficient learning media because it can be accessed anywhere and anytime. several previous researchs have revealed the advantages of websites as learning media. study from ghani and daud (2018) stated that the use of the website as a learning media can increase the effectiveness of the learning process, and most students feel satisfied and play a more active and critical role in developing their skills. in addition, study from astuti et al. (2020) stated web-based technology is often the technology of choice for distance education, given the ease of use of tools to browse web resources from any device, and the relative affordability of accessing them anywhere, websites can also be easily designed using the multiple online platforms available. in addition to having advantages, the use of websites as learning media also has disadvantages. based on andriasari (2017) opinion, the disadvantages of using the website include: (1) it requires a stable internet connection, this is intended so that the website can be accessed properly and smoothly, (2) a good security system is needed, so that the server does not drop when accessed by users simultaneously. the findings from this study will help advance our understanding of the integration of the use of websites as alternative learning media in the midst of the covid-19 pandemic on student learning motivation. therefore, this study adds valuable insight into alternative learning media that can be used during online learning and increases student motivation in learning mathematics. to achieve this goal, this study aspires to answer questions (1) how is the use of the website as an alternative learning media during the pandemic? and (2) how do students' motivation when using the website as an alternative learning media? 2. method 2.1. research design this research is a descriptive study with a quantitative approach. in this study, it will be known how much intensity the students use of the website as a learning media (see figure 1). the website accessed by students is a website developed by the researchers themselves using google sites. the subject matter contained on the website was also developed by researchers in the form of 8th and 9th grade mathematics subject matter. in addition, it will also be known how much student motivation when using the website. in line with seixas et al. (2018), this approach will focus on the latest problems and phenomena that are happening in the form of research results in the form of numbers that have meaning about the use of websites on student motivation. suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 20 figure 1. the example of interface an educational website 2.2. participant in this study, the sampling technique was random sampling. this is because researchers cannot limit someone from other institutions. researchers only limit, the subjects in this study were students in grade 8 and 9 junior high school. researchers do not limit gender, school name, age of respondents, or others. this study involved 25 students from grade 8 and 9 junior high school. among them, 25 students came from 2 different schools, namely nurul falah kulim junior high school, and from 3 pekanbaru junior high school. researchers only conducted research on some 8th and 9th grade students due to limitations in conducting exploration in the midst of the current covid-19 pandemic, requiring researchers to conduct research online. 2.3. research instruments the instrument in this study was made by researchers, research instrument in the form of a questionnaire made from google form and then distributed to research subjects. researchers collected data through questionnaire from google form that consisting of 22 volume 11, no 1, february 2022, pp. 17-32 21 questions. before filling out the questionnaire, the researcher will provide a website link on the research subject. after that, the researcher will ensure that the website is seen by the research subject through his absence from the website. during the online learning process, the researcher will ensure the presence of each research subject through absence. the questionnaire will be filled in by grade 8 and 9 junior high school students. of the 20 question consisted: 2 question contains about participant’s profile such as name and class; 5 questions contained about the intensity of students using the website; 5 questions contained about the purpose of accessing the website; 5 questions contained about desire in learning: and 5 questions contained about interest in learning. before the questionnaires were distributed to the research subjects, the researchers first validated the content of the 8th and 9th grade junior high school mathematics curriculum standards. the next stage, learning tools, media and questionnaires were consulted with experts in this case were 2 mathematics education lecturers. based on the input given, the researcher improved the media according to the suggestions and inputs until the media that had been made by the researcher was declared valid. while the questionnaire that have been consturcted based on indicators of learning motivation according to theoretical studies to be used as a questionnaire item for student learning motivation. the items that have been constructed will then be validated by experts and further revised according to expert advice until the questionnaire is declared valid. 2.4. data collection and analysis data was collected using google form as online survey. google form are used for reasons of the efficiency and flexebility of compability with students’ learning during a pandemic, and because its easy to use. the questionnaire are disseminated through whatsapp groups and emails. after the respondents fills in the questionnaire, the researchers will get a recapitulation results. the data obtained about the intensity of students using website, the purposes of accesing the website, students’ desire learning, and students’ interest in learning. the data obtained is then interpreted and described by researchers. 3. results and discussion 3.1. results the results of a survey of respondents who were taken as participants in this study found that in the usage intensity section, students in general were able to access several mathematics learning websites during the covid-19 pandemic. table 1. the survey results of respondents in usage intensity indicators sub indicators items frequency percentage (%) score maximum amount usage intensity understand student use of websites i use the website in math learning activities 79 100 79% i use the website to find math subject matter 76 100 76% when doing assignments from teachers, i was not interested in finding information on the website 48 100 48% suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 22 indicators sub indicators items frequency percentage (%) score maximum amount understand the time to use the website i access math learning websites when i have free time 67 100 67% i can't take the time to access maths learning websites every day 59 100 59% total 329 500 329% average 65.8% table 1 shows that the intensity of using the website by respondents is quite high. from this table, it can be seen that respondents often use the website when learning mathematics and looking for mathematics teaching materials. however, when using the website, some respondents still did not take full advantage of the website. this can be seen from the results in the table, where only 48% of respondents access learning websites during their spare time and 59% of respondents have time to access learning websites every day. the results of the survey of respondents who were used as participants in this study found that in the section on the purpose of accessing the website, most of the students understood the use of the website as a source of information and independent learning resources, so they always studied independently and found out things related to mathematics learning. especially in conditions of the covid-19 pandemic like this, students are required to learn independently by utilizing existing technology (see table 2) table 2. the survey results of respondents in the purpose of accessing the website indicators sub indicators items frequency percentage (%) score maximum amount purpose of accessing the website using the website as a means of information i use the website as a means of getting the latest learning information 80 100 80% i use the website to find information on the latest movies and games 61 100 61% using the website as a learning resource i use the website as a substitute learning resource when the teacher cannot carry out face-to-face learning 74 100 74% i use the website as a learning resource and media to access math learning materials 77 100 77% i use this website as a complementary media to access learning materials and gain insight and knowledge about mathematics 76 100 76% total 368 500 368% average 73.6% volume 11, no 1, february 2022, pp. 17-32 23 table 2 shows that many students use websites as learning resources during the covid-19 pandemic. this can be seen from the results in the table, where around 74% of respondents access the website as a substitute learning source when they cannot do face-toface learning activities, 80% use the website to access mathematics learning material, and 76% of respondents use the website as a complementary media to add to their insights however, for the use of the website as information, there are still some students who aim to access information on the website for entertainment such as playing games and looking for movies. the results of a survey of respondents who were taken as participants in this study found that in the part of the desire in learning, most of the students showed a great desire to learn when using the website as an alternative to their learning media. the presentation of attractive material and information that is easy to understand is able to attract the attention of students to use the website in the learning process (see table 3). table 3. the survey results of respondents in desire in learning indicators sub indicators items frequency percentage (%) score maximum amount desire in learning students’ attention when participating in mathematics learning i'm excited to use the website to find math learning materials 78 100 78% using the website helps me understand math learning material 75 100 75% by displaying interesting material on the website, it makes me even more excited about the mathematics learning process 81 100 81% learning math using the website makes me feel bored 38 100 38% i feel hopeless while working on math problems 37 100 37% total 309 500 309% average 61.8% table 3 shows that many students expressed their concerns when using the website. website material is presented with interactive multimedia which is able to attract students' attention in learning mathematics. this can be seen from the results in the table that 81% of respondents better understand mathematics learning materials using websites and also only a small proportion of respondents feel bored when using the website as a learning media. this means that students' desire to learn is high enough when using the website as an alternative media for learning mathematics the results of a survey of respondents who were taken as participants in this study found that in the part of interest in learning, most students showed their interest in using the website. the interactive presentation of material on the website can increase students' curiosity and enjoyment in the learning process (see table 4). suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 24 table 4. the survey results of respondents in interest in learning indicator sub indicator items frequency percentage (%) score maximum amount interest in learning students’ curiosity when taking mathematics lessons i feel happy trying to do math problems using the website 81 100 81% if i have trouble understanding the material, i look for more information on the website 79 100 79% i postpone doing assignments given by the teacher even though i use the website 43 100 43% i am interested in finding other study materials on the website 78 100 78% i am not interested in learning mathematics because i am always given assignments 39 100 39% total 320 500 320% average 64% table 4 shows that many students are increasingly curious when using the website as an alternative learning media. many of them want to use the website for subjects other than math. this can be seen from the results in the table, 78% of respondents are interested in using websites for other subjects and 81% of respondents like to use websites when learning mathematics. only a small proportion of them are not interested in using websites and learning math. based on indicators 1 to 4, the total score can be visualized in the table 5. table 5. percentage of website usage on student learning motivation no. indicators percentage (%) 1. usage intensity 65.8% 2. the purpose of accessing the website 73.6% 3. desire in learning 61.8% 4. interest in learning 64% average 66.3% table 5 show the percentage of total indicators of using the website as a media for learning mathematics on student learning motivation. based on the interpretation criteria of the modified students' learning motivation scores from riduwan (2010) (see table 6), the average percentage of student learning motivation is 66.3%. so that based on these results student learning motivation when using the website as a learning media can be categorized as high. volume 11, no 1, february 2022, pp. 17-32 25 table 6. interpretation criteria for student learning motivation scores no. motivation range interpretation criteria for mathematics learning motivation 1. 0% ≤ motivation ≤ 20% very low motivation 2. 21% ≤ motivation ≤ 40% low motivation 3. 41% ≤ motivation ≤ 60% enough motivation 4. 61% ≤ motivation ≤ 80% high motivation 5. 81% ≤ motivation ≤ 100% very high motivation based on these results, the use of websites as a media for learning mathematics can increase student motivation. the interactive presentation of material, easy-to-understand information, and easy access makes the website an alternative learning media which is quite effective to use during online learning. the use of websites in mathematics learning can increase students' enthusiasm and curiosity. so that the website can be used as an alternative learning media in the midst of this pandemic. 3.2. discussion in addition to discussing the use of websites as alternative media for learning mathematics for student motivation, this study also discusses choosing the right alternative learning media in the midst of this pandemic. in the findings of this study, there are two important points. first, this study shows that the website is a learning media to support elearning; several previous studies have also shown that the website is a learning media that is often used in e-learning (astuti et al., 2020; dogan & dikbıyık, 2016; hamdunah et al., 2016; lestari, 2019; usta, 2011), however teachers still have difficulty applying math concepts on the website. this is because teaching math concepts is very difficult for teachers (indriani et al., 2018; maulydia et al., 2017), and also added to the factor of teachers who still have difficulty teaching online (karal et al., 2015; kebritchi et al., 2017; wang & ip, 2010). many teachers have understood the selection of alternative media during the pandemic, including using learning management systems (lms) (eg google classroom and edmodo) as well as facilities such as video conference (eg zoom and google meet). based on this, it means that many teachers have mastered the use of technology even before this pandemic. for the selection of appropriate alternatives during this pandemic, teachers must be able to adapt again and adapt the media to the needs of students. website can be an alternative because it contains multimedia and the information conveyed is easy to understand. based on the results of the study, the intensity of using the website as a media for learning mathematics is quite high, of which about 79% of students use the website for mathematics learning activities. the intensity of the use of the website is quite high, due to the ease of accessing the website freely and also the design on the website is simple but able to attract the attention of students in learning activities. this is supported by opinion (gautam et al., 2020; lestari, 2019; muhardi et al., 2020; permatasari et al., 2019) who said that the website is one of the e-learning media that can be accessed easily compared to other platforms such as moodle, schoology, and edmodo. in addition, learning materials on the website can also be presented in various forms such as word, pdf, powerpoint, html and the advantages of website design with various interactive menus. this is what makes many students interested in using the website as a media for learning mathematics, as evidenced suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 26 by the results of research on the indicators of the purpose of using the website as many as 77% of students use the website to access mathematics learning materials and as many as 76% access the website with the aim of increasing their mathematical knowledge. second, the results showed that many students used the website as an alternative learning media. some of the existing indicators even show that student learning motivation increases with the use of websites as a media for learning mathematics. based on the results of the study on indicators of desire in learning, as many as 78% of students were excited about looking for mathematics learning on the website, in this case the use of interactive multimedia had an important effect in increasing students' enthusiasm for learning. the use of multimedia in the form of audio, text, video, and animation increases student motivation during the learning process (almara'beh et al., 2015; leow & neo, 2014; maria et al., 2019; nasrum & herlina, 2019; setiawan et al., 2015). using this website can better support the online learning system in the midst of this pandemic besides using the learning management system (lms) platform, video conferencing, and others (djamdjuri & kamilah, 2020; handayani & utami, 2020; lubis & sari, 2020; nartiningrum & nugroho, 2020; putri & irwansyah, 2020; wiratomo & mulyatna, 2020). this is evident from the results of the study, as many as 81% of students feel happy when using the website to solve mathematical problems. website is one of the media that can be accessed anywhere and anytime, this is one of the supporting factors to increase student motivation in using the website. the use of the website is also simple, by clicking on the website link students can directly link to the website homepage. based on opinion susanti and suripah (2021), website is one of the effective mathematics learning media used during online learning activities. with these advantages, making the website a more effective learning media than other platforms that can be accessed freely. this can be seen from the results of research that average percentage of student learning motivation is 66.3%. so that based on these results student learning motivation when using the website as a learning media can be categorized as high. this study examines the use of websites as an alternative media for learning mathematics towards student motivation, it is not easy to choose the right media to use during this pandemic, there are many limitations and obstacles encountered in each lesson. however, teachers and students must be able to adapt to these changes, namely by trying to make optimal use of existing information technology, one of which is accessing the website to develop mathematical insights. 4. conclusion this study concludes that the website can be an alternative learning media during this pandemic. based on the intensity of use and intended use, it is known that students are interested in using websites in the mathematics learning process. in addition, the use of websites as a media for learning mathematics can also increase motivation. with an attractive presentation of material, easy access, and easy-to-understand information to make students more motivated in the mathematics learning process. students' attention and curiosity when accessing mathematics learning materials on the website increase their learning motivation. the use of the website is also recommended for use in other subjects. volume 11, no 1, february 2022, pp. 17-32 27 acknowledgements the authors would like to thank the lectures for their support and guidance so far, and also some students who participated in filling out the questionnaire. without them this research will not be done. references almara'beh, h., amer, e. f., & sulieman, a. 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(2020). impact of the covid-19 pandemic on mental health and quality of life among local residents in liaoning province, china: a cross-sectional study. international journal of environmental research and public health, 17(7), 2381. https://doi.org/10.3390/ijerph17072381 https://doi.org/10.3390/ijerph17072381 suripah & susanti, alternative learning during a pandemic: use of the website as a mathematics … 32 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p331-348 331 designing learning trajectory of set through the indonesian shadow puppets and mahabharata stories irma risdiyanti, rully charitas indra prahmana* universitas ahmad dahlan, indonesia article info abstract article history: received june 28, 2021 revised july 28, 2021 accepted july 30, 2021 indonesia has many cultures that can be used as a starting point in learning mathematics. yet, many teachers still use conventional methods to provide explicit mathematical content without connecting with students' culture and daily activities. one of the learning approaches that can solve these problems is realistic mathematics education (rme). this approach uses context as one of its characteristics containing students' culture and their daily activities. on the other hand, wayang (indonesian shadow puppets) and mahabharata stories have the characteristics that can be a context in the learning of set. this research aims to design the hypothetical learning trajectory (hlt) of the set using the rme approach through wayang and mahabharata stories, which are familiar with students' culture in yogyakarta. this hlt will then be tested on students in further research until it becomes the local instructional theory (lit) on set. students can study about set by grouping wayang in mahabharata stories based on their characters. the research result is the hlt of set through the context of wayang and mahabharata stories containing learning goals, learning activities, and the conjecture of every activity. this hlt can be a promising solution to overcome students' difficulties in understanding the concept of sets and values in the cultural context to improve the students' character. keywords: design research, ethnomathematics, indonesian shadow puppets, mahabharata stories, realistic mathematics education, set copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: rully charitas indra prahmana, departement of mathematics education, universitas ahmad dahlan jl. pramuka no. 42, pandeyan, umbulharjo, yogyakarta 55161, indonesia email: rully.indra@mpmat.uad.ac.id how to cite: risdiyanti, i., & prahmana, r. c. i. (2021). designing learning trajectory of set through the indonesian shadow puppets and mahabharata stories. infinity, 10(2), 331-348. 1. introduction in principle, the elements that build mathematical concepts come from several things developed by humans to respond to the surrounding environment, such as seeking explanations, understanding, experiences, and solutions to phenomena or events around them that they experience (ernest et al., 2016; freudenthal, 2006; d'ambrosio, 2007). in addition, mathematics cannot construct itself. still, it is influenced by historical, environmental, social, and geographical aspects or what we call a culture where humans develop and process their lives (risdiyanti & prahmana, 2017; utami, sayuti, & jailani, 2019). as a result, it becomes inflexible and becomes far from culture and everyday life, https://doi.org/10.22460/infinity.v10i2.p331-348 risdiyanti & prahmana, designing learning trajectory of set … 332 where humans develop and process it (alangui, 2010; muhtadi et al., 2017). therefore, mathematics is very close and departs from culture and daily human life, eventually becoming a proper formal education or school form. the mathematics learning problem is inseparable from the efforts of the west to try to hegemony science to dull the thinking ability of individuals who study mathematics so that it can facilitate and perpetuate colonialism and western influence on other civilizations (d'ambrosio, 2007; joseph, 2010). the mathematics form is rigid and far from the origin of the construction of mathematical concepts (risdiyanti & prahmana, 2020a). it's causing mathematics learning to be limited to knowledge transfer. students only accept it without critical and reflective thinking on the knowledge and cannot know its meaning and use in everyday life (d'ambrosio, 2016). thus, a learning approach that is close to culture and human activities is needed to solve these problems in learning mathematics. d'ambrosio (1985), a mathematician from brazil, initiated the ethnomathematics approach as a solution. ethnomathematics is a way to study and combine ideas, methods, and techniques used and developed by socio-cultural or members of different cultures to learn mathematical concepts (d'ambrosio, 2016; rosa & orey, 2016). it is reflecting on how mathematics was developed based on how mathematics is taught in schools. ethnomathematics tries to reconstruct mathematics so that it is rooted in different cultures and accommodates other ideas so that students become able to reason critically, democratically and can be tolerant of various ideas and ideas during teaching and learning activities (d'ambrosio, 2016; risdiyanti & prahmana, 2020a). therefore, ethnomathematics can be used as one of the educational innovations in mathematics learning, aiming to make students love mathematics, be motivated, and increase creativity in mathematics through their culture. indonesia is a country with an abundant culture to instill mathematical ideas, methods, and techniques through mathematical modeling (prahmana et al., 2021). this approach creates opportunities in mathematics learning to use local contexts or cultures that can increase students' critical reasoning and interest by rediscovering mathematics rooted in the culture around students and benefiting from the mathematical concepts it finds. one of the cultures in java, especially in central java, east java, and the special region of yogyakarta, which has mathematical characteristics, is wayang. elements of wayang characters known by the javanese people have several similarities to identify kind, evil, caste, and several things in them. in addition, there are still many javanese people who hold wayang performances at certain events. mathematical modeling in the context of wayang characters and their stories is expected to teach mathematics. learning mathematics using the cultural context used in learning mathematics has been documented by some researchers, i.e., the gadang minangkabau house for learning the pythagorean theorem (rahmawati, 2020), the soko tunggal mosque for learning two-dimensional geometry (putra, wijayanto, & widodo, 2020), megono gunungan for learning cones (nursyahidah et al., 2020), batik cloth motifs for learning reflection (novrika, putri, & hartono, 2016), and bamboo craft for learning translational (maryati & prahmana, 2020). ethnomathematics is mathematics that exists from how society uses mathematics to deal with their everyday life (d'ambrosio, 2007). ethnomathematics can be part of the realistic mathematics education approach, mainly an authentic context used as a starting point in the learning process (risdiyanti & prahmana, 2020a). therefore, students can understand a mathematical concept from the culture, which implicitly contains mathematical concepts. realistic mathematics education approach with characteristics of using real context as a starting point in learning can be a place for ethnomathematics, an authentic context in learning mathematics. real contexts in rme aim to make it easier for students to understand volume 10, no 2, september 2021, pp. 331-348 333 and abstract the mathematical concepts learned from non-formal to formal forms (hadi, 2017; soedjadi, 2007). in this approach, the teacher acts as a facilitator who accompanies the emergence of students' thinking strategies and not as a source that indoctrinates students' thinking (hadi, 2017). in this approach, students are more likely to be given the freedom to think critically, be independent, and find the knowledge and mathematical concepts they want to know and learn (meirisa, rifandi, & masniladevi, 2018; hadi, 2017). through rme, which uses real contexts such as culture and everyday life, students create a pragmatic view of society, and students who view mathematics as a scary and challenging science to learn and far from civilization and everyday life can be minimized (risdiyanti, prahmana, & shahrill, 2019; zulkardi, putri, & wijaya, 2020). in addition, with the actual context used in rme, students can critically take the meaning of the mathematics they are learning and can feel the benefits to solve the problems they face in students' daily lives (hadi, 2017; risdiyanti & prahmana, 2020a). several researchers have documented the students' difficulty in understanding the concept of set, applying the principles, understanding questions, transforming questions, solving problems, including related story questions with students' daily lives (dwidarti, mampouw, & setyadi, 2019; ratnasari & setiawan, 2019). at the same time, the concept of a set is a fundamental concept used to understand other materials such as relations and functions (nurtasari, jamiah, & suratman, 2017). if students have difficulty understanding the idea of a set, it won't be easy to understand other related materials. therefore, it is very urgent to design learning that can be used to understand and make it easier for students to understand this concept. this study aims to design a learning trajectory of set learning using the rme approach and the mahabharata puppet and storie's context. the use of wayang context is because there are several characterizations in the wayang story, namely protagonist, antagonist, and the tritagonist. the wayang characters are depicted in the visuals of the wayang faces to be easily identified as evil puppets and good puppets. concerning the set concept, several characters in the mahabharata story can be seen based on the storyline's learning nature and the method's visualization. this can be used as a starting point in understanding and rediscovering the concept of sets. in addition, wayang is also a culture that contains moral values and a philosophy of life. this is very useful for students to emulate simultaneously for the good character in students (kasim, 2018). as a result, a learning trajectory based on a realistic mathematics education approach is required to facilitate students' understanding of sets. students will study mathematics to solve problems encountered in their daily lives. this learning trajectory is in the form of learning steps that begin with introducing the actual context, then use that context to rediscover mathematical concepts until finally, students can understand the concept in a standard form. additionally, this outcome will comprehend the relationship between mathematics, culture, and students' daily lives, be familiar with mathematics' applications and encourage students to think critically and meaningfully. this learning trajectory may be an alternative solution for increasing student understanding of the set and preserving indonesian culture. 2. method in this study, the researchers designed the alleged set learning trajectory using a realistic mathematics education approach with the context of wayang and mahabharata story. this design is done by analyzing the culture that students are familiar with and then compiling a mathematical abstraction process that can be done using the real context that risdiyanti & prahmana, designing learning trajectory of set … 334 exists around the students. next, the researcher arranges the learning steps along with the conjecture or conjecture of the student's response and the alleged response that the teacher must give to anticipate the answer given by the student. this study is part of design research. the research design is to develop an intervention in teaching and learning activities as a solution to solve educational problems (plomp, 2013; gravemeijer & cobb, 2006). the design research method can answer the problem formulation and achieve the research objectives (prahmana, 2017; plomp, 2013). this method allows researchers to study student learning processes. in addition, knowing to what extent the activities that have been designed can support students' understanding of the circle material. this research is the first phase of design research, namely the preliminary design or research. in the initial design stage, researchers prepare to learn activities through literature review. researchers obtained information about students' difficulties in learning circles and what activities can support students' understanding of sets from the literature. before the learning trajectory becomes a local instructional theory, the steps are formulated in advance in the form of alleged learning steps and the alleged responses of students and teachers called the hypothetical learning trajectory. 3. results and discussion 3.1. results in this study, the researcher implemented the initial idea of using the context of wayang and mahabharata stories in group learning by reviewing the literature. after that, the researcher made observations to the muhammadiyah magelang elementary school regarding the context used and ended by designing a hypothetical learning trajectory (hlt). the development of hlt in every learning activity is an essential part of designing student learning activities. the design of learning activities is inseparable from the learning trajectory, which contains a hypothesis plan for learning materials, where the learning trajectory is a concept trajectory that students will pass during the learning process. furthermore, the learning trajectory, the learning activities, and the context used in the learning of set will become a local instructional theory in the learning process that has been designed (see figure 1). figure 1. learning trajectory for set learning several activities have been designed based on the hypothesized learning trajectory and students' thought processes. this set of instructional activities has been divided into three activities which were completed in 3 meetings. this research is intended to understand one or more basic concepts of sets in everyday life activities. the relationship between student learning paths, learning activities, and the basic concepts of the set can be seen in table 1. understanding of the concept of the universal set and the members of the set understanding of subsets understanding of sets and subsets in formal form and venn diagrams volume 10, no 2, september 2021, pp. 331-348 335 table 1. the relationship between student learning paths, learning activities, and the basic concepts student learning paths learning activities set basic concepts activity base on experience (mode of) activity 1 watching the mahabarata wayang stories by ki seno watching the mahabarata wayang stories by ki seno universal set identify the characters of wayang mahabarata laison activities (mode for) activity 2 help ki seno to compose the mahabarata wayang on kelir screen understanding the pattern of arrangement of the wayang mahabarata on the kelir screen subset understanding the difference wayang between of evil and kind characters seen from the seen of the face classifying wayang based on evil characters classifiying wayang based on kind characters clasifiying wayang based on pandhawa lima group clasifiying wayang based on kurawa group formal knowledge activity 3 write the set of mahabarata wayang define the set definition and formal form of sets, members of sets and subsets define the member of set write the set-in formal form define the subset write the subset in formal from determine subsets and draw them in the form of a venn diagram write the experience of learning a set of learning activities as contained in the hypothetical learning trajectory (hlt) in table 1 consists of 4 activities. the details can be explained as follows. 3.1.1. activity 1: watching the mahabharata wayang show by ki seno in this first activity, students watched the mahabharata wayang show masterminded by ki seno. next, students identify the wayang characters in the mahabharata story. in this activity, the teacher starts the lesson by distributing student books to the students. then ask risdiyanti & prahmana, designing learning trajectory of set … 336 students about their knowledge and experience about wayang and the mahabharata story, then ask students to explain what they know or explain their experience. the teacher then asked the students to read the information about the wayang and the mahabharata story in the student book. next, the teacher asks students to follow the learning steps in the student book. the teacher asks students to play of a wayang animation video with the mahabharata story in the student book. the duration of video is 3 minutes. then, the teacher asks students to identify the wayang characters in the mahabharata story and write their names in the column provided in the student book. finally, students are asked to present their work in front of the class for discussion. purpose of activity 1 this first activity aims to find out the students' knowledge and experience regarding the context of the wayang and the mahabharata story. in addition, to identify the wayang characters in the mahabharata story, which is actually a clearly defined object, namely the wayang object in the mabaharata story, with the hope that a 'student language' will appear for the set, namely a collection of objects that have clearly defined properties. as well as appearing 'student language' for members of the set, namely objects or objects that are clearly defined. student book and conjecture of activity 1 in this first activity, students are guided by the teacher by using a student book. student activity begins with getting to know wayang and the mahabharata story. then, the students watched the wayang show with the mahabharata story. after that, students write down the mahabharata wayang figures in the columns provided in the student books. more details can be seen in figure 2. figure 2. activity 1 in the student book volume 10, no 2, september 2021, pp. 331-348 337 the conjecture in the first activity consists of activities, predictions of student responses, and the teacher's responses in response to the responses given by these students. more details can be seen in table 2. table 2. conjecture of activity 1 no activity predictions of student responses teacher's responses 1 the teacher asks students about their knowledge and experience about wayang and the mahabharata story. know and/or have experience about wayang and mahabharata stories the teacher asks students to share their knowledge and experiences about wayang and the mahabharata story. don't know and/or don't have experience with wayang and mahabharata stories cerita the teacher tells about the wayang and the mahabharata story, then asks the students to read the information about the mahabharata wayang and story in the student book. 2 students write the mahabharata wayang character write all the wayang characters in the mahabharata story the teacher gives a verbal appreciation of the student's work. writing some of the wayang characters in the mahabharata story the teacher guides the students to be able to write down all the mahabharata wayang characters. 3.1.2. activity 2: compose the mahabharata wayang on the kelir screen in this second activity, students arrange wayang characters in the mahabharata story in wayang colors. the steps for compiling the wayang characters in the mahabharata story on screen are from the wayang characters that have been identified in the previous activity and have been written in the student's book. then the wayang characters are separated based on kind and evil characters and based on the groups of wayang pandhawa five and wayang kurawa. then students arrange the wayangs on the wayang screen with the arrangement pattern as determined in the student book. finally, students are asked to present their work in front of the class for discussion. purpose of activity 2 the purpose of this second activity is to encourage students to understand and identify wayang characters who have kind and evil characters, as well as five wayang characters and wayang kurawa characters, which are actually a subset of a universal set of mahabharata wayang figures. the hope is that a 'student language' will appear for subsets, namely sets whose members are included in other sets. and students can write sets and subsets in formal form and venn diagrams. student book and conjecture of activity 2 in this second activity, students are guided by the teacher using student books. this second activity begins with students observing and understanding the pattern of wayang arrangement on the wayang screen. then the students grouped the mahabharata wayangs based on their character as seen from their face shape. the students then grouped the kind wayangs and the evil wayangs and the five pandhawa wayangs and the kurawa wayangs. risdiyanti & prahmana, designing learning trajectory of set … 338 after that, students arrange the names of the wayangs on the color of the wayangs. more details can be seen in figure 3 and figure 4. figure 3. student understanding the pattern of kelir wayang and the character wayang base on the fase shape figure 4. student classifying mahabharata wayang and compose wayang in kelir screen volume 10, no 2, september 2021, pp. 331-348 339 the conjecture in the second activity consists of activities, predictions of student responses, and the responses that the teacher must give in response to the responses given by these students. more details can be seen in table 3. table 3. conjecture of activity 2 no activity predictions of student responses teacher's responses 1 students observe and understand the pattern of the arrangement of the wayangs on the kelir screen students understand the pattern of the arrangement of the wayangs on the screen the teacher gives a verbal appreciation for students students do not understand the pattern of wayang arrangement on the screen the teacher helps students to be able to understand the pattern of the arrangement of the wayangs on the screen 2 students observe the differences in the character of the wayang based on the shape of the face students understand the differences in wayang characters based on face shape the teacher gives a verbal appreciation for students students do not understand the differences in the character of the wayang based on the shape of the face the teacher helps students to be able to understand the differences in the character of the wayang based on the shape of the face by looking at the different parts of the faces of the evil and kind wayangs 3 students write the names of the evil wayang characters students can write down the names of all the evil characters of mahabharata wayangs the teacher gives a verbal appreciation for students students have not been able to write the names of all the mahabharata wayangs with kind character the teacher helps students to be able to identify the evil wayang by looking at the characteristics of its face and also the characters in the mahabharata wayang storyline in the video that has been played. 4 students write the names of the five pandhawa wayang characters students can write the names of the five pandhawa wayangs the teacher gives a verbal appreciation for students students have not been able to write the names of all the five pandhawa wayangs the teacher helps students to be able to identify the pandhawa wayang from the mahabharata wayang storyline on the video that has been played 5 students write down the names of the kurawa wayang characters students can write wayang kurawa the teacher gives a verbal appreciation for students students have not been able to write the names of all the kurawa wayangs the teacher helps students to be able to identify the wayang kurawa from the storyline of the mahabharata wayang on the video that has been played risdiyanti & prahmana, designing learning trajectory of set … 340 3.1.3. activity 3: writing the set of mahabharata wayang in this activity, students write sets in formal form, including the universal set of wayang characters in the mabaharata story, subsets, namely the set of kind mahabharata wayang figures, the evil mahabharata wayang figures, the five pandhawa wayang figures, the kurawa wayang figures that have been compiled in wayang color. students write these sets in the set column in the student book. finally, students are asked to present their work in front of the class for discussion. purpose of activity 3 this third activity aims to encourage students' understanding in writing or representing sets in a formal form. the hope is that students can represent the universal set and subsets in a formal form. student book and conjecture of activity 3 activity 3 in the student book begins with students defining sets when it is known that the results of grouping wayang characters in the mahabharata story are called sets. furthermore, students are given examples of writing sets in formal form, then students write down the results of grouping wayang or wayang sets in formal form. after that, it is known that the five pandhawa wayang sets consisting of yudihistira, bima, arjuna, nakula and sadewa are then called members of the set, students then define the members of the set using students' language. figure 5. student define set, members of set and subset volume 10, no 2, september 2021, pp. 331-348 341 then, students define subsets when it is known that all the members of the five wayang pandhawa set are also in the wayang set with kind character. then, given an example of writing subsets and how to describe them in the form of a venn diagram. students then find other subsets of the results of grouping wayang then written in a formal form and drawn in the form of a venn diagram. finally, students reflect on their learning experiences using the context of wayang and mahabharata stories, especially learning experiences other than mathematical concepts. the illustaration can be seen in figure 5 and figure 6. figure 6. student write set and subset in the formal form the conjecture in the third activity consists of activities, predictions of student responses, and the responses that the teacher must give in response to the responses given by these students. the details can be seen in table 4. table 4. conjecture of activity 3 no activity predictions of student responses teacher's responses 1 let’s think 1 students define 'set' students can define that a set as a clearly defined collection of objects, in their own language the teacher gives a verbal appreciation for students students have not been able to define a set the teacher explores students' difficulties which cause students not to be able to define sets. then the teacher ignites the creativity of students' thinking by inviting students to reflect on the activities that have been done previously. risdiyanti & prahmana, designing learning trajectory of set … 342 no activity predictions of student responses teacher's responses 2 let’s think 2 students write the setin formal form students can write sets in formal form the teacher gives a verbal appreciation for students students have not been able to write sets in formal form the teacher explores students' difficulties which cause students to not be able to write sets in formal form. then the teacher ignites the critical and creativity of students' thinking by giving examples, and then students are asked to try themselves in writing the “set” in formal form. 3 let’s think 3 students define 'set members' students can define that a member of a set as an object that is clearly defined, in their language the teacher gives a verbal appreciation for students students have not been able to define the members of the set the teacher explores students' difficulties that cause students not to be able to define members. then the teacher ignites the critical and creativity of students' thinking by inviting students to reflect on the activities that have been done previously. 4 let’s think 4 students define “subsets” students can define that a subset is a set whose members include members of other sets, with their language the teacher gives a verbal appreciation for students students have not been able to define subsets the teacher explores students' difficulties that cause students not to be able to define subsets. then the teacher ignites the critical and creativity of students' thinking by inviting students to reflect on the activities that have been done previously. 5 let’s think 5 students describe subsets in the form of a venn diagram students can describe subsets in the form of a venn diagram the teacher gives a verbal appreciation for students students have not been able to describe subsets in the form of venn diagrams the teacher explores students' difficulties which cause students to not be able to draw subsets in the form of venn diagrams. then the teacher ignites the critical and creativity of students' thinking by giving examples, and then students are asked to try themselves in writing the set-in formal form. volume 10, no 2, september 2021, pp. 331-348 343 no activity predictions of student responses teacher's responses 6 let’s think 6 students explain learning experiences obtained from the context of wayang and mahabharata stories in addition to mathematical concepts students can explain the social, moral, or cultural values contained in the wayang and mahabharata stories according to their learning experience the teacher gives a verbal appreciation for students students cannot explain learning experiences other than mathematical concepts, especially sets the teacher triggers students' critical and creative thinking by giving examples. then students are asked to try their own writing the set-in formal form. 3.2. discussion the learning design of this set uses an ethnomathematics context, namely wayang and mahabharata stories and the realistics mathematics education (rme) approach. the context of the wayang and the mahabharata story was used in the design of this study because the context is close to javanese culture, which is the culture of students, and also close to students' daily lives (risdiyanti & prahmana, 2020a; d'ambrosio, 2016). furthermore, in some performances, such as at celebrations or government cultural events, wayang and mahabharata stories are often presented to entertain and educate the public (lim, 2017; sabunga et al., 2014). seeing the problems in mathematics education where mathematics lessons are often considered a frightening specter and many students do not understand mathematical concepts because, in schools, they tend to be taught practical formulas without being explained in detail about the concept of sets and their use in everyday life. in comparison, mathematics is a human activity and must be related to culture and human daily live (freudenthal, 2016). therefore, cultural contexts such as wayang and mahabharata stories are urgent to use in the learning process (risdiyanti & prahmana, 2020b). the rme approach is also effectively used as an approach in this design because rme has the characteristics of using a real context as a starting point in learning (hadi, 2017; prahmana et al., 2020). in addition, rme effectively encourages students' activeness and creative thinking skills because rme has characteristics, namely, learning is carried out with the student center or more active students. the teacher is only a facilitator and can bring up student thinking strategies where the strategy arises from critical thinking skills and creative students (hadi, 2017; gravemeijer, 1994; sembiring, hadi, & dolk, 2008). the activity in the set learning design using the context of wayang and the mahabharata story begins with students watching an animation of a wayang show with the mahabharata story masterminded by ki seno. this activity is carried out based on the learning activities using the rme approach (hadi, 2017; prahmana, zulkardi, & hartono, 2012; sembiring et al., 2008). the first activity starts from the mode for the activity to introduce context in an abstract form, namely by how students watching the mahabharata wayang then identify all the wayang characters in the story. this activity still uses the concrete form of this context. in addition to introducing context, students are asked to create a universal set through an risdiyanti & prahmana, designing learning trajectory of set … 344 actual context in this first activity, namely wayang. then enter the second activity, namely connecting or "mode for," where students classify the wayangs based on their characters and based on their groups. after that, they compile the results of the classification on the wayang colors. finally, the formal knowledge activity defines the universal set, members of the set, and subsets and can write it down formally. in this last activity, students could not use the context of the wayang and the mahabharata story again. when students have been able to write sets in standard form and are separated from context, it means that abstracting process of the concept of the set is successful (hadi, 2017; gravemeijer, 1994). the details can be seen from the iceberg illustration in figure 7. figure 7. the iceberg of learning design of set using wayang and mahabharata story the trajectory of the set learning design using wayang and mahabharata stories, as can be seen in the iceberg in figure 7, are adjusted to the student learning flow contained in the mathematics learning curriculum made by the indonesian ministry of education and culture as well as the sequence of achievement of the competency standards included in the education curriculum in indonesia (as’ari et al., 2017). the curriculum states that students understand the concept of the universal set, then set members, then subsets and write them informal form. in addition, some of the questions regarding activity three are adjusted to indicators of critical thinking skills, namely the ability to identify and justify concepts or the ability to provide mastery of ideas, the ability to generalize, the ability to analyze algorithms (hendriana, rohaeti, & sumarmo, 2017; joyner & reys, 2000; komariyah & laili, 2018). in addition, it is also adjusted to indicators of creative ability, namely fluency, flexibility, volume 10, no 2, september 2021, pp. 331-348 345 originality, and elaboration (amidi & zahid, 2017; hendriana et al., 2017; joyner & reys, 2000). this research has an additional reference in mathematics education to complement previous studies using the ethnomathematical context and rme approach in learning mathematics. for example, the story of the wayang barathayudha war and the uno stacko to learn number patterns (risdiyanti & prahmana, 2020b), traditional indonesian games in learning number operations (prahmana et al., 2012), playing one house in learning number operations (nasrullah & zulkardi, 2011), patok lele in measuring learning (wijaya, 2008), the traditional indonesian game kubuk manuk as a stimulated starting point to understand the knowledge of social arithmetic concepts (risdayanti et al., 2019), and the gasing game in measuring learning time (jaelani, putri, & hartono, 2013), and several mathematical activities in estimating, measuring, and making patterns using sundanese culture (muhtadi et al., 2017). therefore, this study takes the role of adding context studies to be used as a starting point for learning mathematics. 4. conclusion the local contexts such as culture can use to understand the concept of sets. this study succeeded in designing the learning trajectory of the set using the rme approach with the context of wayang and mahabharata stories. the learning trajectory consists of three activities, namely watching the mahabarata wayang stories by ki seno, helping ki seno compose the mahabarata wayang on kelir screen, and writing the set of mahabharata wayang. this design allows students to rediscover the concept of sets from real and abstract contexts. it will make students understand mathematical concepts easily because it is fun for them, and most importantly, culture relates to activities in their daily lives. this research can play a role in developing the learning trajectory of the set using wayang and mahabharata stories as local learning contexts. it is also a basis to implement it in teaching experiments and analyze the result using retrospective analysis to construct the local instructional theory on set for further research. acknowledgements the authors wish to express their gratitude to the institute of research and community services, universitas ahmad dahlan, for supporting and funding this research through the penelitian tesis magister (ptm) grant under contract number ptm201/sp3/lppm-uad/vi/2021. additionally, the researcher wishes to express gratitude to the master program in mathematics education at universitas ahmad dahlan for providing the researcher with the opportunity and resources necessary to complete this research. references alangui, w. v. 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(ed.). international reflections on the netherlands didactics of mathematics (pp. 325-340). icme-13 monographs. cham: springer. https://doi.org/10.1007/978-3-030-20223-1_18 https://doi.org/10.26740/jrpipm.v4n1.p10-22 https://doi.org/10.1088/1742-6596/943/1/012032 https://doi.org/10.22342/jme.11.1.10225.157-166 https://doi.org/10.17051/ilkonline.2019.639439 https://doi.org/10.1007/978-3-319-30120-4_3 https://doi.org/10.1007/s11858-008-0125-9 https://doi.org/10.22342/jpm.1.2.807. https://doi.org/10.22342/jme.10.3.7611.341-356 https://doi.org/10.1007/978-3-030-20223-1_18 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p103-114 103 the lakatosian methodology in teaching the surface area of a cone and a student’s conceptual transition ugorji iheanachor ogbonnaya1*, chrysoula dimitriou-hadjichristou2 1university of pretoria, south africa 2ministry of education cyprus article info abstract article history: received oct 15, 2021 revised jan 4, 2022 accepted jan 10, 2022 although many scholars support the lakatosian method of mathematics education for enhancing students learning, the use of the method to foster students’ transition from alternative to scientific conceptions in mathematics does not seem to have been a focus of any research. this study explored if the lakatosian heuristic method of teaching could foster a students’ transition from alternative to scientific conceptions of the surface area of a cone. the study used a qualitative, exploratory single case study design to undertake an in-depth study of a student in the 11th grade at a secondary school in cyprus. data was collected through lesson observation and analyzed using deductive content analysis. the study found that beginning from informal conjectures and proofs to more formal proofs the student was able to transit from alternative conception to the scientific conception of the surface area of a cone. the finding suggests that the lakatosian method of mathematics can foster students’ conceptual understanding of some mathematical concepts. keywords: conceptual change, cone, cyprus secondary school, deductive proof, euclidean geometry, lakatosian heuristic, surface area of a cone this is an open access article under the cc by-sa license. corresponding author: ugorji i. ogbonnaya, department of science, mathematics and technology education, university of pretoria groenkloof campus, corner of george storrar drive and leyds street, groenkloof, pretoria, south africa email: ugorji.ogbonnaya@up.ac.za how to cite: ogbonnaya, u., i., & dimitriou-hadjichristou, c. (2022). the lakatosian methodology in teaching the surface area of a cone and a student’s conceptual transition. infinity, 11(1), 103-114. 1. introduction students often have a misconception about the surface area of a cone. some of them fail to see the relationship between the construction/deconstruction of a cone from a 2-d shape to a 3-d shape and vice versa. the misconceptions could be because euclidean geometry, in general, is taught through the derivation of theorems from axioms in a deductive formalised logical approach. in contrast to the deductive approach, lakatos' heuristic approach to mathematics education advocates the teaching and learning of mathematics through proofs and refutations. lakatos’ philosophy of mathematics tried to make a distinction between the euclidean theories and the ‘quasi-empiricists’ theories. according to lakatos (1978), “euclidean heuristic separates the process of finding the truth and of https://doi.org/10.22460/infinity.v11i1.p103-114 https://creativecommons.org/licenses/by-sa/4.0/ ogbonnaya & dimitriou-hadjichristou, the lakatosian methodology in teaching the surface … 104 proving it” (p.72). he claimed that the discovery of the truth has an element of guessing the necessary axioms or the appropriate statements (that have already been proved), from which we can start the process of deductive proof. the idea of “improving by proving” (lakatos, 1976, p. 37) that is, demonstrating a conjecture via a series of gradual improvements/revisions of proofs and refutations never occurred in the euclidean system. to support his opinion, lakatos (1978) claimed that: “the greeks did not find a process of decision for their geometry though they dreamt of one. however, they found a compromising solution: a heuristic procedure, which does not always produce the desired result, but which is still a heuristic rule, a standard pattern of the logic of discovery. this heuristic method was the method of analysis-synthesis”. (p. 72) according to lakatos (1978), a quasi-empiricist theory as a heuristic method is a method of analysis and synthesis, which he referred to as a rule. the analysis-synthesis rule consists of three parts: (a) make your conjecture, one after the other, assuming that it is true. if you reach a false conclusion, then your conjecture was false, (b) if you reach an indubitably true conclusion, your conjecture may have been true, and (c) in the case of (b), reverse the process, work backward, and try to deduce your original conjecture via the reverse route from the indubitable truth to the dubitable conjecture. if you succeed you have proved your conjecture (lakatos, 1978). lakatos called the first two parts (a and b) ‘analysis’ and the third part (c) ‘synthesis’. lakatos strongly criticised the deductivist approach in mathematics. in the deductivist approach, definitions, axioms, and theorem statements are presented with no explanation about their development and are considered to be eternal, immutable truths (pease et al., 2004). the deductivist approach “is constituted by a system of apodictically certain, a priori grounded euclidean axioms, lemmas, and definitions from which theorems are deductively derived and thereby also secured as certainties” (shaffer, 2015, p.1). the lakatos (1976) heuristic method is based on proofs and refutations as a method for the discovery of mathematical knowledge. discovery is seen as a method of acquiring knowledge (cellucci, 2020). to lakatos, mathematics resembles natural science (ravn & skovsmose, 2019). his theory is that mathematics, “like the natural sciences, is fallible, not indubitable; it grows by the criticism and correction of theories, which are never entirely free of ambiguity or the possibility of error or oversight. starting from a problem or a conjecture, there is a simultaneous search for proofs and counterexamples. new proofs explain old counterexamples, while new counterexamples undermine old proofs” (dimitriouhadjichristou & ogbonnaya, 2015, p. 186). lakatosian heuristic is characterized by thought (mental) experiment (quasiexperiment) (morales-carballo et al., 2018). thought experiment “suggests a decomposition of the original conjecture into sub-conjectures or lemmas, thus embedding it in a possibly quite discrete body of knowledge” (lakatos, 1976, p. 9). thought experiment (deiknymi, ‘δείκνυμι’) was the most ancient pattern of mathematical proof. it prevailed in pre-euclidean greek mathematics according to lakatos (1976). “lakatos considers informal proof as just another name for thought-experiment” (motterlini, 2002, p. 27). in the heuristic methods, the teacher of mathematics, through proper activities and experiences, uses a combination of guided methods to help the students to become involved in ‘the research processes’ of finding the truth and proving it. the help/guidance is gradually minimized so that the student becomes autonomous, through continuous proof and refutation of hypotheses/conjectures. this process is adjusted by the teacher with suitable alternating questions arising from either criticism or refutation of hypotheses/conjectures, leading to the reconstruction of the initial hypotheses/conjectures in the light of new counterexamples, by the logic of proofs and refutations (lakatos, 1976). thus, the method is based on the quasi– volume 11, no 1, february 2022, pp. 103-114 105 empirical system where the typical flow of the process is to bring “lies” back from the false “basic sentences” or “basic statements” (popper, 1959, p. 78) in a down-up direction of the original hypothesis (shaffer, 2015). for students to transit from their alternative to the correct (scientific) conceptions, it is essential to provide alternative views that contradict their previous thinking. students’ concepts that do not correspond with the consensus view of the scientific community are the students’ alternative conceptions (weissová & prokop, 2020). these alternative conceptions are not considered wrong, but rather regarded as models; perhaps in the same sense as used by scientists to simplify the complexity of a problem (laburú & niaz, 2002). studies have shown that conceptual change could be difficult to achieve (bofferding, 2018; lehtinen et al., 2020; vamvakoussi, 2017). according to chinn and brewer (1993), students resist changes in their core beliefs (cf. ‘hard core’ as lakatos (1970) puts it), more strongly than they resist change in the more peripheral aspects of a subject (laburú & niaz, 2002). for this reason, students look for an auxiliary hypothesis to defend their core beliefs. the new/alternative view must appear initially plausible to the students. auxiliary hypotheses used by students to defend their core beliefs may provide clues and guidance for the construction of novel teaching strategies. this is based on the lakatosian thesis that scientists do not abandon a theory based on contradictory evidence alone, and that ‘there is no falsification before the emergence of a better theory (lakatos, 1970, p. 119). hidden lemma (lakatos, 1976) or “guilty lemma” was the one whose replacement leads to the most progressive problem-shift (motterlini, 2002, p. 12). the modification of a theorem is stopped either when no more counterexamples can be found or when the theory has proved a conjecture (pease et al., 2004, p. 13). euclidean geometry is taught deductively in a “proof scheme” (de villiers, 2012, p. 1) in cypriot secondary schools based on “accepted truths” stylianides (2010, p. 41), which is in line with the rationalist epistemological tradition. this approach characterises the euclidean theory (shaffer, 2015), and it does not allow for the development of critical/creative thinking in mathematics. according to sriraman and mousoulides (2020), “lakatos makes the point that this sort of euclidean methodology is detrimental to the explanatory spirit of mathematics; it can also ignore the needs of students as they learn argumentation that constitutes a proof” (p. 2). the educational system in cyprus is based on the principles of encyclopedism which promoted teacher-centric methods. according to persianis (1998) (cited in karagiorgi & symeou, 2006, p. 3), “the greek educational system, was influenced by the french system with its underlying epistemological tradition of encyclopaedism and its extensive centralization and uniformity”. the deductive method of teaching euclidean geometry unlike a reliance on intuition and argumentation in mathematics promotes a lecturing method, which encourages the avoidance of in-depth discussion of the whys (european evaluation committee of the pre-service programme in cyprus, 2009). from the second author’s interaction with students as a curriculum specialist and supervisor of mathematics teaching and learning in cypriot secondary schools, she found that geometry, in general, constitutes considerable difficulties for students. she observed that many students have a misconception about the surface area of a cone because they fail to see the relationship between the construction/deconstruction of a cone from a 2-d shape to a 3d shape and vice versa. the objective of this study was to explore if the lakatosian heuristic method of teaching could foster a students’ transition from alternative to scientific conceptions of surface area of a cone. the findings of this study will advance our knowledge of the lakatosian heuristic method of teaching proofs and refutations and how it promotes students’ conceptual learning of mathematical concepts. ogbonnaya & dimitriou-hadjichristou, the lakatosian methodology in teaching the surface … 106 2. method the study employed a qualitative exploratory single case study design (merriam & tisdell, 2016) this was to gain an in-depth insight into how the lakatosian heuristic method can foster a student’s transition from alternative to scientific conception of the surface area of a cone. a qualitative case study helps researchers to explore a phenomenon in a context (rashid et al., 2019). this study is part of a larger study on the lakatosian method of mathematics teaching in cypriot secondary schools conducted in 2015. in this study, the lakatosian method was used to teach the sac to 11th grade students at a secondary school. data was collected through lesson observation. the data for this study was from a recorded dialog between the teacher (the second author) and a student identified in this paper by the pseudonym alexa.. alexa was purposively and conveniently selected for the study because she was a low achiever in mathematics and lacked interest in learning mathematics. in addition, she showed misconception about the surface area of a cone and she consented that the recorded conversation she had with the teacher during the lesson be analysed in this study. therefore, she was found suitable for the study. in the teaching, the teacher posed the following questions to the class. it was requested that a cone-shaped tall hat be made for the junior school carnival show. circle only one of the following shapes that is the proper one to be used for the model of the cone hat. data presentation involved a verbatim record of the transcribed conversation between the teacher and alexa. deductive content analysis (bass & semetko, 2021) employing laburú and niaz’s (2002) model as the reference, was used to analyse the data to make sense of the student’s transition from the alternative conception to the scientific conception of the sac. figure 1. task about the construction of the sac volume 11, no 1, february 2022, pp. 103-114 107 3. results and discussion 3.1. results from the dialogue between the teacher and alexa, we explored alexa’s transition from her alternative to scientific conceptions of a cone. the transition was analysed and presented in three phases according to laburú and niaz’s (2002) model: from the alternative model (am) to transitory model (tm) and finally to the scientific model (sm). laburú and niaz’s (2002) model was based on the lakatosian framework that one learns not by accepting or rejecting one single theory but by comparing one theory with another for theoretical, empirical, and heuristic progress (lakatos, 1976). alternative model (am) teacher : looking at the question, which one of the four options do you think is the correct one? alexa : the circle. the circle because i can see only the base of the shape, i don’t know how to see the shape in 3-d. i can see it only as its base because the cone has a circular base. teacher : how can it be a cone? alexa : i can hold it up from the point o, the centre of the circle, to be a cone. (the teacher was surprised. if alexa had a rubber material, she could similarly think of how lakatos (1976, p. 7) explained in his utopian class the development of a thin rubber cube in a flat network to prove the formula f+v=e+2). teacher : hm…what do you think about the triangle (figure 1c)? alexa : i think that i see an ‘empty space’ from whichever angle i look at the cone there is a triangle. so, the triangle may be the answer not because this one forms a cone but because i can see it (triangle) inside a cone. (she refuted immediately her first answer about the circle). teacher : a triangle inside a cone? transitory model (tm) alexa : yes! like the traffic light cone in the road. it’s a triangle above a circle. teacher : please draw what you mean. (nb: she drew an isosceles triangle above a circle as an example of what she could “mentally see” (hersh (1978). the shape was exactly the same as what apollonius of perga did and explained why students think like that which is be discussed later in this paper). (nb: she was thinking silently by putting her hands together as a closed shape in 3-d and her hand in her attempt to ‘see’ the new shape in 2-d. by opening and closing her hands she realized how to construct or deconstruct a cone. she tried to convince herself, speaking aloud, that a circle and a triangle do not construct a cone. however, she was confused about what the true answer was. her hands were reacted as a tool of a heuristic method). teacher : how about the other two shapes? alexa : shape a may be transformed into an ‘empty space’ (nb: she means a ‘funnel’ cone to be a hat) but no….it must have a base to stand on like a triangle (nb: she refuted herself again returning to a triangle as the correct answer. she still worked by her hands when she saw the sector as a possible answer. however, she refuted that. finally, the teacher asked her to use a piece of paper to check if a right-angle triangle (figure 1c) could form a cone hat. after many trials and errors, she folded the two edges h and e of a triangle. she was very surprised by the resulting extra paper which prevented the cone hat from standing on its base. she confirmed that a triangle could not construct a cone). ogbonnaya & dimitriou-hadjichristou, the lakatosian methodology in teaching the surface … 108 teacher : how about the shapes a and b? what’s the difference between them? alexa : yes! shape b can also be a cone! and it can stand because i think it has a base to stand on! teacher : what do you mean? alexa : its material (nb: area of a sector) is more than the shape a, b can stand on its base! if i connect point b with point γ of shape a there will be a cone! the same applies to shape b if i connect point δ and point ζ there will be a cone too. i mean if i connect their radius! teacher : excellent! which one will be the tallest? (nb: the teacher believed that alexa comprehended the concept believing that she could find the tallest cone. however, alexa was wrong again. she thought that the radius of a sector in shape a seems to be greater than that in shape b). alexa : shape a is the tallest cone because it depends on its radius. teacher : good. which radius? (nb: unfortunately, alexa means the sector radius which was the same not the base circle radius of a cone) alexa : it is αγ or αβ of the shape a, and in the shape b, it is any point of the circle to the centre o. teacher : these radii are equal as the sectors were cut from the same circle to form a cone hat! (nb: however, the teacher realized from her previous answer that alexa was pseudoconceptualizing (vinner, 1997)). alexa refutes her thinking once more unable to find a new counterexample while she could not improve her proof. however, she was insisting to find the true answer. scientific model (sm) alexa : the shape b has more material (nb: she means that the sector/lateral area when it is folded forms a “funnel” cone with circle base) to form a base, which will be quite wide, and the cone can stand on its base. she demonstrates it by forming a cone using her hand as well as a piece of paper) so it won’t be so tall compared to the shape a. however, because, for the shape a there is less material (i.e., sector/lateral area) to create a cone, it forces me to bring the two sides, i.e. the radii, closer together. this makes the cone (a) taller (nb: the paper hat works as a heuristic tool helping her to change her alternative conceptions about the correct shape used to construct a cone hat). teacher : so? alexa : shape b fulfills the criteria, but it is not tall. teacher : what do you mean? alexa : it is not as tall as a shape a. teacher : what bothers you about the shape (a)? alexa : the idea that i have in my mind is that shape a will not be able to stand as a shape b will. teacher : so, let’s go back to the initial concern, of how it will stand. alexa : if i had a piece of paper (nb: means in her maths lessons) i could cut it to see the shape as it would actually be in reality. it would answer all of my questions. but i feel i would be cheating the process of learning because i feel i ought to be able to come up with the answer by looking at the question without having to construct it using a piece of paper (nb: she was influenced from the traditional teaching method where students cannot use the heuristic tools at all). teacher : why do you think it is wrong to use a piece of paper? volume 11, no 1, february 2022, pp. 103-114 109 alexa : because i feel that as a student, i am expected to have the knowledge and the ability to come up with the answer by just looking at the shape. teacher : is it embarrassing to use a piece of paper to construct it? alexa : no, not embarrassing but it is like cheating–it is like taking a test and having an open book to look for the answers (nb: she was also influenced by the traditional teaching method where students have to learn by watching their teachers’ lecturing concentrating only on the blackboard). 3.2. discussion alexa felt incapable (in her words ‘stupid’) due to her need to ‘see’ it experimentally to reason about the problem, and she thought that mathematical thinking was an abstract thinking capability that she did not possess. as the teacher and alexa discussed the construction /deconstruction of a cone, alexa was initially not sure of the correct answer though she guessed the correct answer twice she could not support it. her pseudoconceptualization (vinner, 1997), when she said shape a is the tallest because it depends on its radius (transitory model) could have been a hindrance to her understanding of the teacher had not asked her to explain the radius she meant. she was unable to see that the radius of a sector was equal to the lateral height of the cone. however, due to the ‘improving by proving’ process of the lakatosian method she could see the hidden lemma that the tallest cone would be made from the sector the less sector material (the smaller surface area). though she was not able to prove the sac, she changed her alternative conception to the scientific concept of a cone through the construct/deconstruct of a cone. by folding a sector paper, she discovered a “key” that helped her to see the relationship between the two shapes, such as the radius of a sector is the same as the lateral height of a cone. in line with laburú and niaz’s (2002) hypothesis, alexa's conceptual transition progressed from alternative model (am) to transitory model (tm), and scientific model (sm). the am reflects alexa’s alternative conceptions of a cone. alexa held strong core beliefs about the shapes in 2-d corresponding to a cone in 3-d, based on the reasoning that when she looks at a cone, she can see only the base of it or a triangle inside a cone. she believed that there is a space inside the cone, and she could not grasp how to transform a 2d shape to a 3-d shape and vice versa, to construct/deconstruct a cone. this misconception could be because of how the students were taught the concept of a cone traditionally. by resisting change in her hard-core, alexa attempted to invent an auxiliary hypothesis to support her belief. she had a “mental model” (hersh, 1978) that a right-angle triangle creates a cone in 3-d when it is rotated about one of its vertical sides, in the alternative model (laburú & niaz, 2002) she could not imagine how the sector constructs a cone. according to laburú and niaz (2002), scientists generally show resistance to changes in the hardcore of their research programmes by postulating auxiliary hypotheses. guidance on what is to be done in the face of anomalies is provided by the positive heuristic of the programme which provides suggestions for developing the refutable parts of the research programme. in confronting anomalies, alexa did her best to comprehend how the cone can stand on a plane: she moved to the tm. she came up with arguments, such as if i connect b with γ of shape a there will be a cone! the same applies to shape b if i connect δ and ζ there will be a cone. i mean their radius! to support her alternative hypothesis, just as the scientists built a protective belt to defend the hardcore of their research programs (lakatos, 1970). in the face of anomalies, the positive heuristic was provided, enabling scientists to build models by ignoring “the actual counter-examples, the available data” (lakatos, 1970, p. 135, original emphasis). the positive heuristic in this research programme is alexa’s understanding (by using experimental methods) that both shapes a and b are cones and can stand on a plane independently of their sector material, helping her to refine her theories into ogbonnaya & dimitriou-hadjichristou, the lakatosian methodology in teaching the surface … 110 the sm. however, she found it difficult to understand that the height of the cone depends on its base radius. as a result, she was unable to understand the relations between the two spaces e.g., the tallest cone depends on the length of the arc of its sector and also the angle of this sector. the counterexample was posed to alexa experimentally, helping her to refute her core belief when she realized that a triangle (figure 1c) could not form a cone. she changed her initial conception (core belief) that the sector of a circle cannot be constructed into a cone, by posing the counterexample that an extra material (meant the segment) prevented it from standing on a base level. by cutting and folding a piece of paper she was led to the ‘new knowledge’ that if i connect point b and γ of the shape a that will form a cone. she understood that a sector constructs an open cone, she realized that was its sac in the scientific model. even though, she was unable to prove the sac’s formula she gained important experiences during the thought experiment. lakatos (1970, p. 133) emphasized that “the hard-core of a program itself develops slowly by a long preliminary process of trial and error and does not emerge fully armed like athena from the head of zeus”. in addition, according to niaz (1998), “just as a hardcore of students’ beliefs is constructed slowly any change perhaps will also follow a similar process” (p. 123). this line of thought explains why after she achieved the sm, insisted that the shape b fulfills the criterion (that the shape can be transformed into a cone), but it is not tall. she was resistant to changing her hard-core concept and insisted that the cone cannot stand on a plane, hence returning to the tm, reluctant to give up all the elements of the am. therefore, she built a protective belt to defend the hardcore of her ‘research programme’ by asserting that she would need a piece of paper to construct the cone. she claimed that she could not imagine a solid cone that was represented by one of the proposed shapes; hence she would try to construct it using a piece of paper. she still reflected on her am by saying …the first thing that comes to my mind is the two-dimensional shapes… or if i were to draw a cone using a pencil, i would draw a triangle and a circle as a base. according to laburú and niaz (2002), the above outline of alexa’s thinking process confirms that at this stage (sm) “even if a student constructs it, it cannot be claimed that a change in the hard-core of students’ understanding has been achieved” (p. 217). 3.3. implications of the findings it is clear in teaching and learning the concept of a cone that teachers must spend time with their students to construct the definition by their own words by using heuristic tools such as the cone hat to see the relationship between the solid cone’s construction/deconstruction and the creation of its surfaces. according to the lakatosian method they could use heuristic tools that can help students to resolve their misconceptions. in contrast, the visual representation of the rotation of the right-angle triangle about one of its vertical sides, on the whiteboard traditionally, prevents them from comprehending a cone in 3 dimensions. as wadhwa et al. (2020) observed, it is necessary for teachers to explore students’ alternative conceptions because it helps teachers to understand the students’ thinking and the subtle source of their conceptions. students’ core belief (cf. lakatos, 1970, ‘negative heuristic’) is that the sac in 2-d is a right-angle triangle. another misconception is how they draw the cone in 3-d as a plane isosceles triangle above a circle. teachers must pay attention to this in teaching. appolonius of perga explains this misunderstanding by constructing a cone from 3-d to 2-d, explaining to us why a cone in 2-d is constructed as a circle and a triangle above it. according to flaumenhaft as cited in densmore (2010), apollonious asserted in his proposition 3 that “if a cone is cut by a plane through the vertex, volume 11, no 1, february 2022, pp. 103-114 111 the section is a triangle” (p. 6). apollonius was engaged in a study of lines obtained by the intersection of a cross-section of a cone, which is neither straight nor circular but closed. the difficulty that alexa encountered, and other students in general, in constructing or deconstructing a cone in 3-d may be due to the new kind of lines which is generated by putting together the straight line and the circle to generate a conic surface–a surface that is neither flat nor spherical; and if we cut that curvy surface with a plane surface, we shall get (as the intersection of the surfaces) a kind of a curvy line (densmore, 2010, p. xxviii). apollonius as cited in densmore (2010), examined this new kind of line concerning the cone surface area. in his first proposition, he showed that a conic surface is not wiggly in any direction: the straight lines drawn from the vertex of the conic surface to points on the surface are on that surface (densmore, 2010, proposition 1). therefore, the locus of the points of a cone which are on the straight line drawn from the vertex to the base circle of a cone when they are rotated about their axis forms the sac. apollonius also showed that a conic surface is like a circle in being everywhere curvy bulging outward (if you go from a point on it toward another point, but without going straight toward its vertex) [proposition 2]. then he explained in his third proposition (figure 2) that (by cutting through the vertex) one obtains a conic section that is straight-lined (being a triangle) and another circular conic section (by cutting to make an angle equal to one of the base angles: parallel [prop. 4] or sub contrariwise [prop. 5] (densmore, 2010, p. xxix). figure 2. apollonius proposition 3 (densmore, 2010, p. 7) the new kind of ‘mixed lines’, as shown in figure 2, is what exactly alexa created in her mind to imagine a cone in a plane: the first thing that comes to my mind is the two dimensional shapes my mind perceives when looking at a cone, just as looking at a cone in the street. for example, a circle (as a base) and a triangle (isosceles) above it. this misunderstanding (mixed-lines in 2-d) might have prevented her from realizing that a cone in 3-d is represented in 2-d, not as a cross-section of a cone by a new line together with a triangle and a circle, but as a sector of a circle. this is a result of a conceptual-embodiment (tall, 2013) based on students’ as well as other people’s perception of the cone which can be overcome by using an experimental (down-up) method in teaching and learning, such as the lakatosian method. it helps students not only to change their core beliefs but also to contribute to the understanding of the concept, and to develop problem-solving skills due to developing higher-order thinking as this study has demonstrated. ogbonnaya & dimitriou-hadjichristou, the lakatosian methodology in teaching the surface … 112 4. conclusion students come to the classroom with knowledge or conceptions that they acquired from previous learnings and experiences. sometimes the conceptions the students hold might be erroneous (misconceptions). in mathematics, many students have a misconception about the surface area of a cone. some of them do not recognise the relationship between the construction/deconstruction of a cone from a 2-d shape to a 3-d shape and vice versa. this case study explored how the lakatosian heuristic method of teaching could enhance students’ transition from alternative conceptions to scientific concepts of a cone. the study showed that through the lakatos proof and refutation approach the student was able to transit from her alternative conception to scientific concepts of the surface area of a cone. this study can be used as a reference for further studies on the effectiveness of the lakatosian methodology in teaching mathematics. references bass, l., & semetko, h. a. 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(2013). how humans learn to think mathematically: exploring the three worlds of mathematics. cambridge university press. vamvakoussi, x. (2017). using analogies to facilitate conceptual change in mathematics learning. zdm mathematics education, 49(4), 497-507. https://doi.org/10.1007/s11858-017-0857-5 vinner, s. (1997). the pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. educational studies in mathematics, 34(2), 97-129. https://doi.org/10.1023/a:1002998529016 wadhwa, m., gahlawat, i. n., & chhikara, a. (2020). understanding students’ alternative conceptions: a mirror to their thinking. integrated journal of social sciences, 7(1), 9-13. weissová, m., & prokop, p. (2020). alternative conceptions of obesity and perception of obese people amongst children. journal of biological education, 54(5), 463-475. https://doi.org/10.1080/00219266.2019.1609549 https://doi.org/10.1007/978-3-030-15789-0_131 https://doi.org/10.1007/s11858-017-0857-5 https://doi.org/10.1023/a:1002998529016 https://doi.org/10.1080/00219266.2019.1609549 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p13-26 13 integrating peer tutoring video with flipped classroom in online statistics course to improve learning outcomes ramadoni1*, kao tai chien2 1universitas pgri sumatera barat, indonesia 2national dong hwa university, taiwan article info abstract article history: received dec 15, 2021 revised nov 21, 2022 accepted feb 22, 2023 online learning has become a solution in the field of education lately. statistics is one of the subjects students must take at the university level. learning statistics takes work for students. based on the author's investigation, there are three obstacles to students in online learning: the constraints of students in understanding the material, the online learning process, and the assignment process. the sample in this study is a student with an indonesian worker background in taiwan, where they study online at night. since an appropriate online learning method is needed to achieve student success, this study analyzes three online learning methods: conventional online learning, conventional flipping classes in online learning, and innovative flipped classrooms in online learning. this study investigates the three learning methods' results in differences in gender, job, and age. the results obtained indicate that there are significant differences in student learning outcomes in the three sample groups. further analysis showed that the innovation of flipped classrooms in online learning significantly differs from the other two learning methods. based on the type of job, there are differences in student learning outcomes taught in the conventional flipped classroom in online learning. it was also seen in the analysis of combining all student learning outcomes taught with online learning that there were differences in student learning outcomes regarding the job. a household assistant is better than a factory worker because students who work as household assistants have more flexible time to watch videos and repeat them than those who work as factory workers. keywords: flipped classroom, online learning, statistics this is an open access article under the cc by-sa license. corresponding author: ramadoni, department of mathematics education, universitas pgri sumatera barat jl. gunung pangilun, padang, west sumatra 25111, indonesia. email: ramadoni.100393@gmail.com how to cite: ramadoni, r., & chien, k. t. (2023). integrating peer tutoring video with flipped classroom in online statistics course to improve learning outcomes. infinity, 12(1), 13-26. 1. introduction online learning has become one of the solutions for many universities in various countries. indonesia has legalized the distance class. by legally acknowledging online-based learning in indonesia has many positive impacts on many students who have financial, time, https://doi.org/10.22460/infinity.v12i1.p13-26 https://creativecommons.org/licenses/by-sa/4.0/ ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 14 and other limitations. this online learning can make it easier for students to learn. they don't need to come into class to learn, and they can know wherever they are. online learning is beneficial for many students who wish to continue to college but have limited costs, and they can work while in college. as in taiwan, many indonesian workers still have a high determination to continue their studies here. most of them work as household assistants and workers in factories. this research needs to be conducted to determine the best learning method for overcoming students' online learning problems (learning outcomes). the students' issues include understanding the material, issues in the online learning process, and assignment problems. researchers try to use flipped classroom learning because it is considered capable of helping students who do not have enough time and are in different places. while on the other hand, they all have a high motivation to get a bachelor's degree. online-based learning still has obstacles for students in understanding statistical material. ramadoni and hafiz (2022) said that many students have difficulty learning statistics online in terms of topics, processes, and assignments. another study conducted by zusmelia and ramadoni (2017) in the city of padang also revealed that mathematics learning was not liked by 70% of junior high school, senior high school, and university level students. many students who do not like learning mathematics make many researchers who conduct experiments to solve this problem. since we know that learning mathematics is essential to understand by all students at every level, particularly in statistical subjects, these subjects are mandatory for all undergraduate students in indonesia. this knowledge is beneficial for research that students must do to complete their degrees. this is in line with research by zheng et al. (2020), which revealed that three factors that influence online learning are students, instructors, and courses. and a study conducted by viano (2018) also revealed that there are factors that influence the success of online learning, namely: learning materials, technological skills, skills for learning. this is the background of researchers conducting experiments on taiwan open university students using flipped classroom in online learning. flipped classroom learning is an exchange of learning process between in class and out class time. students will learn basic knowledge by themselves in the out-class, while in-class stage, students will focus on classroom interaction (bergmann & sams, 2012). several other studies also revealed about online learning flipped classroom conducted by wang (2019) maintains that flipped classroom learning can facilitate learning that is easily arranged by students, increase student involvement in learning and improve student learning outcomes. sojayapan and khlaisang (2020) said that group learning using the flipped classroom method could improve student learning outcomes. zhu et al. (2020) said that flipped classrooms use shows promising results for the improvement of student learning outcomes and independent learning abilities. murillo-zamorano et al. (2019) maintains that flipped classrooms positively impacted students' knowledge, skills, involvement, and satisfaction in learning (wilson et al., 2019). this study says no significant difference between flipped classrooms and didactic methods with active learning. flipped classroom learning's weakness is students' uncontrolled learning in mathematics, especially at the learning stage outside the classrooms (lo et al., 2018). that is why researchers want integrated peer-tutoring video activities. wang (2017) reveal that a good learning management system in online learning flipped classrooms will improve learning outcomes. therefore, in this study, a design of three learning models was carried out. in this study, three sample groups were taken. the first class is taught using online-based learning in a conventional manner. conventional classes are taught like how online teachers teach as usual, where the teacher provides a complete explanation of the material. the second class is taught using online-based learning by applying a conventional flipped classroom. conventional flipped classroom where the volume 12, no 1, february 2023, pp. 13-26 15 teacher provides an explanation video before class and discusses statistical questions in class. the third class is conducted using online-based learning by applying learning innovations to the flipped classroom method. the innovations carried out in the innovation flipped classroom are group learning, students make video tutoring, online discussions, learning using skype, class presentations and discussion questions. 2. method 2.1. setting and participants in this study, three groups of samples were performed. the first class is taught using conventional online learning (col). the second class is taught using flipped classroom in online learning (cfcol). the third class is conducted using innovations flipped classroom in online learning (ifcol). this research was conducted in september-december 2018 for online-based learning in the conventional way of 39 students, february-may 2019 for onlinebased learning by applying conventional flipped classrooms to 21 students, and septemberdecember 2019 for online-based learning by using learning innovations on the flipped classroom method for 26 students. all students in online learning are indonesian workers in taiwan. sometimes, students cannot be too focused on learning when online because some are still working. although the time taken to study is at 22.00-24.00 taiwan time in each meeting, some people are still working. so, the solution that can solve it is in groups. with groups, students can develop their abilities, utilize the time they feel is right to learn together outside learning, and help each other in understanding the material. online learning using skype application is held every week nine times and face-to-face in class twice in taipei. at this learning stage, the focus is on online learning design, where classroom learning means online learning using skype and other steps before and after learning using skype. 2.2. experimental design this study has a learning design that is described in the experimental design are seen in figure 1 as follows. figure 1. experimental design the name of experimental research is post-test only group design. in this study, it is assumed that all students have the same ability because all students in the three sample groups have never taken statistics courses, and all students are freshmen. in this study, the students' learning outcomes were taken, and differences were seen based on age, gender, and job. this test was carried out against age because online learning students' age variation ranged from 20-42 years. this research examines gender because management students taught online have gender differences that are considered to have differences in learning ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 16 outcomes and learning abilities. this study also challenges in terms of jobs because the online-based learning process and work in taiwan have different and uncertain times with each other, so researchers consider testing the differences in work important. 2.3. instruments in online learning, there is some equipment that students must have in learning provided by the university is skype application. while equipment that must be provided by students themselves in learning is laptops, smartphones, internet, textbooks, and worksheets. while some of the equipment added by researchers are videos, power points, student assessment sheets (learning outcomes, assignments, performance, activity, etc.). the final test questions given consist of 7 questions in essay form. 2.4. research procedures next explain the research procedures of learning for each method used in three classes: a. online based learning in a conventional method. 1) classroom online learning a) explanation of the objectives and material discussed. b) explanation of material by the teacher. c) provide opportunities for students to ask questions. d) practice solves questions in textbooks in groups. 2) after learning in online learning class a) students doing homework. b) online discussion using social media. b. online-based learning by applying conventional flipped classroom. 1) before learning in online learning class a) students watch a video provided by the teacher. b) students make 5 important points from videos and one question. c) online discussion using social media. 2) classroom online learning a) explanation of the objectives and material discussed. b) explanation of important points by the teacher. c) provide opportunities for students to ask questions. d) practice solves questions in textbooks in groups. c. online-based learning by applying learning innovations to the flipped classroom method. 1) before learning in online learning class a) group discussion: the teacher divides students into eight groups. b) students make a short video about the explanation of the material in each section (see figure 2). volume 12, no 1, february 2023, pp. 13-26 17 figure 2. students recording explanatory videos in a variety of ways c) the group provide 5 statistics questions with the answer. d) online discussion: each student uploads their part explanation video in the class (see figure 3). figure 3. student online discussion of less understood learning e) students’ feedback: other groups must watch videos, make important points in their notebooks, and give comment about their classmate’s videos (see figure 4). ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 18 figure 4. students feedback from classmates videos 2) classroom online learning a) teacher explained important parts that need to be added and emphasized (see figure 5). figure 5. teacher's explanation of important things in online learning (face-to-face online) b) teacher opens the opportunity for students to convey their questions about the topic of learning. c) the presentation group provide 5 statistics questions given to other groups to discuss and practice solve questions in textbooks in groups (see figure 6). volume 12, no 1, february 2023, pp. 13-26 19 figure 6. students in group presentations discussing questions 3. result and discussion 3.1. result 3.1.1. learning outcomes of the three methods are not same based on the analysis conducted using spss with the anova test, the results obtained are seen in table 1 as follows. table 1. comparison of three methods of learning in online learning n ms f p method 1 (col) 38 78.21 0.062 0.016* method 2 (cfcol) 19 75.95 method 3 (ifcol) 24 84.79 total 81 79.63 * p< 0.05, ** p< 0.01. based on the table 1, we can see that the three learning methods are significantly different. in other words, ho is rejected, then there is a significant difference between conventional online learning, conventional online learning of flipped classroom, and online innovation learning of flipped classroom with α = 0.016*. after that, the post hoc test is performed to see the differences that occur between classes, the results obtained as shown in the following table 2. table 2. post hoc test to see the comparison of each method used method 1 (col) method 2 (cfcol) method 3 (ifcol) method 1 (col) 1.000 method 2 (cfcol) 0.447 1.000 method 3 (ifcol) 0.019* 0.008** 1.000 * p< 0.05, ** p< 0.01. based on the table 2, we can see no significant difference between conventional online learning and conventional online learning of flipped classrooms with α = 0.447. while learning by using online innovation, learning of flipped classroom is significantly different ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 20 from the other two methods (conventional online learning and conventional online learning of flipped classroom) with α = 0.019** and α = 0.008**, respectively. 3.1.2. the learning outcomes of the three methods are not the same in terms of gender based on the analysis conducted using spss with the one-way anova test, the results obtained are seen in table 3 as follows. table 3. differences in learning outcomes of the three methods used seen from gender gender method 1: col method 2: cfcol method 3: ifcol n ms n ms n ms male 12 75.50 2 48.50 4 80.75 female 26 79.46 17 79.18 20 85.60 total 38 78.21 19 75.95 24 84.79 gender method 1: col method 2: cfcol method 3: ifcol f 0.558 0.000 0.725 p 0.275 0.003 0.113 * p< 0.05, ** p< 0.01. table 3 show that in method one, the number of males is 12 people and female is 26 people, with an average of 75.50 and 79.46 respectively. the data obtained by gender in method 1 is homogeneous. there is no significant difference between males and females by using method 1 with α = 0.275. whereas in method 2, there were 2 males and 17 females, with an average of 48.50 and 79.18, respectively. data obtained by gender in method 2 is not homogeneous. and in method 3 with 4 males and 20 females, with an average of 80.75 and 85.60, respectively. the data obtained by gender in method 3 is homogeneous. and the conclusion there is no significant difference between male and female by using method 3 with α = 0.113. 3.1.3. the learning outcomes of the three methods are not the same in terms of job based on the analysis conducted using spss with the one-way anova test, the results obtained are seen in table 4 as follows. table 4. differences in learning outcomes of the three methods used seen from job job method 1: col method 2: cfcol method 3: ifcol n ms n ms n ms household assistant 21 79.14 11 83.09 13 85.62 factory worker 17 77.06 8 66.13 11 83.82 total 38 78.21 19 75.95 24 84.79 job method 1: col method 2: cfcol method 3: ifcol f 0.415 0.082 0.858 p 0.541 0.010** 0.442 * p< 0.05, ** p< 0.01. volume 12, no 1, february 2023, pp. 13-26 21 table 4 show that in method 1 there are 21 household assistants and 17 factory workers, with an average of 79.14 and 77.06, respectively. the data obtained based on the job in method 1 is homogeneous. and conclusion, there is no significant difference between a household assistant and a factory worker using method 1 with α = 0.541. whereas in method 2, there were 11 household assistants and 8 factory workers, with an average of 83.09 and 66.13, respectively. the data obtained based on the job in method 2 is homogeneous. and in conclusion, there is a significant difference between household assistants and factory workers by using method 2 with α = 0.010**. and in mothod 3 with 13 household assistants and 11 factory workers, with an average of 85.62 and 83.82, respectively. the data obtained by gender in method 3 is homogeneous. there is no significant difference between household assistants and factory workers using method 3 with α = 0.442. 3.1.4. the learning outcomes of the three methods are not the same in terms of age based on the analysis conducted using spss with the anova test, the results obtained are seen in table 5 as follows. table 5. differences in learning outcomes of the three methods used seen from age age method 1: col method 2: cfcol method 3: ifcol n ms n ms n ms <= 25 years 17 76.53 12 73.92 14 86.57 >= 26 years 21 79.57 7 79.43 10 82.30 total 38 78.21 19 75.95 24 84.79 age method 1: col method 2: cfcol method 3: ifcol f 0.343 0.128 0.535 p 0.371 0.456 0.062 * p< 0.05, ** p< 0.01. in method 1 the number of small-age students is equal to 25 years, as many as 17 people, and students of older age are equal to 26 years as many as 21 people (see table 5). the average of each learning outcome is 76.53 and 79.57. whereas in method 2 (see table 5), with the number of small students equaling 25 years by 12 people and students aged greater than 26 years were 7 people. the average of each learning outcome is 73.92 and 79.43. and in method 3 (see table 5), with the number of small students equal to 25 years as many as 14 people and students aged greater than 26 years were 10 people. the average of each learning outcomes is 86.57 and 82.30. all data obtained based on method 1, method 2, and method 3are homogeneous. and the conclusion there is no significant difference between students of different ages using method 1 with α = 0.371, method 2 with α = 0.456 and method 3 with α = 0.062. 3.1.5. the learning outcomes of the online learning are not the same in terms of gender, job and age based on an analysis of all students who study online learning based on gender classification, the results can be seen in table 6, as follows. ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 22 table 6. learning outcomes by online learning viewed from gender, job and age criteria n ms f p gender male 18 73.67 0.03 0.01 female 63 81.33 job household assistant 45 81.98 0.05 0.03* factory worker 36 76.69 age <= 25 years 43 79.07 0.08 0.63 >= 26 years 38 80.26 * p< 0.05, ** p< 0.01. table 6 show that for all students who learn by online learning based on job classification. the number of male students is 18 students and 63 female students, with 73.67 and 81.33, respectively. data obtained by gender is not homogeneous. so, it can't be used. table 6 show that that the number of students who work as household assistants is 45 people and factory workers are 36 people, with an average of 81.98 and 76.69, respectively. data obtained by a job is not homogeneous. and the conclusion there is a significant difference between students who work as household assistants and factory workers using online learning with α = 0.03*. table 6 can be analyzed for all students who study online learning based on age classification. the number of small-age students is equal to 25 years, as many as 43 people and students aged greater than 26 years are 38 people. the average of respectively 79.07 and 80.26. data obtained based on age are homogeneous. and the conclusion there is no significant difference between students of different ages using online learning with α = 0.63. 3.2. discussion in this study, online learning using conventional flipped classrooms is no different from conventional online. this is caused by the absence of control over students outside the classroom. this was also expressed by elledge et al. (2018) that there was no difference between learning with flipped classroom and didactic learning. but students prefer learning flipped classrooms because it can increase broader knowledge by using e-learning. and learning flipped classroom can also increase student confidence. in learning with innovation, the flipped classroom has a significantly different result from the others because of students' control outside the classroom. this was also expressed by lo et al. (2018) that in learning statistics using the flipped classroom method with a learning design outside the classroom can improve learning outcomes and develop student potential. flipped classroom is group learning methods. rawas et al. (2020) argues that the design of flipped classroom learning with group is better than individual. teaching students in group can make learning interactive and collaborative (reynolds & muijs, 1999). the group size is made in small numbers to prevent laziness in the group (trytten, 2001). innovation flipped classroom is peer tutoring students through videos. videos were made by students through various sources. students has to understand the topics in depth before making a video (eugenia, 2018). the various videos in flipped classroom provides an opportunity for students to apply and gain much knowledge (obradovich et al., 2015). flipped classroom learning conducted using videos, online quizzes, and group learning in class can increase student satisfaction and better learning experiences (awidi & paynter, 2019). furthermore, flipped classroom that involves students in editing videos both individually and in groups can positively impact learning outcomes (eugenia, 2018). volume 12, no 1, february 2023, pp. 13-26 23 students become more sensitive to the performance feedback given, and their perceptions become more realistic, constant, and stable (dweck, 2002). peer tutoring flipped classrooms create better interaction between teacher and students, peer interaction, make students more creative, make learning fun, enthusiastic, make maximum use of time in class, students participate in making decisions and make conclusions (graziano, 2017). peers will facilitate before class to have prior knowledge (graziano, 2017; tsai et al., 2020). activities in peer tutoring flipped classrooms allow students to be more involved and active in learning by utilizing technology (nerantzi, 2020). peer tutoring engages students and encourages learning outcomes (schell & butler, 2018). peer tutoring is interactive learning activities, student-centered paradigm, play more active roles in driving instruction (bishop & verleger, 2013). peer tutoring gives flexibility, indepth, students’ self-learning, interactive instruction, efficiency, practical learning, and empowers students to teach and learn from each other (baepler et al., 2014). peer tutoring is useful for improving students’ learning outcomes, conceptual understanding, problemsolving, and decision-making (nicol & boyle, 2003). peer tutoring is useful for promoting diversity in the background so that they are easy to blend in (chubin et al., 2005). peer evaluation is another commonly employed peer-to-peer learning approach (hersam et al., 2004). peer evaluation is more widely employed in high education (lee, 2009). peer evaluation is useful for making students more critical thinking and learning outcomes (boud et al., 2014). 4. conclusion the conclusions of this study are differences in student learning outcomes in the three sample groups (conventional online learning, conventional flipped classroom in online learning, and innovation flipped classroom in online learning). further analysis was conducted that conventional online learning and conventional flipped classroom in online learning did not differ significantly. whereas innovation flipped classroom in online learning differs considerably from the two other learning methods. this is due to the absence of strict control of students before learning by using conventional online learning methods and conventional flipped classroom in online learning. in conventional online learning, students are not prepared before studying in class. while using conventional flipped classrooms in online learning, students are indeed given the task to watch learning videos before learning online learning, but there is no strict control over students. this is different from students using the innovation flipped classroom method in online learning. they must make a short video of their explanations to learn and understand the material before learning. and for other students watching the video, their classmates must also provide feedback by giving comments and questions to their classmates. if reviewed in more depth, there is no difference in student learning outcomes in conventional online learning classes, and innovation flipped classroom in online learning between the two genders. whereas in the conventional flipped classrooms in online learning data, the data are not homogeneous, the data cannot be used. this explains that the learning outcomes of male students are no different from female students. this happens because they have the same desires, abilities, and motivation in learning. because to study while working is their decision, they are among those who have high awareness in their studies. furthermore, there are differences in student learning outcomes in conventional flipped classrooms in online learning seen from students' jobs. whereas in the conventional online learning and innovation flipped classroom, online learning classes are not different between the two jobs. student learning outcomes using conventional online learning are no different ramadoni & chien, integrating peer tutoring video with flipped classroom in online statistics … 24 because they study in class as usual and do homework after online class, so no time effect affects the learning outcomes of both types of jobs. on the other hand, students using conventional flipped classrooms in online learning methods that work as household assistants have more flexible time at work, so they have more time watching videos and repeating them. the learning outcomes are better than students who work as factory workers. where students who work as factory workers cannot watch videos during work. this is different from the learning of flipped classroom innovation when time is no longer a differentiator between the learning outcomes of the two types of jobs because students are required to make videos, provide comments and questions to their classmates' videos. therefore, students are sure to understand the subject matter and are matured in the classroom. so, students using the flipped classroom innovation method utilize the time they must understand the subject matter. when we search into it further, we can see that there is no difference in student learning outcomes in students' three methods based on age differences. it shows students using the three methods even though they are of different ages but still have the same abilities, wills, and motivations. because with their desire to learn while working it indicates an immense desire from them to study again. moreover, there are differences in student learning outcomes in all taiwan open university students with online learning in student jobs. this is because students who work as household assistants are more flexible than students who work as factory workers. while seen from the difference in the age of students, there is no difference. this is because they have a very high will and motivation in learning even though, on the other hand, they must work. while in terms of gender, the data cannot be used because the data are not homogeneous, so it can't be used. acknowledgements the authors would like to thank the all ut taiwan students. this research was carried out with personal funds. references awidi, i. t., & paynter, m. 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(2017). tantangan profesionalisme guru pada pembelajaran matematika melalui 4c’s ditinjau dari perspektif sosiologi. in prosiding seminar nasional dan workshop matematika dan pendidikan matematika stkip pgri. https://doi.org/10.1016/j.hpe.2019.06.002 https://doi.org/10.1080/13632439969032 https://doi.org/10.3389/feduc.2018.00033 https://doi.org/10.1002/j.2168-9830.2001.tb00572.x https://doi.org/10.1016/j.tsc.2020.100747 https://doi.org/10.1080/08923647.2018.1412554 https://doi.org/10.1016/j.compedu.2017.06.012 https://doi.org/10.1016/j.compedu.2019.103653 https://doi.org/10.1016/j.cptl.2019.09.017 https://doi.org/10.1016/j.compedu.2020.103851 https://doi.org/10.1016/j.nedt.2019.104262 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p183-194 183 analysis of students’ mathematical communication ability by using cooperative learning talking stick type dwi maulida sari universitas pendidikan indonesia, jl. setiabudi no.229, bandung, west java, indonesia dwimaulida20@student.upi.edu received: may 12, 2017; accepted: july 02, 2017 abstract this research aims to describe the students’ mathematical communication ability by using cooperative learning talking stick type. this research conducted at one of junior high school in asahan-north sumateranorth sumatera. this research used posttest experimental class design as the method to obtain data. the indicators used to measure mathematical communication ability in this research arranged in three, as follows: 1) the ability of explaining a mathematical problem into figure; 2) the ability to explain mathematical problem situations by own words; 3) the ability of stating a mathematical problem in writing into mathematical models and doing calculation to solve the problem. the result found from this research is there are the differences in the students’ way of answering the problem even though the learning model and the instrument used is same. keywords: cooperative learning, mathematical communication ability,talking stick type abstrak penelitian ini bertujuan untuk mendeskripsikan kemampuan komunikasi matematis siswa dengan pembelajaran kooperatif tipe talking stick. penelitian ini diadakan di salah satu sekolah menengah pertama di asahan-sumatera utara. penelitian ini menggunakan desain kelas eksperimen dengan posttes untuk mengumpulkan data. indikator yang digunakan untuk mengukur kemampuan komunikasi matematis dalam penelitian iniyaitu: 1) kemampuan untuk menjelaskan persoalan matematika kedalam bentuk gambar/ visual; 2) kemampuan untuk menjelaskan persoalan matematika kedalam bahasa sendiri; 3) kemampuan untuk menentukan dan membuat model matematika yang sesuai dari persoalan matematika tertulis serta melakukan perhitungan untuk menyelesaikan persoalan matematika tersebut. hasil dari penelitian ini adalah terdapat perbedaan karakteristik cara siswa menjawab soal, walaupun model pembelajaran dan instrumen yang digunakan sama. kata kunci: kemampuan komunikasi matematis, pembelajaran kooperatif, tipe talking stick. how to cite: sari, m. d. (2017). analysis of students’ mathematical communication ability by using cooperative learning talking stick type. infinity, 6 (2), 183-194. doi:10.22460/infinity.v6i2.p183-194 mailto:dwimaulida20@student.upi.edu sari, analysis of students’ mathematical communication ability by using cooperative … 184 introduction educational researchers state that students need to learn effectively how to communicate their thinking both orally and in writing (national middle school association [nmsa], 2004; secretary’s commision on achieving necessary skills [scans], 1991; national council of teachers of mathematics [nctm], 1989, 2000; cobb, yackel, & wood, 1992). kist (clark, jacobs, pittman, & borko, 2005) stated that in general effective communication ability should be possessed by students in all subject matters and not only in mathematics, that is communication is important in all subject also in life. mathematics as a vehicle for education, not only can be used to achieve one goal such as to educate students but can also form the personality of students and develop certain skills. mathematics is abstract and deductive science, mathematics is a knowledge that studying patterns, shapes, and structures and mathematics are the human activity. according to permendiknas no. 22 (2006) the purpose of learning mathematics are: “(1) memahami konsep matematika. menjelaskan keterkaitan antar konsep dan mengaplikan konsep atau algoritma, secara luwes, akurat, efisien dan tepat dalam pemecahan masalah, (2) menggunakan penalaran pada pola dan sifat, melakukan manipulasi matematika dalam membuat generalisasi, menyusun bukti, atau menjelaskan gagasan dan pernyataan matematika, (3) memecahkan masalah yang meliputi kemampuan memahami masalah, merancang model matematika, menyelesaikan model dan menafsirkan solusi yang diperoleh, (4) mengkomunikasikan gagasan dengan simbol, tabel, diagram atau media lain untuk memperjelas keadaan dan masalah, (5) memiliki sikap menghargai matematika dalam kehidupan, yaitu memiliki rasa ingin tahu, perhatian, dan minat dalam mempelajari matematika serta sikap ulet dan percaya diri dalam pemecahan masalah”. so, as said by the permendiknas no.22 that one purpose of learning mathematics is communicate ideas with symbols, tables, diagrams or other media to clarify the situation and problems. pugalee (2001) also state that to improve students’ mathematical communication ability need to motivated them to give relevant reason on their answer. students were able to understand mathematics concepts being learned meaningfully.it’s clearly showing that the ability of mathematics communication is needed. national council of teachers of mathematics (2000) state that mathematical communication abilities include expressing mathematical thought by using mathematical language clearly, precisely, and succinctly, understanding others’ mathematical equations and concepts (lin, shann, & lin, 2008) and evaluating others’ mathematical concepts by asking meaningful questions and explaining the reasons for others’ incorrect mathematical thought. in general, the communication can be interpreted as the process to achieve the purpose of communication itself which is to conveying a message from one person to another either directly (orally) or indirectly. nctm (2000) state that through communication, ideas become objects of reflection, refinement, discussion, and amendment. the communication process also helps build meaning and permanence for ideas and makes them public. the process of learning and teaching also contributes to developing students' communication ability, by convey their ideas and opinions teachers can find out how much the students learned and understand the material. students are challenged to think and give a reason about mathematics also communicate the results of their minds to others orally or in writing, they learn to be clear and convincing and this is the process of communication. volume 6, no. 2, september 2017 pp 183-194 185 communication in mathematics is called mathematical communication ability is the ability to deliver something that is known through speech or writing dialogue about the concepts, formulas, or problem-solving strategies in mathematics. the mathematical communication ability reflect students' understanding. the concept or information of mathematics given by a teacher to the student, or students gets it by themselves that is when the transformation of mathematical information from the communicator to the communicant happen. in mathematics, the quality of interpretation and the response was often a special problem. it is as a result of characteristics of the mathematics itself which loaded with terms and symbols. therefore, the mathematical communication ability became a special requirement in learning mathematics. it is happen because mathematics language different from the mother language, layzer (baroody, 1993) said the language of mathematics is fundamentally different from natural languages, though, in that it describes ideal situations rather than ordinary situations. the description above shows the importance of mathematical communication ability in learning mathematics, it has an important role in building the knowledge of mathematics and expresses mathematical ideas from students' various perspectives. through communication, students can submit their ideas to the teacher and to other students. based on preliminary observations in class vii on one junior high school at asahan-north sumatera obtained that the lesson is still centered on the teacher, not on students. the result in this kind of learning made students passive in developing their minds or deliver their idea. from this preliminary observation also found that students’ mathematical communication ability is still low.for example, problem number one; write down every sentence below into a mathematical expression by using a variable. (a) the result from multiple of two natural numbers and add it by 2 is 9, (b) amount of ikhsan and bayu books is 11, while the difference of their books is 1. figure 1. one of student’s answer for problem number 1 for problem number one, from the figure can be seen that student can not state the model for the two natural number, student think the two natural number on as the number and the other as the power of it and also give power to number 2, this might happen because student remember about exponential that they learn before or maybe students did not understand the problem or student ability to state the words into mathematical expression is low. it also happens in b question, student confused to state the sum and also the difference of ikhsan and bayu book. this show the student ability in state words into mathematical expression is in an unsatisfactory manner. for the problem number two the problem is state into figure form (you can choose the shape that you want). sari, analysis of students’ mathematical communication ability by using cooperative … 186 figure 2. student’s answer to problem no.2 some students are able to state in right figure form but some are not. show in the figure above student wants to state the problem in apple form, but students still confused how to divide the apple in right half form, maybe that is student erase the first figure. it also show that students ability to make the right figure from is still low. the problem number three is: a company will deliver package to their 60 employees, which consists of 2 bottles of syrup and 12 pieces of instant noodles. explain how many dozen syrup and instant noodles are required by the company. figure 3. one of student answer to problem no. 3 the indicator of mathematical communication ability in problem number three stated by ansari (2012) is explaining problem situations by own words and doing the calculation. from figure 3 can be known that student can do the calculation but they can not explain clearly meaning of the number in their answer sheet. also, students make the syrup and instant noodle in the same amount both on a dozen, even syrup and instant noodle are different type. some students knew the way to solve the problem but can not state the reason about their way to answer the problem, this is probably because students find it difficult to express their idea or students not trained to giving a logical reason for the answer that they serve, it means students ability in explaining problem by own words and doing calculation is also weak. interviews were conducted with mathematics teachers of grade vii, concluded that the level of students' mathematical communication ability in the first grade have not developed optimally. most students find difficulty in writing, explaining, and presenting mathematical ideas. students lacking in interact to establishing communication with the teacher and other students. interviews also conducted with some students, found that some students not interested in learning mathematics, they consider learning mathematics is difficult and all material about mathematics are hard to understand. in addition, they wanted the learning process is more varied, some of them want to have shared in learning, students which have a high ability want to help other students. lack of mathematical communication ability can lead to misunderstanding the mathematical concept or the problemand in the end students can not solve the problem, that is the reason needed to find an appropriate approach which will help student’s to develop their mathematical communication ability and stimulating them to achieved their goals in learning. volume 6, no. 2, september 2017 pp 183-194 187 a good way of learning is learning that gives students full opportunity to express their opinions and ideas also alearning approach that is more effective, creative, and fun. on this basis, the authors try to apply cooperative learning talking stick type to see the improvement of students' mathematical communication ability. cooperative learning talking stick type has an aims to expand students' knowledge and accuracy in understanding a concept. as suprijono (2009) said talking stick teaching methods encourage students to dare to express opinions. agreed by istarani (2012) that talking sticks learning model encourage students to dare to express their opinions, when teachers give an explanation about the material then students have time to read and write things they know after that the talking stick will be given to students and student that hold the stick must answer the question that teachers give or give the idea about thing that discussed, that is why cooperative learning talking stick type is one of an appropriate approach to develop student’s mathematical communication through mathematical understanding which stimulated by the talking stick which going around the whole class to provide the opportunity for students to give their opinions.the syntax of talking stick approach in this research is modified from the syntactic of talking stick from suprijono (2009) can be seen as follows: 1. introduction a. delivering the learning objectives b. motivating students c. delivering the learning method that will be used, which is talking stick type d. prepare the talking stick and the music e. give students the sas 2. exploration and elaboration a. asking students do sas b. asking students to discuss with membersof their own group c. teachers guiding and motivate groups of studentswhile doing sas d. teachers subsequently asked the students to close the book. teachers take the stick which has been prepared in advance. the stick was given to one of the students e. students who received the baton required answering questions from the teacher, onwards. when the stick rolling of the student to other students and accompanied by music. 3. confirmation a. teachers provide opportunities for students to reflect on the material that has been learned. teacher gives a review of all answers given by the student b. together with the students formulate conclusions 4. evaluation a. teachers give quizzes b. teachers evaluate learning outcomes based on the syntax, students are given the opportunity to discuss with their friends and then need to express their idea, opinion or the answer to the problem that teacher give. after that with the teacher discuss the material again to obtain a summary. the syntactic that cooperative learning talking stick type has, give a lot of opportunities for students to develop their mathematical communication ability in groups or their self. by using cooperative learning talking stick type concept made studied mathematics in a more meaningful way because the students are trying to understand the material by their own and then communicating back by giving the feedback. sari, analysis of students’ mathematical communication ability by using cooperative … 188 the indicators that used to see the mathematical communication ability in the concept of quadrilateral in this study are: (1) the ability of explaining mathematical problem into figure, (2) the ability of explaining problem situations by own words, and (3) the ability of stating mathematical problem into mathematical model and doing calculation to solve it (ansari, 2012). based on the explanation, the issues to be discussed are the analysis indicators of students' mathematical communication ability that showed during the implementation of cooperative learning talking stick type and also to see if talking stick can be one of the learning approach that can be used to increase students' mathematical communication ability. the purpose of this study was to analyze the indicators of mathematical communication ability of students in class vii on one junior high school at asahan-north sumatera that appeared during the implementation of cooperative learning talking stick type and to find that cooperative learning talking stick type can increase students' mathematical communication ability. method this research uses quasi-experiment with the post-test experimental class design with descriptive analysis of qualitative. in this research, the subject is one class in grade vii in one junior high school at asahan. the independent variable in this study is cooperative learning talking stick type and the dependent variable is the students' mathematical communication ability. primary data in this research is students' post-test results that made based on mathematical communication ability indicators that have been determined beforehand. the instrument in this research is post-test of mathematical communication skills that given at the end of the implementation of cooperative learning talking stick type. post-test results were analyzed by determining the suitability of the students' answers to the indicators of the ability of mathematical communication that expected. from the data will be seen the students' average post-test results to be compared with mathematics minimum score that determined by the school to see the cooperative learning talking stick in improving students' mathematical communication ability. results and discussion results based on the results of the research that conducted in class vii in one junior high school at asahan in the material quadrilateral with mathematical communication indicators used in the post-test are: (1) the ability of explaining mathematical problem into figure, (2) the ability of explaining problem situations by own words, and (3) the ability of stating mathematical problem into mathematical model anddoing the calculation to solve it, found that there are differences instudents’ ways of answering the question even the treatment and the instrument used is same. the problem is mr. syamsuddin has 3 lands. the first land is side by side north and south to the second garden with a total area of 136 m 2 . the first land is a square with sides is 8 m, the second land is a rectangular shape with a width 6 m. the third garden is on the right side of the second land with also a rectangular shape and the width is 4m and a length equal to first land sides plus the second land width. from the situation above; (a) by your own words find the length of the second land (give your reason); (b) illustrate it into figure form; (c) determine the mathematical expression to find the total area of land that owned by mr. syamsuddin and solve it.this question has all the three indicators that going to be analyzed. volume 6, no. 2, september 2017 pp 183-194 189 question number 1 part a, the indicator that used is indicator number (2) the ability to explain problem situations by own words, found that there are differences in students’ ways of answering the question even the treatment and the instrument used is same. there a re some differences in students’ ways of answer this part, as shown below: a b figure 4. student’s answer for problem number 1 part (a) for problem part b the indicator used is the ability to explain mathematical problem or situation into a figure. from 30 students there are different answers that found, the most answer shown in the figure below: a b c figure 5. students’ answer to the first question part (b) for part c, the indicator that going to measure is the ability to state mathematical problem into a mathematical model and doing the calculation to solve it. the way students answer shown in picture below: a b c figure 6. students’ answer for problem no.1 part c. sari, analysis of students’ mathematical communication ability by using cooperative … 190 discussion in problem part a, as shown in the picture in discussion student (a) can state what on their mind on what need to do clearly but did not answer a full question, they do not find the length size of the second land and it make the answer just circulating in information that already given in the problem. this may be happening because the student already knows the concept of the problem and know how to solve it but student just has a habit of solving problem by using mathematical expression but cannot change that into their own words. some students understand the concept or the problem it is just they still not able to make flow to conclude the answers to their own sentence. this kind of answer can also happen because students has difficulty in language, maybe students do not know which word they need to choose to lead the words to answer. the problem students can finish can happen also because they did not have interest in learn mathematics that has a lot of word, students already have paradigm that mathematics is all number. in students (b) answer clearly show that student already know the concept and can communicate it rightly, even though the answer not perfect because students still use a mathematics expression such as subtraction and equal, but this kind of answer is close to what is wanted by the teacher as one of a good answer. the mathematical expression still in students answers in term of make situation into own words because students already familiarized with giving a good reason to complete an answer but still in not perfect ways. this can happen to because maybe students need more time to answer. student (b) can clearly communicate what they want to tell and finish it until they found the answer. it is shown that some students that has a similar answer to the student (b) mathematical communication ability in part can make reason by using their own word is in good term. in the learning process cooperative learning talking stick type all students have the same chance to give their thought, idea and also an opinion about the subject. students which can make the right word or still stuttering in answer or give opinion getting help by their friend and also get the encourage from all class members such give applause. student (a) has less confidence but when he gets courage from his classmate he can tell what on his mind and give a good way in answering the question, maybe in the exam he just does not know to choose the right words when he writes the answer. researcher conduct an interview with student (a), when research asks if he understand the question number 1 part a, he can explain that the question is to find the length of the second land, he know how to solve it by using mathematical expression, he just not confidence to answer in words and loose what he want to write, that is why he just write the information on the problem. the researcher also conducts an interview with student (b), as the answer to her question she already knows the concept and the question, so she just writes down what she knows and what word she need to choose. able to use mathematical expression is also a good skill in completed the mathematical communication ability, but students need to familiarized with giving a reason in a problem so students can grave the ways to answer in their long time memory storage because they give a meaning to the answer. some students that have an answer like student (a) need more training to make they accustomed with giving answer or reason by their own words. in question part b, all students answer rightly in this question. it is shown that students already understand the concept of north and south, also the concept of the right side, this can happen because student already learns about the compass and understand it completely, also student volume 6, no. 2, september 2017 pp 183-194 191 clearly understand to differentiate square with a rectangle, even all students answer are differences in the way their answer this question. this differencesoccur because students have a different perception of the land shape, this happen because student has their own experience about land also the shape of square or rectangle. students that have an answer that similar to student (a) maybe wants to show that rectangle can be placed horizontally also vertical, it is just the partition between land i, i and iii cannot see clearly, but the division of each length and side is good. interview with student (a) found that her father has the land that has the same form as the land i and ii. for students that have similar answer with student (b), they divided the land in good portion even for land i and ii the size of 8 m and 6 m is same, maybe this happening because students do not pay full attention in subject scale when they learn in elementary school or student just assume them is different because they going to label it. but the students who answer similar like student (b) give a label for each land, it means students understand the position of each land. interview with the student (b) found that she never known or see the land that has the same or similar form with the land in the problem, also the reason sided size of square and the width size of the rectangle is same because she just needs to figure it not to scale in the right measurement. for students that have similar answer with student (c) answer in a good way, they give all the size of the square and rectangle rightly and they use the information that the length of land iii is total amount of sided of square and width of the land ii rectangle and make the sided of square above the width of the land ii, so they just draw length of land iii in the same position with the incorporation of sided land i and width land ii. all students in this experiment answer the question as the teacher expected they to do. the mathematical communication ability in indicator the ability to explain mathematical problem into figure is in good term. this happen maybe because in the teacher-learning process of cooperative learning talking stick method all students have the same opportunity to draw the rectangular and presenting it in front of the class as personal or group, so the concept of square and rectangle already understood by them. in part c, from 30 students more than half answer this question rightly. some students do wrong in the calculation and the other did not finish the calculation and did not make the mathematical model or expression that describe the problem. this may happen because students did not understand the problem or student do not get the concept of the problem. as for students which have a similar answer to student (a), they do the calculation wrongly and did not state the mathematical model that shown the question. this may happen because students feel the need to finish the test fast and make they lose their concentration. in the learning process of cooperative learning talking stick type this kind of problem might happen because students have more time to talk about their opinion but a little time to write what is on their mind. as shown in student (a) answer, he understands the problem which is to find the total area of mr. syamsudiin land, maybe he thinks is no need to made the mathematical expression because he already knew, even though the mathematical expression is in the question. interview conduct with student (a) found that he feels do not need to explain more because he already knows the way to find the total land area and for the miscalculating, he states that he rushing in calculate it because of he thinks the time is almost done. it is shown that students with a similar answer to student (a), their mathematical communication ability in indicator the ability to state mathematical problem into a mathematical model and doing the calculation to solve it is in an unsatisfactory manner. sari, analysis of students’ mathematical communication ability by using cooperative … 192 for students that have a similar answer to student (b) clearly, understand the problem and knew how to solve it. they state the mathematical expression rightly to describe the way to find mr. syamsuddin total land area by adding all the area from land i, ii and iii. they also use the information rightly that the total amount of land i and ii already knew so they just need to calculate the area of land iii and adding it together, this may happen because students understand the problem and all the information that given. this also can happen because maybe student already do this kind of problem before in their elementary school or they have experienced to calculate the total amount of area before. this also can be the effect of learning by using cooperative learning talking stick type, in the learning process when students given the sas they found the material by their self and also try to understand it fully to be able to present it when the talking stick come to them. interview with student (b) found that she already knew the total amount of land i and ii, so she thinks it does not need to calculate it again, and she already has similar problem before. the same for students that have the similar answer to student (c), they clearly state the mathematical model to find the total area of the land. they do the checking part by calculation the land area one by one, this also can be the effect learning by using cooperative learning talking stick type in part when the teachers provide opportunities for students to reflect on the material that has been learned and the teacher gives a review of all answers given by the student. they also give the explanation at the end of their answer to conclude the calculation. this kind of answer might happen same as the student (b) because students that answer similarity with student (c) already understand the problem and also the concept. also, this can happen because students already experienced this kind of problem before. when students do the checking part by calculating it one by one the area is because they do not want to do the wrong step. the interview conducted with student (c) found that he wants to check again is he do the right calculation when he finds the length of land ii, and he wants to show the teacher that he understands the different of square and rectangle also the formula that uses for each of them. the answer of student (b) and (c) are the closest to answer that teacher expected they to do. based on students answer from the three indicator measured after the implementation of cooperative learning talking stick type approach can be sorted from easiest indicator to the most difficult for students in question are as follows: (1) the ability of explaining mathematical problem into figure, (3) the ability of stating mathematical problem into mathematical model and doing the calculation to solve it and (2) the ability of explaining problem situations by own words. in other word in the learning process noted that the indicators the ability to make the situation into a picture is the highest position. it can be concluded most students master this indicator, and the indicators that difficult for the student is giving a reason or evidence for the correctness of the solution from a situation. to see the effect after implementation of cooperative script talking stick type approach can effect to students’ mathematical communication ability researcher compare the average of students’ test value with the minimum mathematics score that assigned by the school. minimum score assigned by the school for mathematics is 72, from the data that researcher get found the average score of the students is 74.91, with maximum score is 91.67 and minimum score is 61.11, it shown that students mathematical communication ability by using cooperative learning talking stick type approach is more high than the minimum score even though the different did not so big. from all data also found that 17 students pass the minimum score, 5 students score is exactly the minimum score and 8 students did not pass the volume 6, no. 2, september 2017 pp 183-194 193 minimum score, it is shown that cooperative learning talking stick type has a good impact on students’ mathematical communication ability but not the really fine approach to use. conclusion analysis of students’ mathematical communication ability after the implementation of cooperative learning talking stick type show that there are the differences in students’ characteristics ways of answering the mathematical problem. other than that found that studentsmost difficult indicator is explaining the way to answer the problem situations by own words. it is also in line with the opinion of hidayat (2017) who argued that one's difficulty in explaining the process of the problem given due to the habit of the less innovative way of teaching. references ansari, b. i. (2012). komunikasi matematik dan politik suatu perbandingan: konsep dan aplikasi. bandaaceh: penerbit pena baroody, a. j. (1993). problem solving, reasoning, and communicating k-8 helping children think mathematically. new york: merril an in print of macmillan publishing company clark, k. k., jacobs, j., pittman, m. e., & borko, h. (2005). strategies for building mathematical communication in the middle school classroom: modeled in professional development, implemented in the classroom. current issues in middle level education, 11(2), 1-12. cobb, p., yackel, e., & wood, t. (1992). interaction and learning in mathematics classroom situations. educational studies in mathematics, 23(1), 99-122. depdiknas, r. i. (2006). peraturan mendiknas nomor 22 tahun 2006 tentang standar isi. jakarta: depdiknas. hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. istarani (2012). 58 model pembelajaran inovatif. medan: media persada national council of teachers of mathematics (1989). curriculum and evaluation standards for school mathematics. reston, va: nctm. national council of teachers of mathematics (2000). principles and standards for school mathematics. reston, va: nctm national middle school association (2004). position paper of the national middle school association middle level curriculum: a work in progress. westerville, oh: nmsa pugalee, d. k. (2001). using communication to develop students' mathematical literacy. mathematics teaching in the middle school, 6(5), 296. secretary’s commission on achieving necessary skills (1991). what work requires of schools: a scans report for america 2000. washington, dc: u.s. department of labor. sari, analysis of students’ mathematical communication ability by using cooperative … 194 lin, c. s., shann, w. c., & lin, s. c. (2008). reflections on mathematical communication from taiwan math curriculum guideline and pisa 2003. retrieved from http://www.criced.tsukuba.ac.jp/math/apec/apec2008/papers/pdf/ 16.lin_su_chun_taiwan.pdf suprijono, a. (2009). cooperative learning teori & aplikasi paikem. yogyakarta: pustaka pelajar. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p111-120 111 effect of learning with abductive-deductive strategy towards the achievement of reasoning ability of high school students ali shodikin universitas islam darul ulum, jl. airlangga no. 3 sukodadi lamongan, indonesia aliandr4@gmail.com received: february 08, 2017 ; accepted: may 12, 2017 abstract the purpose of this study was to investigate the effect of learning with abductive-deductive strategy towards the achievement of mathematical reasoning abilities of high school students. research carried out an experimental pretest-posttest design and the control group was not randomized in class xi student at one high school in pati, central java, indonesia. data analysis was conducted quantit ative research based on early mathematical ability categories (kam) and overall. the results showed that the achievement of mathematical reasoning abilities that students acquire learning abductive-deductive strategy better than students who received the expository learning. in more detail of kam categories, only middle category that show achievement of mathematical reasoning abilities better. while in upper and under categories have the same reasoning abilities achievements. this research is expected teachers can encourage students to do abduction and deduction in the learning achievement of students’ mathematical reasoning abilities. keywords: abductive-deductive strategy, achievement, reasoning. abstrak tujuan penelitian ini adalah menginvestigasi pengaruh pembelajaran dengan strategi abduktif-deduktif terhadap pencapaian kemampuan penalaran matematis siswa sma. penelitian yang dilakukan merupakan eksperimental dengan desain pretes-postes dan kelompok kontrol tidak acak (nonrandomized control group, pretest-posttest design) pada siswa kelas xi di salah satu sma di kabupaten pati, jawa tengah, indonesia. analisis data penelitian dilakukan secara kuantitatif berdasarkan kategori kemampuan awal matematis (kam) maupun keseluruhan. hasil penelitian menunjukkan bahwa pencapaian kemampuan penalaran matematis siswa yang mendapatkan pembelajaran dengan strategi abduktif-deduktif lebih baik daripada siswa yang mendapat pembelajaran ekspositori. secara lebih rinci dari kategori kam, hanya pada kategori tengah yang menunjukkan pencapaian kemampuan penalaran matematis yang lebih baik. sedangkan pada kategori atas dan bawah memiliki pencapaian kemampuan penalaran yang sama. dari penelitian ini diharapkan guru dapat mendorong siswa untuk melakukan abduksi dan deduksi dalam pembelajaran dalam pencapaian kemampuan penalaran matematis siswa. kata kunci: strategi abduktif-deduktif, pencapaian, penalaran. how to cite: shodikin, a. (2017). effect of learning with abductive-deductive strategy towards the achievement of reasoning ability of high school students. infinity, 6 (2), 111120. doi:10.22460/infinity.v6i2.p111-120 shodikin, effect of learning with abductive-deductive strategy … 112 introduction mathematical reasoning ability is the main characteristic that can’t be separated from the activities of studying and developing or solving mathematical problems. reasoning in mathematics plays a critical role in developing mathematical understandings. (bragg, loong, widjaja, vale, & herbert, 2015). in fact, the implementation of learning that emphasizes the existence of reasoning is very recommended (nctm, 2000). there were strong positive relationships between the students’ spatial reasoning and mathematics performance (lowrie, logan & ramful, 2016). however, many studies show that the reasoning ability of students is still low (nataliasari, 2014). their thinking of algebraic reasoning required only procedural knowledge and did not include generalisation or functional thinking (glassmeyer & edwards, 2016). though reasoning ability is needed in the mastering and solving of mathematical problems. but this reasoning ability is often overlooked in learning (nizar, 2007). therefore, in learning mathematics mathematical reasoning ability need attention given to be able to solve a math problem required students’ reasoning abilities. report the results of other studies, showed similar findings. reasoning ability which a part of high order mathematical thinking abilities (sumarmo, 2013). study reports mullis, martin, ruddock, o’sollivan & preuschoff (2009) show that learning mathematics is generally not focused on developing high order mathematical thinking abilities. student more dominant solve problems from the textbook and get less non-routine problems that can train this high order mathematical thinking abilities. thus the need for efforts to develop mathematics learning oriented to the development of high order thinking abilities. based on a preliminary analysis of reasoning ability is necessary to develop a learning that can improve the understanding of essential concepts. as a general framework in solve a problem in mathematics is the ability to identify the given facts (data) and formulate what is asked in the problem (final target). in determining the final target is based on data provided, it is necessary to elaborate the ability to apply the essential concepts that are relevant with the given data to obtain intermediate target before finding the answer to the final target. not a few problems in mathematics can be more easily solved by adding a condition (intermediate target) that is based on a concept relevant essential to arrive at the final target in question. general framework as described above has been developed at the research shodikin (2016) in a learning with abductive-deductive strategy. abductive is a mathematical thinking skills (reasoning) that can’t fully answer the problem but the process of offering a reason as the basis for a specific action (aliseda, 2007). this general framework was originally developed to develop the proving ability the beginner student learning of proof. the results showed that student who learn with abductive-deductive strategy have the proving ability better than students who learn with conventional learning. possible application of this strategy has been reviewed by sun, finnie & weber (2005) for the problem of reasoning and problem solving ability. the possibility of applying this framework to the wide range problems (mathematical literacy) for students in secondary schools has also been studied theoretically (shodikin, 2013). based on the notion of learning with abductive-deductive strategy, in this study developed learning syntax abductive-deductive strategy more operational as shown in figure 1. volume 6, no. 2, september 2017 pp 111-120 113 figure 1. schematic of learning with abductive-deductive strategy stages of learning with abductive-deductive strategy above in more detail is shown in table 1 (shodikin, 2016). table 1. syntax of learning with abductive-deductive strategy phase teacher behavior phase 1 orientation of problem • teacher discusses the problem of learning objectives • teachers describe various important logistics needs • teachers motivate students to be directly involved in learning activities • teachers provide apperception phase 2 organize for learning • teachers help students to define and organize the tasks of learning and information related to the problem phase 3 analyze and process evaluate • analyze and evaluate the teacher directs students to find their own solutions from information already possessed by students • teachers encourage students to do transactive reasoning as to criticize, explain, clarify, justify and elaborate a proposed idea, either initiated by students and teachers • teachers assist students in planning and preparing materials for presentations and discussion • teachers help students to reflect on the investigation process and other processes used in solving problems phase 4 generalize the findings • teachers help generalize the findings obtained phase 5 discussion of strategies to more problems • teachers assist students in finding strategies to the problems are much more • teachers provide training and evaluation orientation of problem organize for learning generalize the findings discussion of strategies to more problems analyze and process evaluate key process d e d u c ti v e p r o c e ss a b d u c ti v e p r o c e ss shodikin, effect of learning with abductive-deductive strategy … 114 to be involved in transactive discussion, early mathematics ability (kam) student plays a very important, where an idea that appears to develop gradually so as to build a comprehensive mathematical concept of information obtained. the kam students are categorized into three categories: upper, middle and under. this grouping is used to see if there is mutual effect between the learning is done with early mathematics ability of the students’ reasoning abilities. besides that, it can be obtained more detail the effect of learning in each category of early mathematical ability. based on the background and formulation of the problem described above, this study aims to investigate the influence learning abductive-deductive strategy towards the achievement of high school students’ mathematical reasoning abilities. method the method applied in this study is experimental with pretest-posttest design and the control group was not randomized. with this design, subjects initially performed pretest, and then treated with a form of learning abductive-deductive strategy and then performed post-test to measure students’ mathematical reasoning abilities in polynomial of matter. this design is chosen according to the purpose of research to show the effect of the application of learning with abductive-deductive strategy towards the achievement of students’ mathematical reasoning ability. in the chart of design used are presented in figure 2. experimental class o pretest x1 treatment abductive-deductive strategy o posttest control class o pretest x2 treatment expository o posttest figure 2. design research the study was conducted at one high school in pati, central java, indonesia. the samples have been two classes that have the same initial capabilities of the eight classes by random sampling, each totaling 34 students. grouping students by category early mathematical ability (kam) is obtained from the average value of two daily tests, mid semester test and semester test. results and discussion results the selection of the class which is used as a sample study in addition seen early mathematical abilities seen from the initial reasoning abilities of students obtained from the pretest scores, both overall and by category kam. it has been shown that the students is learning with abductive-deductive strategies and students learning expository no difference in early mathematical ability of reasoning, both in terms of overall and by category kam (upper, middle, under). volume 6, no. 2, september 2017 pp 111-120 115 achievement of mathematical reasoning ability obtained through posttest scores. based on calculations, the achievement of mathematical reasoning abilities obtained an average value based on class research (experimental and control) and kam (upper, middle, under) are presented in the following bar chart. figure 3. achievement score bar chart reasoning ability figure 3 show that the students who received learning with abductive-deductive strategy (experimental class) shows the overall average achievement of mathematical reasoning abilities greater than students who received the expository learning(controlclass). judging from kam category is upper the level students’ kam, greater the average achievement of mathematical reasoning ability. to find out the reasoning abilities of learning achievement of which one is better,do mean difference test. before the test the average difference, the normality test and homogeneity tests. mean difference test used the t-test for normally distributed data and homogeneous. while the pair is not normally distributed data were analyzed using mann-whitney u nonparametric test. results mean differences test are presented in table 2. table 2. test results mean differences pos-test score mathematical reasoning ability kam comparison of average (e:k) t mannwhitney u sig. (2 tailed) sig. (1 tailed) ho upper 30.00 : 30.00 0.000 1.000 0.500 accept middle 25.81 : 15.88 130.5 0.006 0.003 reject under 13.67 : 12.83 0.166 0.871 0.435 accept overall 24.53 : 17.00 304.5 0.001 0.000 reject ho: the average student achievement reasoning abilities experimental class lower or equal to the control class in terms of kam (upper, middle, under) as well as overall. table 2 shows that the mathematical reasoning ability students acquire learning with abductive-deductive strategy (experimental class) better than students who acquire expository 30 30 25,81 15,88 13,67 12,83 24,53 17 experimental control upper middle under overall shodikin, effect of learning with abductive-deductive strategy … 116 learning (control class). seen more detail from the kam category, only in the middle category, achievement of students’ mathematical reasoning ability that acquire learning with abductive-deductive strategy better than students who acquire expository learning. but in the upper and under category of kam, the achievement of mathematical reasoning ability students acquire learning abductive-deductive strategy (experimental class) is lower or equal to the students who acquire learning expository (control class). after seeing the average achievement, the upper and under category gained an average the experimental class greater than average control class, so concluded the achievement of mathematical reasoning abilities that students acquire learning with abductive-deductive strategy (experimental class) equals students who acquire the expository learning (control class). discussion specifically indicator mathematical reasoning ability as measured focused on three skills namely (1) make logical conclusions; (2) estimate answers and solution processes; and (3) use patterns and relationships to analyze mathematical situations. it has been shown that the students who acquire learning with abductive-deductive strategy and expository learning no difference in mathematical ability early of reasoning, both in terms of overall and by category kam (upper, middle,under). this is normal, because both classes have not been subjected to different learning. achievement of mathematical reasoning ability students who acquire learning with abductivedeductive strategy is better than students acquire expository learning. these results are consistent with the hypothesis proposed previously and showed that indeed the phases of learning with abductive-deductive strategy to support and facilitate the improvement of students’ reasoning abilities. the results of this study as well as the findings of the study other researchers which states that students who acquire learning with the mathematical process thinking has the reasoning abilities better than conventional learning, specifically for metacognitive (noto, 2015) and reflective (rohana, 2015). although the research conducted at the different levels of students and inductive approach, but its similarity to learning with abductive-deductive strategy is equally a kind of learning that emphasizes the mathematical process thinking. average achievement scores (post-test) on the reasoning abilities of students acquire learning with abductive-deductive 24.53 of the ideal 40 score. from this data it can be concluded that the reasoning ability in students who acquire learning with abductive-deductive still less than optimal. the reason for this is related to adjustments in thought relatively difficult students. in fact, think hard into the main capital in constructing knowledge in view of constructivism based learning (ormrod, 2008). another reason is a test that is used in this study was relatively difficult. it is recognized by some of the current students do interviews, that the test items in this study is more difficult than the usual questions given by the teacher in the learning prior to the study. recognition of students is in line with the test results that the questions used most difficult category. whatever the reason is related to the achievement of the results are still far from optimal, it is of the low mathematical reasoning abilityhigh school studentsbased on this sample. descriptions make it clear that high order mathematical thinking ability (reasoning) is not an easy job. however, it is undeniable that the students who acquire learning with abductivedeductive strategy is able to demonstrate better achievement than students who obtain volume 6, no. 2, september 2017 pp 111-120 117 expository learning. this indicates that if the learning abductive-deductive strategy consistently applied it is possible to increase students’ reasoning abilities optimally. reviewed in more detail by category kam, only in the middle category that shows achievement of mathematical reasoning ability students acquire learning with abductivedeductive strategy better than students who acquire the expository learning. while the upper and under category achievement same ability. this suggests that learning with abductivedeductive strategy has been facilitated by both students with middle categories so as to improve mathematical reasoningability. while the student with upper categories, similarity results obtained in improving the ability of reasoning is possible for the students have been great motivation and ability to accept the learning that have been good too, so despite the lack of supporting learning though still able to obtain good results. not much better reasoning ability enhancement students acquire learning with abductive-deductive strategy compared with expository learning does not mean that students do not improve ornot facilitated above, but with both of these learning both increased and facilitated. similarly, the students with under categories, similarity results obtained in improvingreasoning ability in learning with abductive-deductive strategy and expository learning because the students with under categoryhave motivation and ability to accept the lesson less, so that although the learning support though still obtain less results. based on that, in general learning with abductivedeductive strategy has been able to facilitate the achievement of better reasoning ability. the following description seems to reinforce reasons the learning with abductive-deductive strategy has been able to facilitate the achievement of mathematical reasoning ability students better than the students who received learning expository. the following reasons are described by indicators measured reasoning ability. indicators (1) make logical conclusions. this indicator, in learning with abductive-deductive strategy facilitated the phase generalize the findings obtained. learning activities that encourage students to generalize the findings obtained from the problems or the data obtained,was to familiarize and understand students to be able to make conclusions from a logically statement. this is in accordance with the opinion of vygotsky (jones & thornton, 1993), which is the process of improving the understanding and reasoning on students occurred as a result of learning. while the ability to generalize the findings needed reasoning abilities. in other words, the phase generalize the findings obtained in learning with abductive-deductive strategy has been able to facilitate the indicators make logical conclusions. if compared to expository learning, as experienced students the opportunity to learning with abductive-deductive strategy tends to be less. this is because the characteristics of expository learning that make it so. indicators (2) estimate answers and solutionsprocess in the learning with abductive-deductive strategy greatly facilitated in analyze and process evaluate phase. stages of this phase, the teacher directs students to find their own solutions of the information that has been owned by the student. teachers encourage students to do transactive reasoning as to criticize, explain, clarify, justify and elaborate a proposed idea, either initiated by students and teachers. teachers assist students in planning and preparing materials for presentations and discussions. teachers help students to reflect on the investigation process and other processes used in solving the problem of habituation to give students the ability to estimate answers and solution processes. compared with expository in every phase of learning where the teacher presents the material in a way giving a lecture or reading material that students were prepared from a textbook or instructional materials are less certain to develop the ability to estimate shodikin, effect of learning with abductive-deductive strategy … 118 answers and solution processes. this is supported by learning theories expressed by peaget, where knowledge is not passively received. mathematical knowledge is constructed by the children themselves should not be given in the form of so. it should students become active seekers and processors of information, not a passive recipient (davis & murrell, 1994). in other words, students are given the opportunity to learn independently and connect the concepts that have been previously owned, and become involved in meaningful learning. opportunity to explain the idea also be one of the factors supporting the increase in students’ reasoning ability (baig & halai, 2006). students are involved in assessment for learning and of learning (taylor & parsons, 2011). this is the value given to learning with abductivedeductive strategy compared with expository learning. indicators (3) use patterns and relationships to analyze mathematical situations in learning with abductive-deductive strategy facilitated in discussion of strategies to more problems phase. activities of the students in finding strategies to the problems that require more students to see patterns and relationships between a problem with another problem. students will construct a new mathematical knowledge through reflection on actions undertaken both physically and mentally. they made observations to find patterns and relationships, and forming generalizations and abstractions (dienes, 1964). with the investigation of the objects, comparison and analysis of the similarity systemic or non-similarity (pattern) will enhance the students’ reasoning ability (christou & papageorgiou, 2006). therefore, this phase is very help familiarize students use patterns and relationships to analyze mathematical situations. seeing the advantages of learning with abductive-deductive strategy than expository learning in facilitating the development of students’mathematical reasoning ability as described above reinforce that learning with abductive-deductive strategy better than expository learning in the achievement and improvement of students’ reasoning abilities. conclusion based on the finding of research and discussion, it was stated conclusion that the achievements of mathematical reasoning ability students acquire learning with abductivedeductive strategy better than acquire expository learning an overall. seen more detail by category kam, only in the middle category that showed an achievementsof mathematical reasoning abilities students better. while the upper and under categories, both of learning show the achievement of mathematical reasoning ability students is same. recommended for teachers used learning with abductive-deductive strategy in materials with abductive-deductive characteristics to improve mathematical reasoning ability. further research needs to be done for the development of learning with abductive-deductive strategy on other materials in accordance with the characteristics of abductive-deductive such linear program, logarithmic, and trigonometric. there should also be extended to the level of its application such as vocational schools and junior high schools. extended of study and research for the improvement of the other mathematical ability to use learning with abductivedeductive strategy can also be done. for comparison also necessary to do research on the comparison with the strategy of inductive, deductive, inductive-deductive or other extension. references aliseda, a. (2007). abductive reasoning: challenges ahead. theoria. revista de teoría, historia y fundamentos de la ciencia, 22(3), 261-270. volume 6, no. 2, september 2017 pp 111-120 119 baig, s., & halai, a. (2006). learning mathematical rules with reasoning. eurasia journal of mathematics, science & technology education, 2(2). bragg, l. a., loong, e. y. k., widjaja, w., vale, c., & herbert, s. (2015). promoting reasoning through the magic v task. australian primary mathematics classroom, 20(2), 10. christou, c., & papageorgiou, e. (2007). a framework of mathematics inductive reasoning. learning and instruction, 17(1), 55-66. davis, t. m., & murrell, p. h. (1994). turning teaching into learning. the role of student responsibility in the collegiate experience. eric digest. dienes, z. p. (1964). mathematics in the primary school: macmillan. glassmeyer, d., & edwards, b. (2016). how middle grade teachers think about algebraic reasoning. mathematics teacher education and development, 18(2), 92-106. jones, g. a., & thornton, c. a. (1993). vygotsky revisited: nurturing young children's understanding of number. focus on learning problems in mathematics, 15, 18-28. lowrie, t., logan, t., & ramful, a. (2016). spatial reasoning influences students' performance on mathematics tasks. mathematics education research group of australasia. national council of teachers of mathematics (nctm). (2000). principles and standards for school mathematics (vol. 1): national council of teachers of mathematics. mullis, i. v., martin, m. o., ruddock, g. j., o'sullivan, c. y., & preuschoff, c. (2009). timss 2011 assessment frameworks: eric. nataliasari, i. (2014). penggunaan model pembelajaran kooperatif tipe think pair share (tps) untuk meningkatkan kemampuan penalaran dan pemecahan masalah matematis siswa mts. jurnal pendidikan dan keguruan, 1(1). nizar, a. (2007). kontribusi matematika dalam membangun daya nalar dan komunikasi siswa. jurnal pendidikan inovatif, 2(2), 74-80. noto, m. s. (2015). efektivitas pendekatan metakognisi terhadap penalaran matematis pada matakuliah geometri transformasi. infinity journal, 4(1), 22-31. ormrod, j. e. (2008). psikologi pendidikan. jakarta: erlangga. rohana, r. (2015). peningkatan kemampuan penalaran matematis mahasiswa calon guru melalui pembelajaran reflektif. infinity journal, 4(1), 105-119. shodikin, a. (2013). abductive-deductive strategy: how to apply it in improving student mathematics literacy in junior high school. paper presented at the international seminar on mathematics, science, and computer science education, bandung. shodikin, a. (2016). peningkatan kemampuan pemecahan masalah siswa melalui strategi abduktif-deduktif pada pembelajaran matematika. kreano, jurnal matematika kreatif-inovatif, 6(2), 101-110. doi:http://dx.doi.org/10.15294/kreano.v6i2.3713. sumarmo, u. (2013). kumpulan makalah berpikir dan disposisi matematik serta pembelajarannya. jurusan pendidikan matematika: fmipa upi. sun, z., finnie, g., & weber, k. (2005). abductive case‐based reasoning. international journal of intelligent systems, 20(9), 957-983. http://dx.doi.org/10.15294/kreano.v6i2.3713 shodikin, effect of learning with abductive-deductive strategy … 120 taylor, l., & parsons, j. (2011). improving student engagement. current issues in education, 14(1). sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p297-310 297 infinity mathematics critical thinking ability materials social arrithmatic class vii assisted video animation in the era of covid-19 mely della pratiwi, ratu ilma indra putri*, zulkardi universitas sriwijaya, indonesia article info abstract article history: received dec 9, 2021 revised sep 14, 2022 accepted sep 21, 2022 this study aims to determine students' mathematical critical thinking skills after the implementation of learning using pmri and lslc on social arithmetic material with the help of animated videos in class vii. this type of research is descriptive. the subjects of this study were students of smp negeri 1 palembang class vii.1, totaling 32 students. the data collection technique used is the provision of test questions which amount to 2 questions in the form of descriptions, observations, and interviews. after doing the research, it was found that the mathematical critical thinking skills of class vii.1 students on social arithmetic material, the subject of sales, purchases, profits, losses, and percentages had appeared a lot. however, some students still did not show indicators of their mathematical critical thinking abilities. the indicators that seem the most are the analytical indicators, while the indicators that appear the least are the interpretation indicators. keywords: animated video, lslc, mathematical critical thinking, pmri, social arithmetic this is an open access article under the cc by-sa license. corresponding author: ratu ilma indra putri, department of mathematics education, universitas sriwijaya jl. ogan, ilir bar. i, palembang, south sumatra 30139, indonesia. email: ratuilma@unsri.ac.id how to cite: pratiwi, m. d., putri, r. i. i., & zulkardi, z. (2022). mathematics critical thinking ability materials social arrithmatic class vii assisted video animation in the era of covid-19. infinity, 11(2), 297-310. 1. introduction one of the competencies for learning mathematics in social arithmetic material in the minister of national education of the republic of indonesia number 22 of 2006 concerning content standards is to explain mathematical problems in everyday life (kemendikbud, 2006). according to rahmawati and apsari (2018), social arithmetic is a mathematical science that discusses sales, purchases, profits, losses and percentages that are often encountered in everyday life. the importance of this subject as a knowledge base to solve problems that are often experienced. although the subject looks easy, in reality it becomes a difficult subject and causes many problems (ridwan et al., 2016). yanti et al. (2019) said that mathematics is a science that can train a person's way of thinking logically, critically and creatively. one of the competencies contained in regulation https://doi.org/10.22460/infinity.v11i2.p297-310 https://creativecommons.org/licenses/by-sa/4.0/ pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 298 of the minister of education and culture of the republic of indonesia number 20 of 2016 is critical thinking skills (kemendikbud, 2006). according to widiantari et al. (2016), students' critical thinking skills are needed to understand and solve a problem they face by being able to analyze, evaluate, and interpret their own thinking for the better so as to allow errors in working on mathematical problems to be minimized. however, the reality is that students' mathematical critical thinking skills are still low. in the research conducted by pane (2019), the low mathematical critical thinking skills of students in the answers to practice questions given were unsatisfactory, students were unable to understand the problems indicated by writing down what was known and asked questions. appropriately. according to sholihah and mahmudi (2015) research, the low mathematical critical thinking ability of students in learning mathematics is motivated by teacher-centered learning, presentation of material that is not practice-oriented, learning resources are only from textbooks, there are no teaching materials that can help students. in solving problems. this type of learning is not interesting for students should start from what they understand (ahmad, 2018). from these facts, it is necessary for teachers to be able to create innovative learning, one of which is using a mathematics learning approach that is in accordance with the 2013 curriculum. the appropriate approach is the indonesian realistic mathematics education (pmri) approach, because it has the potential to develop democratic, creative, and independent characters. students (johar et al., 2016). pmri is one of the learning approaches carried out in the process of seeking knowledge that is relevant to real problems or everyday life, suitable as a starting point in learning mathematics (putri, 2016). mathematics must be close to students and must be lived with everyday life. in pmri, students should have the opportunity to rediscover mathematical ideas and concepts through various situations and practical problems under adult guidance (fauziah et al., 2021). the following are some of the main principles of pmri, namely guided discovery, mathematical discovery, educational phenomena and independent development models (putri & zulkardi, 2018). in addition, pmri has 5 characteristics, namely using context, using various models, student contributions, interactivity and linkage (putri & zulkardi, 2018). the ability to think mathematically with the indonesian realistic mathematics education approach (pmri) is very much needed by students in solving a mathematical problem. so that the pmri approach is suitable for use in mathematics learning to improve students' mathematical critical thinking skills. this research is supported by using lesson study for learning community. lesson study requires members to learn collaboratively with the aim that students are able to understand, exchange ideas, opinions and also discuss and build their own understanding well by collaborating with friends (nuraida & putri, 2019). lesson study can increase the effectiveness of student learning, the class becomes more effective and the teacher as a facilitator, so that learning becomes better (rusiyanti et al., 2021; shimizu, 2020). currently the world is being shocked by a global pandemic called corona virus disease (covid-19) which causes learning to be carried out online or distance learning (kuntarto, 2017). distance learning (pjj) is learning when students and teachers do not always face to face physically and simultaneously at school (setiawan, 2020). according to budiman (2017), the increasingly rapid development of information technology in the current era of globalization has an effect on the world of education. during the covid-19 period, one of the effective learning media that can used for distance learning, namely learning animation. the benefits of this animated video in the learning process are that it can improve students' critical thinking skills in the form of increasing abilities in terms of: focusing questions, analyzing, inducing and considering results, evaluating and giving reasons, it can be concluded that learning with the help of animated videos can increase volume 11, no 2, september 2022, pp. 297-310 299 infinity students' learning motivation and student attractiveness, so that learning can increase (munandar et al., 2018; octriana et al., 2019). based on this background, a study will be conducted that aims to determine students' mathematical critical thinking skills after being given pmri and lslc learning assisted by animated videos in the covid-19 era. 2. method the type of research used is descriptive qualitative research. the subjects in this study were class vii.1 students of smp negeri 1 palembang, totaling 32 students. data collection techniques used are observation, giving test questions, and interviews. this research was carried out in accordance with the lslc stages, namely plan, do, see and redesign. next, the step to analyze the data from the written test results is to determine the students' mathematical critical thinking skills. there are 6 indicators of mathematical critical thinking skills used in research as shown in the table 1. table 1. indicators and descriptors of mathematical critical thinking ability no indicators descriptors 1. interpretation students can understand the problem by writing down what is known and what is asked of the problem correctly. 2. analysis learners can identify the relationships between the questions, concepts given in the problems shown by making mathematical formulas correctly and giving proper explanations. 3. evaluation students can write problem solving. 4. inference students can draw conclusions from what is asked correctly 5. explanation students can write down the final results and write down the reasons for the conclusions drawn. 6. self-regulation students can review written answers to find out the emergence of indicators of students' mathematical critical thinking skills, it can be done by calculating the scores on each indicator of the questions given. the minimum score for each indicator is 0 and the maximum score for each indicator is 4. if the student gets a score of 4 from each indicator, the student can be said to have brought up indicators of mathematical critical thinking skills properly and correctly. if students get a score of 3 and 2, then the student can be said to have brought up his mathematical critical thinking skills but has not been maximized. however, if students get a score of 1 and 0 then it can be said that students are still not able to bring up their mathematical critical thinking skills. pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 300 3. result and discussion this research was carried out on march 22, 2021 until mid-april 2021. this research was carried out according to the stages in the lslc (lesson study for learning community), namely plan (planning stage), do (implementation stage), see ( observation and reflection stage), and re-design. in the planning stage (plan) the researcher and the math teacher of smp negeri 1 palembang collaborate and discuss to prepare several learning tools, such as lesson plans, lkpd sharing tasks and jumping tasks, as well as test questions to determine students' mathematical critical thinking skills. this plan phase activity involved 3 students of sriwijaya university mathematics education and 2 mathematics teachers from smp negeri 1 palembang (see figure 1). then at the implementation stage (do) learning activities are carried out 2 times, each meeting in 1 meeting is carried out within 2 x 30 minutes. figure 1. planning stage process (plan) the first meeting was held on march 30, 2021 in class vii.1 through the zoom meeting application. the research subjects were divided into 8 groups where one group consisted of 3-4 students and the group consisted of students with low, medium and high abilities (see figure 2). figure 2. online learning process through zoom meeting at the implementation stage, learning activities are divided into 3 parts, namely asynchronous (scheduled pre-study), synchronous (scheduled study), and asynchronous (scheduled post-study). in the scheduled pre-learning learning process (asynchronous) students will be given a video link containing an animated video of student learning on social arithmetic material with the subject of sales, purchases, profits, losses and percentages sent via whatsapp group, then the student can watch the learning video to complete and record volume 11, no 2, september 2022, pp. 297-310 301 infinity any important things that have not been understood which will then be discussed together during the learning activities. there are three learning activities implemented in the scheduled learning process (syncronous), namely preliminary activities, core activities and closing activities. in the introductory activity, the teacher provides a zoom meeting link to students, then students can access it to join the class. then the teacher opens the lesson by greeting, asking students' readiness to carry out learning activities, sending attendance, informing the learning objectives, and providing motivation to students. then the teacher gives appreciation to the students and continues with the core activities. in the core activity there are several activities that must be done by students, namely working on lkpd sharing tasks and jumping tasks and test questions. when working on the lkpd students can write their answers on their respective sheets of paper and the teacher provides opportunities for students to have discussions with their group of friends through a breakout room at a zoom meeting. after completing the lkpd students are returned to one room. after all the students gathered, the teacher gave the opportunity for students to present their answers. if the students have presented their answers, then the teacher gives the students' mathematical critical thinking ability test questions that will be done by each student individually. then the results of these answers can be sent via google classroom which the teacher has prepared. in the last asynchronous activity (scheduled post-study) the teacher made sure all students had collected the answers in google classroom. at this stage of observation and reflection, teachers are asked to convey their impressions during teaching activities. then the observer who served as an observer was asked to convey his findings during the learning activities and provide comments or suggestions. comments or suggestions from the observer will be used by the teacher to redesign the next lesson so that learning activities can be better. at this stage of observation and reflection, observer group 2 said that there was one student who became the focus of observation, namely student a. in his observations, the observer focused on student a who was very confused and did not focus on working on the lkpd. student a seemed to hold and scratch his head repeatedly. but student a is silent and does not ask his group friends for help. figure 3. student a is not focused initially, student a was not focused and felt confused (see figure 3). however, the group was very active in discussing so that one of his friends invited him to discuss and student a began to ask his group friends. the re-design stage was carried out to improve the design of the learning process and documentation. at this stage the teacher and other presenters agree again to improve the lesson plan or something else. the result of the pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 302 recapitulation of the students' mathematical critical thinking ability test questions presented in the table 2. table 2. recapitulation of mathematical critical thinking ability completeness indicators indicators question number 1 2 writing down what is known (interpretation) 8 6 writing formulas or concepts (analysis) 20 16 write down the solution (evaluation) 19 17 making conclusions (inference) 18 16 write the final result correctly (explanation) 19 21 reviewing the answers that have been written (self-regulation) 12 15 question number 1 sfa students (high ability) sfa students are one of the students who answer and write down the information in the problem and can solve the problem correctly and completely (see figure 4). when learning takes place, sfa students are also active participants. figure 4. results of sfa students' answers volume 11, no 2, september 2022, pp. 297-310 303 infinity based on the results of the students' answers (see figure 4), sfa students are able to write answers according to the questions given in the questions. so that sfa students get a maximum score of 4 from each indicator. sfa students are able to bring up 6 indicators of students' mathematical critical thinking skills, namely in indicator 1, sfa students are able to write down information that is known and asked correctly, then indicator 2 is analysis. sfa students can identify the relationships between the questions, concepts given in the problems shown. in indicator 3, namely evaluation, sfa students can write down the steps of completion correctly to get the appropriate final result. the 4th indicator is inference, sfa students can write conclusions from the answers that have been completed. then the 5th indicator, namely explanation, sfa students are able to write the final results correctly. the 6th indicator of self-regulation is that students can review the answers they have written. alm students (medium ability) alm students are one of the students who can solve problems and perform calculations correctly, but there are alm students' answers that are still incomplete (see figure 5). figure 5. the results of alm students' answers pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 304 alm students have involved indicators of mathematical critical thinking skills but have not been maximized (see figure 5). students do not write down information that is known from the problems given so that they get a score of 0 for this indicator. for other indicators, alm participants are able to write and explain completely and precisely so that they get a score of 4. la students (low ability) la students are one of the students who have not been precise and complete in solving the problems given. the following are answers from la students (see figure 6). figure 6. results of la students' answers from the answers of these students, la students have not been able to meet all the indicators of students' mathematical critical thinking skills. the answers written by la students are also still inaccurate, this is because students are less focused on working on the questions (see figure 6). question number 2 in question number 2, there are a variety of student answers, there are student answers that identify the elements of the question and operate correctly, can identify the elements of the question but there are errors in performing calculations. sfa students (high ability) the following is the answer from one of the high-ability students on question no. 2 (see figure 7). volume 11, no 2, september 2022, pp. 297-310 305 infinity figure 7. answers of sfa students students can write down information from the questions (interpretation) analysis inference inference evaluation students are able to write the final result correctly (explanation) inference evaluation analysis explanation inference pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 306 from interviews that have been conducted, sfa students are able to write answers according to the questions given to the questions and indicators requested. so that sfa students get a maximum score of 4 from each indicator. alm students (medium ability) figure 8. answers of alm students from interviews that have been conducted, alm students have been able to solve the problems given correctly and bring up indicators of critical thinking skills but are not complete. alm students do not bring up all the indicators requested, as does the interpretation indicator. alm students do not write down what they know and are asked about from the problem, then at point b, alm students do not show an analysis indicator, that is, they do not write down formulas or concepts (see figure 8). la students (low ability) the following is the answer from one of the low-ability students on question no. 2 (see figure 9). volume 11, no 2, september 2022, pp. 297-310 307 infinity figure 9. answers of la students la students have been able to understand the problems given, but la students have not been able to solve the problems given in accordance with the requested indicators (see figure 9). la students do not write down the information asked so that they get a score of 0, then la students do not write down formulas or concepts so that they get a score of 0, but students are able to make solutions to the problems given so that they get a score of 4. the research that i did was a research on students' critical thinking mathematical abilities after learning using the pmri and lslc approaches on social arithmetic material in class vii.1. the research data taken is from the results of the mathematical critical thinking ability test which was carried out at the evaluation of the second meeting. two test questions given to students are mathematical critical thinking ability test questions that are compiled based on pmri characteristics and indicators of mathematical critical thinking skills. indicators that appear a lot are analytical indicators (writing formulas or concepts). this indicator appears more often than other indicators. this is because students are quite good at analyzing questions, also because they have been trained in working on lkpd questions, sharing tasks and jumping tasks at the first meeting. meanwhile, indicators that do not appear much are interpretation indicators (writing what is known). the appearance of this indicator is the least compared to other indicators. after conducting interviews with students, this happened because students were too focused on solving the problem to completion. then considering the time given is not too much and not enough time to write it down. apart from the research that has been done the teacher's role here is to guide students so that students can solve problems with their knowledge. there is a pmri principle called guided reinvention and has involved student contributions in completing, there is interaction between students and teachers in completing them (zulkardi, 2002). from the results of student work, in activity 1 the teacher used lkpd which had been specially designed with pmri characteristics. the learning process is carried out following the pmri principles where students are guided to find concepts individually (putri & zulkardi, 2018; rahayu & putri, 2018). this research is supported by using lesson study for learning community (lslc), using the stages in the implementation of lesson study, namely plan, do, see, redesign (nuraida & putri, 2019). lesson study can increase the effectiveness of student learning, the class becomes more effective and the teacher as a facilitator, so that learning becomes better (rusiyanti et al., 2021; shimizu, 2020). do not write down the concept/formula not writing down known and asked information miscalculation pratiwi, putri, & zulkardi, mathematics critical thinking ability materials social arrithmatic … 308 from the research that has been done, it shows that after implementing learning with pmri and lslc using the help of animated learning videos, students' mathematical critical thinking skills are quite good. it can be seen from the number of students who succeeded in bringing up the requested indicators. learning with pmri and lslc approaches using animated learning videos can indirectly guide students to develop and improve mathematical critical thinking skills. so that learning mathematics using the pmri and lslc approaches using the help of animated learning videos can be applied in schools. 4. conclusion based on the research that has been done, the results of research on mathematical critical thinking skills of class vii at smp negeri 1 palembang through the pmri and lslc approaches assisted by animated videos in the covid-19 era, the mathematical critical thinking ability of class vii.1 students of smp negeri 1 palembang on social arithmetic material has emerged, although not all achieve the maximum score. the conclusion from the research that has been done is that the mathematical critical thinking ability of grade vii students through the pmri and lslc approaches on social arithmetic material assisted by animated videos is said to be quite good with several details as follows: the indicators that most often appear are analytical indicators, while the indicators that appear the least are namely the indicator of interpretation. the use of learning animation videos as learning media is very helpful for students in learning in the current covid-19 era, which requires students to learn from home online. to improve and develop students' mathematical critical thinking skills, teachers can apply pmri learning and the lslc system with the help of learning animation videos. acknowledgements the authors would li to thank universitas sriwijaya who has given support. this article is part of a research project funded by a professional grant from sriwijaya university with the rector's decree number 0014/un9/sk.lp2m.pt/2021 and research contract number 0127/un9/sb3.lp2m.pt/2021. references ahmad, h. 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(2018). noticing students' thinking and quality of interactivity during mathematics learning. in proceedings of the first indonesian communication forum of teacher training and education faculty leaders international conference on education 2017 (ice 2017). https://doi.org/10.2991/ice-17.2018.118 rahayu, a. p., & putri, r. i. i. (2018). learning process of decimals through the base ten strips at the fifth grade. international journal of instruction, 11(3), 153-162. https://doi.org/10.12973/iji.2018.11311a rahmawati, p., & apsari, n. (2018). analisis kemampuan pemecahan masalah aritmatika sosial pada siswa sdn no. 05 suruh tembawang (perbatasan indonesia-malaysia) [analysis of social arithmetic problem solving ability of students at sdn no. 05 tell tembawang (indonesia-malaysia border)]. jurnal pendidikan informatika dan sains, 7(2), 209-218. https://doi.org/10.31571/saintek.v7i2.1044 ridwan, r., zulkardi, z., & darmawijoyo, d. 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(2020). lembar kegiatan literasi saintifik untuk pembelajaran jarak jauh topik penyakit corona virus 2019 (covid-19) [scientific literacy activity sheet for distance learning on the topic of 2019 corona virus disease (covid-19)]. edukatif: jurnal ilmu pendidikan, 2(1), 28-37. https://doi.org/10.31004/edukatif.v2i1.80 shimizu, y. (2020). lesson study in mathematics education. in s. lerman (ed.), encyclopedia of mathematics education (pp. 472-474). springer international publishing. https://doi.org/10.1007/978-3-030-15789-0_91 sholihah, d. a., & mahmudi, a. (2015). keefektifan experiential learning pembelajaran matematika mts materi bangun ruang sisi datar [the effectiveness of experiential learning in mathematics learning in subject matter of flat side construct]. jurnal riset pendidikan matematika, 2(2), 175-185. https://doi.org/10.21831/jrpm.v2i2.7332 widiantari, n. k. m. p., suarjana, i. m., & kusmariyatni, n. 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(2002). developing a learning environment on realistic mathematics education for indonesian student teachers. doctoral dissertation. university of twente, enschede. retrieved from https://repository.unsri.ac.id/871 https://doi.org/10.2991/assehr.k.211122.034 https://doi.org/10.31004/edukatif.v2i1.80 https://doi.org/10.1007/978-3-030-15789-0_91 https://doi.org/10.21831/jrpm.v2i2.7332 https://doi.org/10.26877/aks.v10i2.4399 https://repository.unsri.ac.id/871 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.239 59 development of teaching materials algebraic equation to improve problem solving sri adi widodo mathematics education, sarjanawiyata tamansiswa of university, yogyakarta, indonesia sriadi@ustjogja.ac.id received: october 27, 2016 ; accepted: january 5, 2017 abstract problem-solving skills are the basic capabilities of a person in solving a problem and that involve critical thinking, logical, and systematic. to solve a problem one-way necessary measures to solve the problem. polya is one way to solve a mathematical problem. by developing teaching materials designed using the steps in solving problems polya expected students could improve its ability to solve problems. in this first year, the goal of this study is to investigate the process of learning the hypothetical development of teaching materials. this study is a research & development. procedure development research refers to research the development of thiagarajan, semmel & semmel ie 4-d. model development in the first year is define, design, and development. the collection of data for the assessment of teaching materials algebra equations conducted by the expert by filling the validation sheet. having examined the materials of algebraic equations in the subject of numerical methods, reviewing the curriculum that is aligned with kkni, and formulates learning outcomes that formed the conceptual teaching material on the material algebraic equations. from the results of expert assessment team found that the average ratings of teaching materials in general algebraic equation of 4.38 with a very good category. the limited test needs to be done to see effectiveness teaching materials on problem-solving skills in students who are taking courses numerical methods. keywords: teaching materials, problem-solving, equation of algebra abstrak kemampuan memecahkan masalah adalah kemampuan dasar seseorang dalam memecahkan masalah dan yang melibatkan pemikiran kritis, logis, dan sistematis. untuk memecahkan suatu permasalahan dibutuhkan suatu cara atau langkah-langkah untuk memecahkan permasalahannya. polya adalah salah satu cara untuk memecahkan masalah matematika. dengan mengembangkan bahan ajar yang dirancang menggunakan langkah-langkah dalam pemecahan masalah polya diharapkan siswa dapat meningkatkan kemampuannya untuk memecahkan masalah. pada tahun pertama ini, tujuan dari penelitian ini adalah untuk menyelidiki proses belajar pengembangan hipotetis bahan ajar. penelitian ini merupakan penelitian dan pengembangan. penelitian pengembangan prosedur mengacu penelitian pengembangan thiagarajan, semmel & semmel yaitu 4-d. pengembangan model pada tahun pertama adalah mendefinisikan, desain, dan pengembangan. pengumpulan data untuk penilaian pengajaran persamaan bahan aljabar yang dilakukan oleh ahli dengan mengisi lembar validasi. telah memeriksa bahan persamaan aljabar dalam subjek metode numerik, meninjau kurikulum yang selaras dengan kkni, dan merumuskan hasil belajar yang membentuk materi pengajaran konseptual pada materi aljabar persamaan. dari hasil penilaian tim ahli menemukan bahwa peringkat rata-rata bahan ajar secara umum persamaan aljabar dari 4,38 dengan kategori sangat baik. tes terbatas perlu dilakukan untuk melihat bahan ajar efektivitas keterampilan pemecahan masalah pada mahasiswa yang mengambil kursus metode numerik. kata kunci: bahan ajar, problem-solving, persamaan aljabar widodo, development of teaching materials algebraic equation to improve … 60 how to cite: widodo, s. a. (2017). development of teaching materials algebraic equation to improve problem solving. infinity, 6 (1), 59-68. introduction problem-solving skills are the basis of a person's ability to solve a problem involving critical thinking, logical, and systematic. importance is given a mathematical problem can not be separated from its role in life, which is to develop one's ability to face problems. in mathematics, problem-solving skills have an important role, namely as an initial capability for students in formulating the concept and capital success for students in solving mathematical problems. in addition, students can develop an idea or ideas they have. troubleshooting is also becoming an important thing to be imparted to learners. one of the goals of mathematics learning is the learners are expected to foster critical thinking skills, logical, systematic, thorough, effective, and efficient in solving problems (bsnp, 2006). whether or not the purpose of learning mathematics one of which can be seen from the success of learners in understanding mathematics and utilize this understanding to resolve the problems of mathematics and other sciences. with problem-solving, mathematics becomes not lose meaning. for a concept or principle be meaningful if it can be applied in problem solving, as expressed by widjajanti (2009) which states that solving the problem is the process used to resolve problems. anisa (2014) problem-solving ability is strongly associated with the student's ability to read and understand language about the story, present in the mathematical model, plan calculation of the mathematical model, and complete the calculation of the questions that are not routine. almost as disclosed by windari, dwina & suherman (2014) states that the study of mathematics student should be able to solve problems that include the ability to understand the problem, devised a mathematical model, solve the model, and interpret the obtained solution. to solve the math problem required a method or systematic measures so that the process becomes the easy and effective solution. as expressed by bransford (purnomo & mawarsari, 2014), stated that the measures to solve the problem, namely (1) the identification of the problem, (2) defining the problem through the process of thinking about the problem and perform sorting the relevant information, (3) to discover its solutions through a search for an alternative, brainstorming, and the checking of different points of view, (4) implementing alternative strategies chosen, and (5) to review and evaluate the consequences of the activities undertaken, while solving problems step by polya (1973), is understand the problem or understand the problem, make a plan, or a plan, carry out our plan or carry out the plan, and look back at the completed solution or check answers. of these two opinions, in principle measures to solve the problem boils down to solving the problem of polya. in solving the problem, students sometimes do not write what is known and asked of the problems encountered. educators so difficult to guess whether learners have understood the problems encountered or not. when educators have indicated that learners do not understand the problem, it turns out that learners are able to solve the problem correctly. but if educators have indicated that learners understand the problem, such learners have not written what is known and what is being asked. it is as expressed by widodo (2013) states that there are some students in solving problems, which are not written what is known and what is being asked. volume 6, no. 1, february 2017 pp 59-68 61 the same thing also expressed by widodo and sujadi (2015) who found that a small percentage of learners in solving the problem, do not write what is known and what is asked, but learners are able to resolve the problems faced by correct. in addition, the students are sometimes not able to tell what steps should be done to resolve the problem. learners do not understand the steps necessary to plan to resolve the problem. terms of necessary and sufficient to resolve the already can describe plans are sometimes not performed by a student. accepted by widodo (2013) states that learners are not able to deliver a sufficient condition and a necessary condition so that students have not been able to plan your to solve their problems. in the phase or step to re-examine the answers, students are almost entirely not the process. learners consider that this step makes time to solve the problem is not short (time-wasting). as expressed by widodo (2013) learners did not do anything at this stage of checking back. in fact, if the students are able to use the stage of checking back with good, small mistakes made by learners can reduce. different things revealed by utomo (2012) states that at the stage of checking back in a stage that weighs most in the classification level of thinking, this is because at this stage to re-examine learners only checks the accuracy of the calculation results she had done, check systematics funds stages of its solution if it's good or not. the thought is what causes most learners do not re-examine stages in solving the problem. based measures such polya, learners are only able to able to solve the problem using only the third step. for the first step, the second and fourth learners are rarely used to use in solving problems. this is why the ability to solve problems and student achievement be not optimal. though anisa (2014) stated that learning mathematics is successful if it produces students who have problem-solving skills, communication skills, reasoning ability, comprehension ability and the ability of other well and are able to utilize mathematical usefulness in life. numerical methods is one of the subjects who learn about the techniques to solve the mathematical problems that are not able to be completed in general. one of the materials studied in numerical methods is algebraic equations. in the material algebraic equation given the problems learners including shaped polynomial equations and learners should be able to determine the roots of the polynomial equation. for example, problems in numerical methods is to determine the roots of the equation 4x 7 1,25x 6 + 120x 4 + 15x 3 120x 2 x + 100 = 0, learners will be difficult to resolve the problem of the equation. this is what led to the achievement of students in the subject of numerical methods have not been satisfactory. learners still think that to solve algebra problems are not common can still use methods that are commonly used. in fact, by using numerical methods, the problems are not generally be solved. as expressed by widodo (2014) which states that to solve algebraic equations of degree two polynomial can still use the formula abc, but to an algebraic equation with a polynomial of degree more than three will be trouble if using formula abc. instructional materials have a very important position in learning, ie as a representation of the explanation educator in front of the class. on the other hand, teaching materials serve as the means to achieve competence. so the preparation of teaching materials should be guided competencies to be achieved. teaching materials prepared without referring competence, certainly will not provide many benefits to the learners. making teaching materials are part of the development process of innovation in education. teaching materials used do not always have to be conventional but as educators, at least take action to repair paradigm, perspective, widodo, development of teaching materials algebraic equation to improve … 62 thinking, attitudes, habits, professionalism, and behavior in teaching. thus educators deliver innovation in making teaching materials because it will have an impact on the smooth operation of student learning that is not likely to feel bored. development of teaching materials that originated from conventional towards innovative becomes very important because it will greatly help the learning process itself, especially teachers to assist students in learning to become interested and feel pleasant. if students have had the pleasure of learning thus studied spirit will increase. the key to the development of innovative teaching materials lies in the creativity of teachers themselves. it thus should not be an obstacle but a challenge for teachers to be able to continue to upgrade the capability to develop her potential, especially in the development of innovative teaching materials. along with the modern education system and the demands are growing, not infrequently schools still use conventional ways to implement the learning process. learning in the conventional way this is usually done in solitary, which means that the learning process from the planning, implementation, to the students' learning evaluation conducted by one teacher. planning done by teachers usually set up a book or instructional materials is nearly the same as in the previous year, even learning plan which will be used during the learning process is still the same as years previous. whereas the preparation of plan at least have to adjust the characteristics of learners. therefore, when teachers are still using the same plan with previous years, the teachers are indirectly learners are considered equal characteristics, whereas humans are born into the world has unique differences. suyono in nugroho (2011) stated that the weakness of mathematics instruction is done by teachers at the school include the lack of ability of teachers using a variety of learning, teaching capabilities limited only to answer the questions, teachers did not want to change the learning is already considered to be true and effective, and teachers simply using conventional learning without attention to think of learners. one of the learning tools that are used during the learning process includes instructional materials or textbooks. instructional materials become a determining factor for learners to participate in the learning and create interest in the material to be taught. as expressed by supriyono, setiawan & trapsilasiwi (2014) states that to produce active learning, easy to understand, and fun for students requires a learning model that makes students actively participate in the learning process, the learning process which makes the students active participation during the learning process and students interested in learning that can be created by using the device. based on this background it is necessary to develop a learning device in this case algebraic equations teaching materials adapted to the troubleshooting steps. the hope is to develop teaching materials, academic achievement and problem-solving skills of students in the subject of numerical methods can be improved. method the method used in this study is a model of research and development continued the experiment. the research model is the development of research methods used to produce a specific product and test the effectiveness of product (sugiyono, 2009). model development in this research aims to acquire problem-solving based teaching materials on the subject of volume 6, no. 1, february 2017 pp 59-68 63 algebraic equations. having obtained the problem-solving based teaching materials, the next step is to conduct research experiments to see the effect of the instructional materials to the student achievement in order to obtain teaching materials based on the final solution. rnd procedure as an activity process used to develop the various aspects related to education to produce or develop. the main purpose of research and development as proposed gay (1990) is not to test the hypothesis but rather to produce a product that can be used in education. under these conditions, the main purpose of this research and development is to obtain teaching materials based on the material problem solving algebraic equations. model development of teaching materials in this study refers to the research model development of the 4-d developed by thiagarajan, semmel & semmel (1974), which define, design, development, and dissemination. according to yusnita (2011), the advantages of model 4-d, among others: (a) more appropriate to use as the basis for developing a learning device is not to develop a learning system, (b) the description seems more complete and systematic, (c) in its development involves the assessment of experts, so that prior to being field-tested learning device has been revised based on assessment, advice, and input of experts. phase activities define done to establish and define the terms of development. in general , at this defining stage, activities of development needs analysis, requirements development of products that fit the needs of users as well as research and development models suited to develop products. the analysis can be done through the study of literature or preliminary research. in determining and establishing the terms of the learning device starts with (a) the analysis of the curriculum, (b) analysis of the material in the course of algebraic equations numerical methods, and (d) formulate learning objectives or learning outcomes. at the stage of design, has prepared prototype learning device or product design (hypothetic teaching materials). at this stage to make teaching materials in accordance with the framework of the contents of the analysis results, curriculum, and materials. prior to the design of the product proceed to the next stage, then the hypothetical teaching materials need to be validated. validation of product design is done by a team of expert judgment as lecturers with qualifications (1) s2 mathematics education ever taught courses in numerical methods or (2) s2 applied mathematics. at this stage of development aiming to produce learning tools which have been revised based on feedback from the experts (team of experts). the hope of teaching materials that really meet the needs of users. while the pilot phase is limited and expanded trials can not be conducted at this year due to the time of distribution and adoption of instructional materials have to adjust the distribution of curriculum by a department of mathematics education in sarjanawiyata tamansiswa of university. conceptual teaching materials have been obtained, it is necessary to assess its feasibility. to see the feasibility of teaching materials is done with content validity. the validity of the contents shows that the teaching materials are not developed at random but must be able to be justified scientifically and correctly in terms of science. comic assessed the feasibility of teaching materials with a range of 1 5. the indicators or aspects of an assessment (validation) comic teaching material refer to (1) structural aspects of teaching materials, (2) the material aspects of teaching materials, (3) organization, presentation and writing on teaching materials, and (4) aspects of language and legibility on teaching materials. widodo, development of teaching materials algebraic equation to improve … 64 data from the analysis of the validity of teaching materials referring to the opinion of suswina (2011) that mean> 3.20 = highly valid, 2.40 <= valid average ≤ 3.20, 1.60 <averages ≤ 2,40 = quite valid , 0.80 <mean ≤ 1.60 = less valid, mean ≤ 0.80 = invalid. because the validation sheet these materials scoring range between 0-5 then the criterion of meiriza suswina (2011: 44-51) was adapted into a mean of> 3.75 = excellent, 2.92 <averages ≤ 3,75 = good, 1.67 <averages ≤ 2,92 = pretty good, 1.25 <averages ≤ 1,67 = less good, average ≤ 1,67 = not good. as the criteria that the teaching materials created learning algebraic equations can be used for further processing (limited trial) if obtained an average score of at least 2.92. results and discussion results the results of the validation of teaching materials are done by a team of experts to complete a validation that has been prepared. with this validation, a sheet is expected (1) to provide feasibility assessment or instructional materials algebraic equations that have been developed and (2) obtain the inputs used to revise teaching materials algebraic equations. table 1. structural aspect indicator average information the structure of teaching materials in general interest to students 4,50 very good see generally for learners 4,00 very good average 4,25 very good table 2. creative aspects indicator average information material support study materials 4.50 very good material support learning outcomes 5.00 very good the material is made to support understanding of the concept 4.00 very good truth concepts (definitions, formulas, laws, etc.) 4.50 very good teaching materials support the ability of learners to problemsolving 4.50 very good average 4,50 very good volume 6, no. 1, february 2017 pp 59-68 65 table 3. organizational, presentation, and writing aspect indicator average information proportional font to format or pictures or graphics 4.00 very good the material is presented in a simple and clear 5.00 very good pictures on teaching materials presented with clear and attractive 3.50 very good average 4,17 very good table 4. language and readability aspect indicator average information materials have been using language that is easily understood by student 4.50 very good teaching materials have been using communicative 4.50 very good teaching materials have been using the term in accordance with the concept of the subject 4.50 very good average 4.50 very good discussion the first year is the development of research that aims to produce teaching materials algebraic equation as one of the subjects in the course of numerical methods. the teaching materials are instructional materials containing material algebraic equations, questions, and exercises. sample questions given on teaching materials is solved by using the steps in solving problems of polya (1973), namely (1) the stage of understanding the problem, (2) phase plan to resolve the problem, (3) the stage of implementing the plan, and (4) the stage concludes the solution or the answer to solving the problem. utomo (2012) stated that the measures implementing the plan have the greatest weight compared with other troubleshooting steps, while on stage recheck answers weigh the lowest when compared with other troubleshooting steps. in addition the results of widodo (2012; 2013); widodo and sujadi (2015) most students solve problems using only the first step that is the first step polya that understand the problems that are implemented with length known and asked and the third step of carrying out plan implemented by step answer. referring to the opinion of polya (1973) and utomo (2012) and the findings in previous studies, the troubleshooting steps polya adapted into three steps is to understand the problem, plan to resolve the problem, and implement plans to solving a problem. while in the fourth stage polya which checking back answers, adapted to conclude the answer. having obtained the final ability possessed by the student and the material to be studied by the students, the next step is to design teaching materials conceptual. the design of teaching widodo, development of teaching materials algebraic equation to improve … 66 materials conceptually refers to the ability of the end possessed by students, namely (1) students are able to apply some numerical method to solve the algebraic equation correctly, (2) students are able to solve problems of algebraic equations using several numerical methods, and (3) the students were able to deduce which method is effectively used to solve the problem of algebraic equations, and the material to be studied by the students that real root layout, biseksi method, method of any false, iteration method, newton-raphson method, and the secant method. teaching materials algebraic equation prepared attention steps polya adapted: (1) understand the problem, (2) plan for a plan to resolve the problem, (3) implement a plan to resolve the problem, and (4) concludes solve the problem, it this aims to make students familiar in solving algebraic equations hopes students have the ability to solve problems and learn quite a good achievement. in addition, algebraic equations teaching materials prepared by the concept of guided problem solving, that means teaching material given sample questions along with the answers, sample questions with answers partially covered, and exercises. the assessment of the expert team concluded that in general the teaching materials algebraic equation has been good, interesting ideas, and can be performed limited testing for effectiveness. but there are some mistakes that should be revised related to (1) grammar equations polynomial should have been written using microsoft equation but not yet written using microsoft equation, (2) subtitles methods one counterfeit is recommended to be replaced with a method of positioning a fake, and (2 ) the calculation process in problem solving algebraic equations still wrong. from this assessment of the expert team, then it is necessary to consider these suggestions so that errors that occurred writing that occurs on conceptual teaching materials can be improved. from table 1 to table 4 shows that the average score ratings were obtained from several aspects: (1) structural aspects of teaching materials gained 4.50 to excellent category, (2) a material aspect obtained 4.50 with very good categories, ( 3) organization, presentation and writing gained 4.17 with very good categories and (4) aspects of language and legibility obtained 4.50 with very good category. from the results of the validation of teaching materials, it can be concluded that the teaching material algebraic equation can be expressed has been very good. his result is also supported by the overall average (mean in general) in the amount of 4.38 with a very good category. to look at the effectiveness of teaching materials algebraic equations need to do limited testing prior to wider research. validity conducted in these studies underscore the validity of the content. the validity of the content may be declared invalid by the very good category by the validator. this is because the teaching materials developed algebraic equations in accordance with the material that should have been presented. based on the results of the validation expert teaching materials algebraic equation of the fourth aspect of the indicator is stated that the teaching materials algebraic equations in the excellent category. in the aspect of teaching language, the structure shows that the overall structure of the teaching materials presented very well, appealing to students. in the aspect of the material showed that the material used is very supportive of study materials and achievement of learning on the subject of numerical methods, material presented supports the understanding of the concept, the truth of the concept (definition, formula, law, and so on) is very good, teaching materials developed to support the ability of participants students in solving mathematical problems, especially algebraic equations. volume 6, no. 1, february 2017 pp 59-68 67 depdiknas (2008) states that the teaching material is a set of materials arranged in a systematic, whether written or not so as to create the environment or atmosphere that allows students to learn. according to depdiknas (2008) is a set of instructional materials arranged in a systematic matter, whether written or not so as to create the environment or atmosphere that allows students to learn. the validity so that the content or truth of the contents in science and alignment system based on values shared by a society or a nation. the contents of teaching materials algebraic equations were developed based on the concept and the prevailing theory in the field of science as well as in accordance with the development of the field and the results of empirical research conducted in the discipline. in the aspect of the organization, presentation and writing showed that the size of the letters is in proportion with the format as well as images or graphics, the material on teaching materials are presented in a simple and clear picture on teaching materials are presented with a clear and compelling. the high rating validator on organizational aspects, presentation and writing as well as aspects of the language and legibility due to the process of development of teaching materials simple algebraic equations attention to aspects of his performance but easy to read. as expressed by nurseto (2011) states that in developing learning tools, developers must pay attention to the principle of "visuals", which is visible (easily seen), interesting (attractive), simple (simple), useful (it useful or beneficial), accurate (correct and accountable), legitimate (reasonable), and structured (structured / structured conclusion the process of developing teaching materials algebraic equations only through 3 stages: (1) step of defining (define) do analyzing curriculum, materials and formulate competency and indicators of achievement of competencies, (2) the design phase (design) done by creating a prototype teaching materials algebraic equations, (3) the development stage (develop) the validation tests conducted by experts who have a background in master in applied mathematics or mathematics education master who taught methods of numerical mathematics materials obtained an average of 4.38 with the criteria very well. from these results can be suggested that the result of the development of teaching materials algebraic equation can be used for experimental research or research limited. references anisa, w. n. 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(2015). analisis kesalahan mahasiswa dalam memecahkan masalah trigonometri. jurnal sosiohumaniora, 1(1), 51-63. windari, f., dwina, f., & suherman. (2014). meningkatkan kemampuan pemecahan masalah matematika siswa kelas viii smp n 8 padang tahun pelajaran 2013/2014 dengan menggunakan strategi pembelajaran inkuiri. jurnal pendidikan matematika, 3(2), 25-28. yusnita, e. (2011). pembelajaran kontekstual berlatar pondok pesantren pada materi garis dan sudut di kelas vii mts. seminar nasional penelitian, pendidikan dan penerapan mipa (pp. pm 11 pm 18). yogyakarta: fmipa uny. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p163-176 163 studying student statistical literacy in statistics lectures on higher education using grounded theory approach lukman, wahyudin, didi suryadi, dadan dasari, sufyani prabawanto* universitas pendidikan indonesia, indonesia article info abstract article history: received sep 27, 2021 revised jan 6, 2022 accepted jan 11, 2022 the purpose of this study was to obtain an overview of the student's statistical literacy model in statistics learning in higher education. researchers conducted an in-depth study of student statistical literacy, how they understand and apply statistics, how statistics are used as a tool for reliable data that can be trusted as scientific works. the research method uses qualitative research methods with a grounded theory approach. participants involved in this study were 114 participants from several universities in west java, indonesia. the results of this study found a student statistical literacy model consisting of 2 dimensions and 5 elements. dimensions of statistical knowledge: descriptive statistics, inference statistics, statistical communication and statistical reasoning. attitude dimensions: confidence and critical attitude. keywords: grounded theory, statistical literacy this is an open access article under the cc by-sa license. corresponding author: sufyani prabawanto, department of mathematics education, universitas pendidikan indonesia jl. dr. setiabudi no.229, isola, sukasari, bandung city, west java 40154, indonesia email: sufyani@upi.edu how to cite: lukman, l., wahyudin, w., suryadi, d., dasari, d., & prabawanto, s. (2022). studying student statistical literacy in statistics lectures on higher education using grounded theory approach. infinity, 11(1), 163-176. 1. introduction in the era of fast and easy digital information, statistical information is easy to obtain through digital media. statistical information is always used by various institutions, such as: research and educational institutions, statistical institutions or bureaus, data information centers, and others. many news stories from the mass media, research journals, and government agencies use statistical information to strengthen their argument and reliability. the data and statistical analysis used must be true and good, and meet reliability and have been tested in theory and practice (bellinga & gillebaart, 2018). the development of information technology triggers everyone to get fast-paced and valid information. no exception with statistical information. almost all institutions in reporting their performance results use statistics because they are more concise and information dense. therefore, many institutions hold statistical analysis training using statistical analysis software to improve https://doi.org/10.22460/infinity.v11i1.p163-176 https://creativecommons.org/licenses/by-sa/4.0/ lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 164 their performance (noha, 2013). the need to understand statistical information has been highlighted by various organizations in the united states, including the american library association. despite the increasing need for teaching statistics, statistical education has historically been viewed by many students as difficult and unpleasant to learn, and by many teachers as a frustrating and unprofitable course. as more students enroll in introductory statistics courses, teachers are faced with many challenges in helping these students succeed in their study of statistics (ben-zvi & garfield, 2004). understanding statistical information is not just knowing the average, standard deviation, graphs, etc. but must have statistical thinking skills such as understanding how the data was obtained. statistical thinking involves understanding why and how statistical investigations are conducted and the ideas underlying statistical investigations. the foundation of statistical investigation rests on the assumption that many real situations cannot be assessed without proper data collection and analysis. statistical thinking includes how to understand data, model, choose analytical methods, estimate probabilities, test hypotheses, make conclusions, understand the context of problems, criticize and evaluate the results of statistical analysis (ben-zvi & garfield, 2004). statistical models must capture elements of real situations; so that the resulting data will lead to the context of statistical knowledge (cobb & moore, 1997). the topics that can be taught to students for students' statistical thinking and reasoning are the topic of data distribution (bakker & gravemeijer, 2004), the concept of average (konold & pollatsek, 2004), concepts of variance and covariance (moritz, 2004; reading & shaughnessy, 2004), normal distribution (batanero et al., 2004), sample and sample distribution (chance et al., 2004; watson & callingham, 2003). some of the problems in statistics learning are summarized by tishkovskaya and lancaster (2012), namely: how to apply statistics content in problem solving (allen et al., 2010; garfield, 1995), learning is carried out by teachers who do not understand statistics (meng, 2009; smith & staetsky, 2007; verhoeven, 2006), anxiety and lack of interest in learning statistics (gal & ginsburg, 1994; garfield, 1995), difficulties for students of other disciplines to study probability theory and statistics (garfield, 1995; garfield & ben‐zvi, 2007), weak knowledge of basic statistics and mathematical reasoning (batanero et al., 1994; garfield & ahlgren, 1988). from the problems above, statistical literacy is an urgent need for students to have. statistical literacy is a basic ability that must be possessed by students so as not to cause misinterpretations and make wrong decisions. improper representation of data will risk problems and journalists' misunderstandings about statistical issues in their academic press releases (spiegelhalter & riesch, 2008). in indonesia, most people do not yet have awareness of the importance of statistical literacy, except for a small number of the academic community. therefore, the indonesian government includes literacy (including statistical literacy) into the national curriculum that must be taught to all students and students. based on the needs above, there needs to be a minimum competency of statistical literacy, especially for students as a scientific community who thinks scientifically based on valid data. the purpose of this study was to examine students' statistical literacy in depth in order to obtain a student statistical literacy model that is in accordance with the demands of today's era. for this purpose, the researcher uses a qualitative research method with a grounded theory approach. 2. method this study uses a qualitative research approach grounded theory (gt). the reason the researcher uses the gt approach is because in this approach the expected result is a volume 11, no 1, february 2022, pp. 163-176 165 theory (glaser & strauss, 1967). the basic principle of the gt approach is that there are three coding stages that can be developed by researchers (creswell, 2014; goulding, 1999), namely: open coding, axial coding and selective coding. the gt research procedure that the researcher developed is depicted in the flowchart diagram (see figure 1). figure 1. flowchart of research methods and procedures gt approach the participants involved in this research were 114 students at several universities with education, engineering, economics, and social study programs. research locations in several universities in west java, indonesia. data collection and analysis used open questionnaires, in-depth interviews, observations and learning videos. open questionnaire data in the form of written participant answers, in-depth interview data in the form of transcripts of voice, observation and learning data in the form of videos. 3. results and discussion as is usual in qualitative research, the gt research instrument is the researcher himself (charmaz, 2014; creswell, 2014), this is different from quantitative research instruments. theoretical sampling in gt research (generally qualitative research) is in the form of data in the form of interview transcripts, questionnaires, video transcripts, notes, public documents, diaries, participant journals, and researcher reflective notes. researchers used atlas.ti in qualitative data analysis. lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 166 3.1. results 3.1.1. data collection and open coding data collection and open coding were carried out simultaneously until the data was saturated. the data are grouped based on the results of questionnaires, interviews, observations and learning videos. open coding goes through stages: first, grouping words or phrases using word blob, counting words or phrases that often appear, and finally determining the right code. the results of the open coding of questionnaire data, interviews, observations and learning videos were as many as 145 codes from 301 quotations. one quote can be a sentence, one or more paragraphs. in open coding, researchers make memos to find out the relationship between codes and find core categories. this is important in the axial and selective coding stages. the relationship between open coding and axial coding is depicted in figure 2. figure 2. relationship of open coding and axial coding causal conditions are categories of conditions that affect the core category. context is the special conditions that affect the strategy. core categories are the phenomenal ideas central to the process. the intervening conditions are the general conditions that influence the strategy. strategies are specific actions or interactions that arise from core categories. consequences are anything that arises with the implementation of strategies, strategies and consequences are used in the selective coding stage to find theories or research conjectures. the results of the open coding show that statistical literacy is the core category, because this category always appears in every open coding and all categories are connected to it (see figure 3). therefore, the core category of open coding is statistical literacy. other categories that often appear in open coding are descriptive statistics, inference statistics, statistical knowledge, statistical reasoning and statistical communication. some important notes for researchers when open coding is: "there is a strong relationship between descriptive statistics and inference statistics" "statistical communication and statistical reasoning are indispensable in interpreting data contextually and in making conclusions." “students have high confidence when they understand statistics well.” volume 11, no 1, february 2022, pp. 163-176 167 (a) open coding of questionnaire data (b) open coding of interview data (c) open coding of learning video (d) open coding of observation data figure 3. open coding for all data 3.1.2. axial coding based on open coding and researcher notes, the core category “statistical literacy” was obtained. based on the records, the results of the axial coding found 6 strategies consisting of 2 interventions: “descriptive statistics” and “statistics inference”, and 4 contexts: “statistical reasoning”, “statistical communication”, “statistical knowledge” and “attitudes” (see figure 4). figure 4. axial coding with statistical literacy as the core category axial coding is the stage of comparing codes (comparing codes), comparing codes with each other and trying to understand how one code relates to another (chametzky, 2016). lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 168 when comparing code and writing memos a pattern develops. descriptive statistics is the ability to read data, determine models and variables, calculate, analyze and interpret data. reading data consists of reading tables and graphs, reading text and context. calculating statistics based on statistical formulas and rules, students must have good basic counting skills. analyzing and interpreting statistics are important things for students to understand before they make conclusions about the phenomena that occur. philosophically statistics is the applied science of how humans can interpret and predict statistical data inductively (de finetti, 1989; hotelling, 1951; neyman, 1956)generalizing a sample to a population inductively is called inductive inference. therefore, students must understand inference statistics in making decisions and conclusions on the sample data. inference statistics is the ability to choose the right statistical test based on research hypotheses, comparing theoretical and empirical statistics, the ability to make decisions to accept or reject statistical hypotheses (h0) and the ability to make appropriate conclusions based on inductive inference. statistical literacy that must be possessed by students based on the researcher's notes is statistical communication, statistical knowledge, statistical reasoning, and attitudes. statistical communication is the ability to read and interpret statistical language into a language that is understood by the general public. one of the weaknesses of students in studying statistics is statistical communication, this is due to the fact that most students view statistics as an abstract mathematical science, containing symbols and formulas. the ability to analyze and draw conclusions inductively requires good statistical reasoning skills. statistical knowledge which includes understanding the context of the problem to be studied, understanding statistical terms and symbols is a sufficient requirement for students to have good statistical literacy. finally, good statistical literacy will increase students' confidence and confidence in the results of their research. the attitude category consists of subcategories of self-confidence and critical attitude. 3.1.3. validity and reliability validity ensures that the statements made by the researcher reflect the reality he claims to be. validity concerns correctness, while reliability ensures that research findings are reproducible and that there are no or almost minor external disturbances that contaminate the data. the relationship between reliability and validity is that the more unreliable a data is, the less likely it is that the researcher can draw valid conclusions from the data. on the other hand, reliability does not always guarantee validity. two coders with a high coefficient of relevance are objectively likely to be wrong if the two coders each have a unique perspective or have different academic disciplines (krippendorff, 2004). validation and reliability in qualitative research are different from quantitative research, in qualitative research using interpretive concepts while in quantitative research using measurement theory. data validation uses constant comparison analysis on triangulated data, in grounded theory constant comparison analysis emerges inductively from different data sources: questionnaire data, interview data, learning video data and observation results. the researcher uses the atlas.ti program to see the consistency of the codes appearing in each different data (see figure 5). volume 11, no 1, february 2022, pp. 163-176 169 figure 5. constant comparison analysis using atlas.ti constant comparison analysis using code-document table analysis, data were normalized to equalize coding density for all documents. the learning and observation video data are combined in the video column, because they are both video data. the number in the cell indicates the frequency with which the same code appears in different documents, while the percentage indicates the relative frequency of the code in each document that has been normalized. based on figure 5 shows the codes appear constantly in each different data document. the reliability test used the inter-codes agreement (ica) test with the krippendorff coefficient. the ica test was used to find out that the coding by the researcher was not significantly different if the coding was done by someone else. the ica test does not have to be tested for all the code, it is enough to take a representative sample using the krippendorff coefficient. the researcher uses 2 semantic domains, namely: statistical communication and statistical knowledge. the size of the semantic domain is 174.638 words. the results of the ica test analysis can be seen in figure 6. (a) domain semantik komunikasi statistik, 𝛼𝐶𝑈 = 0.971 (b) domain semantik pengetahuan statistik, 𝛼𝐶𝑈 = 0.900 figure 6. ica test using krippendorff α coefficient the proportion of statistical communication sample size for coder-1 and coder-2 was 56.0% and 53.1%, respectively. with a value of αcu=0.971 and based on the krippendorff significant level table with a significant level of 0.05, it can be concluded that the relevance of the agreement between coder-1 and coder-2 on the text has high reliability. 2, 92.2% and 87.4%, respectively. with the value of αcu=0.900 and based on the krippendorff significant level table with a significant level of 0.05, it can be concluded that lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 170 the relevance of the agreement between coder-1 and coder-2 on the text has high reliability. not significantly different when performed by other coders. 3.1.3. selective coding the final stage of data analysis using the gt approach is selective coding, which is to find theories or conjectures of students' statistical literacy models. at this stage the researcher focuses on core categories and categories and focuses on recurring problems and the categories associated with them (glaser & strauss, 1995). the results of the theory or conjecture found by researchers confirm the statistical philosophy and statistical literacy models found previously. consequences and strong relationships in general statistical literacy as a core category is the ability to read and produce statistics of natural and social phenomena where students can represent these phenomena into statistical models and can analyze them. these abilities are part of the descriptive statistics category. a good understanding of descriptive statistics will have consequences for the ability to choose the right statistical test and generalize the sample to the population that is part of inference statistics. so, the general consequences of statistical literacy are descriptive statistics and inference statistics. these relationships were obtained consistently and repeatedly in each coding. an example of this relationship can be seen in figure 7 in the axial coding and the notes made by the researcher. descriptive statistical skills and inference must also be supported by statistical reasoning skills, good statistical communication. the relationship is evident from each coding data source. students who have good statistical reasoning and statistical communication skills tend to have good data interpretation, analysis and conclusion skills. for example, when they choose a statistical test that is appropriate for the sample to be tested, students who have the ability to communicate statistics and the ability to apply good statistical rules tend to have the ability to generalize the sample well. as they answered the questions in figure 7. figure 7. the results of students' answers to statistical literacy skills the relationship between statistical literacy and descriptive statistics and inference statistics when students read analysis of research data from international journals. students can replicate data analysis into different contexts. students' critical ability and confidence in statistical information from various survey results encourage them to study more deeply in volume 11, no 1, february 2022, pp. 163-176 171 statistics. therefore, the three categories of statistical reasoning, statistical communication and attitudes are factors that interfere with student statistical literacy. 3.2. discussion based on the results of selective coding, statistical literacy can be grouped into 2 groups called dimensions (gal, 2004; sharma, 2017), namely: statistical knowledge and attitudes. elements of statistical knowledge are descriptive statistics, inference statistics, statistical reasoning and statistical communication. while the elements of attitude consist of self-confidence and critical attitude (see figure 8). figure 8. student statistical literacy model the relationships between the categories are grouped in table 1 which describes the elements of each dimension and their competencies. the competencies of each element are derived from the consequences of applying statistical literacy strategies, where the grouping of competencies is based on repeated relationships during selective coding. atlas.ti helps researchers to see these relationships by using query analysis. table 1. dimensions, elements and competencies of student statistical literacy models dimension element competence statistical knowledge descriptive statistics the ability to read statistical information and the ability to produce statistics correctly, determine statistics (mean, median, standard deviation, frequency tables, diagrams, etc.), determine sampling techniques. inference statistics ability to make statistical hypotheses and research hypotheses, ability to test hypotheses, determine alpha, make correct decisions and conclusions. statistical communication the ability to communicate statistical information contextually, the ability to communicate the results of data analysis (hypothesis testing) contextually. ability to use statistical tools. statistical reasoning ability to apply statistical rules correctly and logically, ability to relate statistical concepts to mathematics. lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 172 dimension element competence attitudes self-confidence the ability to argue correctly on the reliability of the data, and the ability to argue correctly on the results of his research. critical attitude the ability to analyze statistical information, and the ability to criticize statistical information with correct arguments. based on the results of selective coding, a statistical literacy model was obtained for students: statistical literacy has two domains, namely: statistical knowledge and attitudes. the statistical knowledge domain has 4 elements: descriptive statistics, inference statistics, statistical communication, and statistical reasoning. the attitude element has 2 elements: self-confidence and critical attitude (see figure 8). the respective competencies of the elements are described in table 1. 4. conclusion statistical literacy is the ability to model natural and social phenomena into statistical models to determine population characteristics based on generalized sample data analysis so that they can describe the population or predict phenomena that will occur in the future. philosophically statistics use inductive inference in its proof. therefore, describing the character of the population or predicting future phenomena based on samples using inductive inference. statistical literacy students in general are the ability to interpret and criticize statistical information produced by others and the ability to produce statistics as a tool to explain the reliability of their scientific reports. along with the development of it technology, the ability to use statistical tools is part of statistical literacy that students must have. the literacy model found in this study is almost similar to the gal model (gal, 2004) the difference is in the elements of descriptive statistics and inference statistics. in the knowledge dimension of the gal model, most of it is spread on the elements of descriptive statistics and inference statistics, while the disposition dimension is mostly on the attitude elements and inferential statistics elements. gal does not include the ability to use statistical tools in the model, it is possible that at that time statistical tools had not developed rapidly at this time. this model does not find any statistical literacy level as the model found by watson and callingham (watson & callingham, 2003; 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(2004). developing reasoning about samples. in d. ben-zvi & j. garfield (eds.), the challenge of developing statistical literacy, reasoning and thinking (pp. 277-294). springer netherlands. https://doi.org/10.1007/1-4020-2278-6_12 https://doi.org/10.1111/j.1740-9713.2008.00277.x https://doi.org/10.1111/j.1740-9713.2008.00277.x https://doi.org/10.1080/10691898.2012.11889641 https://doi.org/10.1007/1-4020-2278-6_12 lukman, wahyudin, suryadi, dasari, & prabawanto, studying student statistical literacy … 176 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p169-176 169 analysis of students’ learning obstacles on learning invers function material krisna satrio perbowo 1 , restu anjarwati 2 1 university of muhammadiyah prof. dr. hamka, jl. tanah merdeka, jakarta timur, indonesia 2 sma it al marjan, jl. perintis i, jatikramat, jatiasih, bekasi, indonesia 1 krisna_satrio@uhamka.ac.id, 2 restu.sukamatika@gmail.com received: march 04, 2017 ; accepted: july 11, 2017 abstract this research is based on the presence of obstacle in learning mathematics on inverse function. this research aims to analyze the learning obstacle, to know the types of error that is suffered by the students in learning inverse function. kind of this kualitative research descriptive with data triangulation. the research subjects are high school students which is contained of 74 students and was taken 6 students to be main sample. the data of students’ error is obtained from the writen test result, the students’ false answers are identified into the type of error. then it was chosen several students to be interviewed. which the analysis result finding data in this research showed there are 4 types of errors, which are concept error, procedure error, counting error and concluding error. an obstacle which appear in learning inverse function is influenced by two factors, i.e internal factor and eksternal factor. internal factor is showed by the students’ motivation in following learning and students’ skill in receiving learning material. while the eksternal factor is showed by the curriculum which applied in school with acceleration class caused many narrow learning time, teaching materials that is less complete with the discussion of question sample. keywords: inverse function, learning obstacle, learning process. abstrak penelitian ini didasarkan pada keberadaan hambatan dalam pembelajaran matematika materi fungsi invers. tujuan penelitian ini adalah untuk menganalisa hambatan belajar dan jenisnya, sebagaimana dialami siswa saat mempelajari fungsi invers. penelitian ini menggunakan metode deskriptif kualitatif dengan triangulasi data. subjek penelitian ini adalah 74 siswa sekolah menengah atas dan diambil 6 siswa sebagai sampel utama untuk diwawancara. tes digunakan untuk memperoleh data kesalahan siswa, selanjutnya tipe kesalahan diperoleh dari identifikasi jawaban yang salah. berdasarkan hasil temuan, diperoleh 4 tipe kesalahan yaitu kesalahan konsep, kesalahan prosedur, kesalahan perhitungan, dan kesalahan penyimpulan. hambatan yang muncul dalam mempelajari fungsi invers dipengaruhi 2 faktor, yaitu eksternal dan internal. faktor internal terlihat dari motivasi siswa dalam mengikuti pembelajaran dan kemampuan siswa menerima materi pembelajaran. sedangkan faktor eksternal adalah penerapan kurikulum kelas akselerasi yang diterapkan sekolah mengakibatkan pendeknya waktu belajar, bahan ajar yang kurang lengkap dalam mendiskusikan contoh soal. kata kunci: fungsi invers, hambatan belajar, proses belajar. how to cite: perbowo, k. s., & anjarwati, r. (2017). analysis of students’ learning obstacles on learning invers function material. infinity, 6 (2), 169-176. doi:10.22460/infinity.v6i2.p169-176 mailto:email-penulis-1@ymail.com perbowo & anjarwati, analysis of students’ learning obstacles on learning … 170 introduction the problems that arise in the education system in indonesia are very complex, including mathematics learning problems. mathematics is basic knowledge that is required to proceed to every level of education (nuriadin & perbowo, 2013). at the high school, there is math standard material components of which are a function in mathematics. in this concept is specified in the discussion of inverse function. at school learning in indonesia, material inverse functions listed in the competency standards graduates in mathematics secondary level is "understanding and determine the conditions for a function has an inverse, determine the rules of the inverse function of a function, identify the nature of inverse functions, drawing graphs inverse function of graph of the function of origin ". in fact, to achieve the graduate competency standards there are some obstacles. in the short interview with one of the high school students who learns the inverse function, think that the material is easy because they can use the quick way. however, students complain of difficulties in terms of translation algebraically in determining the inverse function. according to ignacio, barona, & nieto (2006), teachers need to recognize and analyze about how students hold certain beliefs when learning mathematics and interacting with their environment, the success or failure were leaded by these beliefs. students’ error in response to any math problems is often not followed up by tracking the occurrence of obstacles during the learning process. in addition, it seems that teachers’ view of sources of the students’ learning obstacle have received less attention in mathematics education research and literature (bingolbali, akkoç, ozmantar, & demir, 2011). it is possible there are still other difficulties related with material that can be found inverse function which if necessary to find the causes. in this case, teachers must help their students to develop persistence and broader students’ view of mathematics (schackow & thompson, 2005; saija, 2012). the urge to break the barriers in learning which is experienced by students should be developed by one element of the development of the teaching profession. learning mathematics took place regarding to teachers, students and mathematics itself. the learning process is based solely on textual understanding of teaching materials such as only books that will lead to poor learning the meaning and context, as well as the learning process results oriented will result in the student passively. student learning difficulties indicated by the presence of certain obstacles to achieve the learning outcomes can be psychological, sociological, and physiological, which in turn can lead to decreased academic achievement accomplished. mathematics became an essential part of life, not only became one of the requirements t o proceed to every level of education, even mathematics is widely used for everyday life. e.g. market sellers who use scales, builder, driver and almost all aspects of life require mathematics. on a wider scope of mathematics is the queen and servant of science knowledge of the other. jhonson (suherman, 2003) says that mathematics is the study of patterns and relationships, a road or a pattern of thought, an art, a language and a tool. the researchers concluded that mathematics is a science that is obtained by reasoned that using the terms defined carefully, clearly and accurately, represented by the symbol of language and symbols that have meaning. volume 6, no. 2, september 2017 pp 169-176 171 benefits of mathematics are not just limited to the knowledge and calculation, but more importantly when each individual can master math well, the pattern of their thinking became more rational and critical. according to the permendiknas no.22 in 2006 on content standards, mathematics courses should be treated to all students with the ability to think logically, analytical, systematic, critical, and creative, and able to work together. competence is needed so that learners can have the ability to acquire, manage, and use information to survive in a state that is always changing, uncertain and competitive. the purpose of learning is to acquire new knowledge. in the process of development of knowledge, an individual often encounter obstacles or barriers. barriers are anything that obstruct, hinder, impede or individual human encountered in everyday life that come and go, creating barriers for the individuals who experience it. barriers in learning are essentially a phenomenon that appears in different types of manifestations of behavior. symptoms of obstacles are directly manifested in a variety of behaviors. behavior manifested by the presence of certain barriers, will usually be seen in aspects of motoric, cognitive and affective, well into the process and learning outcomes are achieved. cornu (2002) distinguishes between four types of obstacle, namely: cognitive barriers, genetic and psychological barriers, and barriers didactic and epistemological barriers. in this study, researcher sought to focus on the students' learning barriers in understanding the concept of inverse functions, from the aspect of epistemology. basically, the establishment of knowledge occurs through the interaction of subsystems. brousseau (2002) explains that one of the learning sub-systems consists of teachers, students, and system knowledge. the third involvement in the learning subsystem supports the creation of knowledge. interaction among the three becomes an important part of learning. nothing is in focus or emphasis; all three must go hand in hand because if there focusing on one of the subsystems will be an imbalance. in explaining the epistemological obstacle, it is indicates that: barriers epistemology occurs in the development of scientific thought and in practical education (cornu, 2002). cornu and bachelard (tall, 2002) said that epistemological obstacles has two important advantages: a) they can not respond and understand the new knowledge gained b) they are only able to respond a little bit of a concept that has been previously understood. this means that the barriers epistemology greatly affect the learning concepts that have been studied previously. effect of prior knowledge will support the new knowledge to be gained. barriers epistemology of scientific knowledge may lead to stagnation, and even decline in someone’s knowledge. method based on the problems studied, the researchers chose descriptive analysis as type of this research. researchers collected data based on observations of natural situation as it is without being influenced and deliberated. researchers as a research instrument "human instrument". in this case the researchers are a key instrument, or so-called "key instrument", held their own observations or interviews were unstructured, often just using a notebook and camera. descriptive data were collected as many as outlined in report form. perbowo & anjarwati, analysis of students’ learning obstacles on learning … 172 in this study, the research subject are science class majors students in senior high school. in this study to determine the barriers to student learning are in depth after analyzing the test results. the researchers conducted interviews to the students of class 11 th grade were taken from 3 and of 12 th grade were taken 3. as explained in table 1 below: table 1. number of samples research subject amount code students of 11 th grade 3 a1, a2, a3 students of 12 th grade 3 b1, b2, b3 total 6 66 this study emphasizes the filter information in a narrative form. instruments which are used such as: a) test problem-solving ability to get research subjects. b) interview guide to locate the difficulties of students who received a low score based on problem-solving abilities in solving the problem of the inverse function. c) the validation of matter, to determine whether the instrument is made by the researcher actually valid or not, and the instrument must be validated by the validator. d) to test the validity of an instrument to strengthen eligibility to be tested, the validity of the instrument consists of 10 questions that include indicators on the mathematical problem solving material inverse function. data collection techniques which were used in this study included the identification of barriers to learning and categorization of barriers to learning. for a more in-depth data reveal researchers also conducted observations and interviews. in qualitative research would be found on data obtained from various sources, using the techniques of data collection manifold (triangulation), and carried out continuously until data saturation. data saturation is a state of the data does not change when the researchers took samples of up repeatedly. results and discussion results results of analysis tests students' skills in solving problems associated with the function and the inverse function, error grouped students by using tables and classified into types of errors. many errors appear in the data presented in the form of a percentage. based on the test results of students' skills in solving problems related to learning functions and inverse functions derived from all 74 students were given the test, it appears that the errors appear different in each child. this type of error is classified into four types of errors, the errors concept, procedural errors, miscalculations and concluding errors (widdiharto 2008). the details of the problem-solving test results based on the type of error can be seen in table 2 below. table 2. precentaage of students’ errors in solving function problem and invers function types of errors 11 th grade 12 th grade concept errors 32% 8,20% procedural errors 15,30% 33,50% miscalculation 28,90% 21,20% concluding errors 23,60% 36,90% volume 6, no. 2, september 2017 pp 169-176 173 discussion the instrument were used in this study consisted of 10 questions related to problem-solving abilities to seek resolution of functions and inverse functions. the strategy, which is used by students in problem solving functions and inverse functions was demonstrated in figure 1 for question 1, namely students solve problems by determining the value of the function of the operation of algebra functions obtainable strategies used by students a33 is by substitution of data into the equation and calculated based structure. problem:  22 21321       solution:   231 123321 21321 22         figure 1. case 1 interview 1 researcher : how do you solve question 1? a33 : expand it algebraicly, then proceed the operation researcher : do you know about principle of exponent? a33 : i don’t remember the conclusion is the answer is not correct because it does not use the principle of exponent characteristic and the mistake in operating the algorithm. in figure 2, question number 5 test students' ability to solve problems related to the composition of functions, obtained strategies used by students a33 i.e, with substitute function f into a function g, but at later stage does not give the appropriate conclusions perbowo & anjarwati, analysis of students’ learning obstacles on learning … 174 problem: ( ) ( ) ( ( )) solution: 1 1 1 1 ))((           x x x x x xgxfg figure 2. case 2 interview 2 researcher : how do you solve question 5? a33 : simply substitute the function and find the value of x the conclusion is the result is not appropriate because there is a mistake in algebraic operation, especially in substituting the function form. on the student a33, test results which have been obtained, shows that student misconceptions a33 appeared 9 times, 3 times procedural errors, miscalculations and mistakes concluded 5 times as much as 8 times. this suggests that students experience an error in the working section selecting information incorrect data, using the principle of a formula or procedure incorrectly, students lost one of data or more, less accurate in the calculation and use of the proper way, but did not manage to conclude well, and showed a lower operating to draw conclusions. this explanation is reinforced by interviews conducted by researchers. students a33 coupled with the observation of the results of the answers that students provide, it is known that students have difficulty in concluding the final result because there is a concept that does not remember that the rules regarding the nature of exponents. based on the above data can be explained that many students have difficulty in mathematical problem solving process on inverse function subject. the students do not mastering the concept of algebraic function, more because of they do not have enough fundamental prerequisites ability for algebraic function. on the other hand, the students used algebraic properties carelessly. furthermore, the students were experiencing internal resistance, i.e. do not know the benefits from studies, learning materials may be accepted depending on emotional condition and external constraints, the teacher gives questions that has never been taught, teachers are sometimes less interactive, and curriculum foundations make students become less exercise in the discussion of each section in which one section is given training in schools is only one time. all the issues above are causing many problems not solved corectly with appropiate algebraic prosedures. this proves that many obstacle in learning, happened both from internal and volume 6, no. 2, september 2017 pp 169-176 175 external. this is in line with hidayat research (2017) which suggests that high school students in mathematics subjects still tend to lack the novelty of the mathematical reasoning ability so that completion is still following routine procedures. conclusion from this research, it can be concluded that, students solved invers function problems using quick and short algorithm, so they ignore the rules of mathematics itself. students become less rigorous in algebraic computational operations. but, from this study, we did not found any different strategy in solving the functions problems and inverse functions problems. so it is monotonous. according to this study, also found some types of errors that appear from students in related to inverse function subject. namely, misconceptions that causing errors in understanding mathematics concepts, which become the prerequisites and concepts being taught; errors by discrepancy procedure steps in order to responding the problem, so that there is no clear layout in the process of answers finding; and calculation error caused by unappropiate numeracy ability and knowledge, so sutdents can not do the calculation correctly. while, the epistemological barriers found in this study are students did not understand the material prerequisites in order to studying the function and the inverse function, namely the rules of exponents. else, students have lack of understanding on the concept of algebraic functions, so they found it difficult in solving problems. in addition, the students were not understand on how to simplify fractions form, which causing poor intact in understanding algebraic functions. references bingolbali, e., akkoç, h., ozmantar, m. f., & demir, s. (2011). pre-service and in-service teachers‟ views of the sources of students‟ mathematical difficulties. international electronic journal of mathematics education, 6(1), 40-59. brousseau, g. (2002). theory of didactical situations in mathematics. (n. balacheff, m. cooper, r. sutherland, & v. warfield, eds.) (1st ed., vol. 19). dordrecht: kluwer academic publishers. https://doi.org/10.1007/0-306-47211-2 cornu, b. (2002). limits. in d. tall (eds.), advanced mathematical thinking, 153-166. dordrecht: kluwer academic publishers. euis, s. (2011). hambatan epistemologi (epistemological obstacles) dalam persamaan kuadrat pada siswa madrasah aliyah. proceedings international seminar and the fourth national conference on mathematics education. yogyakarta: yogyakarta state university hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. ignacio, n. g., barona, e. g., & nieto, l. b. (2006). the affective domain in learning mathematics. electronic journal of research in educational psychology, 4(1), 47-72. nuriadin, i., & perbowo, k. s. (2013). analisis korelasi kemampuan berpikir kreatif matematik terhadap hasil belajar matematika peserta didik smp negeri 3 lurangung kuningan jawa barat. infinity journal, 2(1), 65-74. perbowo & anjarwati, analysis of students’ learning obstacles on learning … 176 saija, l. m. (2012). analyzing the mathematical disposition and its correlation with mathematics achievement of abstract senior high school students. infinity journal, 1(2), 148-152. schackow, j. b., & thompson, d. r. (2005). high school students' attitudes toward mathematics. academic exchange quarterly, 9(3), 12-19. suherman, e. (2003). strategi pembelajaran matematika kontemporer. bandung: universitas pendidikan indonesia. tall, d. (2002). the psychology of advanced mathematical thinking. in advanced mathematical thinking (pp. 3–21). dordrecht: springer netherlands. widdiharto, r. (2008). diagnosis kesulitan belajar matematika smp dan alternatif proses remidinya. paket pemberdayaan kkg/mgmp matematika. yogyakarta: p4tk matematika sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p1-12 1 development of pisa types of questions and activities content shape and space context pandemic period risda intan sistyawati, zulkardi*, ratu ilma indra putri, samsuriyadi, zahra alwi, sisca puspita sepriliani, ayu luviyanti tanjung, riszky pabela pratiwi, shinta aprilisa, duano sapta nusantara, meryansumayeka, jayanti universitas sriwijaya, indonesia article info abstract article history: received nov 30, 2021 revised sep 14, 2022 accepted sep 21, 2022 this research belongs to the type of development research, which consists of the main stages, including preliminary design and formative evaluation. this study aims to obtain valid and practical pisa-type development questions consisting of initial design, self-evaluation, expert review, one-to-one, and small group. the emergence of this research is due to the low mathematical literacy of students in indonesia. this study took a particular research subject for grade ix junior high school students in palembang city. from this study, the results obtained include the development of pisa-type questions and activities using the context of social distancing during the pandemic. keywords: covid-19, development, literacy, pisa, questions and activities this is an open access article under the cc by-sa license. corresponding author: zulkardi, department of mathematics education, universitas sriwijaya jl. raya palembang prabumulih km. 32 indralaya, south sumatra 30128, indonesia. email: zulkardi@unsri.ac.id how to cite: sistyawati, r. i., zulkardi, z., putri, r. i. i., samsuriyadi, s., alwi, z., sepriliani, s. p., tanjung, a. l., pratiwi, r. p., aprilisa, s., nusantara, d. s., meryansumayeka, m., & jayanti, j. (2023). development of pisa types of questions and activities content shape and space context pandemic period. infinity, 12(1), 112. 1. introduction mathematical literacy is one of the most important skills to have in the 21st century. mathematical literacy is closely related in everyday life, especially in the problems that arise. (stacey, 2015). stacey (2015) argues that mathematical literacy is needed in various fields of expertise and in various age ranges. the main idea that is often carried out in media literacy is related to real-world problems and mathematical problems. media literacy is very necessary as a provision in knowing the role of mathematics in everyday life, as needed in the 21st century (stacey & turner, 2015). https://doi.org/10.22460/infinity.v12i1.p1-12 https://creativecommons.org/licenses/by-sa/4.0/ sistyawati et al., development of pisa types of questions and activities content shape and space … 2 based on the results of pisa, especially for indonesian students, it is stated that the mathematical literacy ability of school students in indonesia is relatively low (putri & zulkardi, 2020; rawani et al., 2019; zulkardi et al., 2021). the scores obtained in 2015 and 2018 showed a significant decrease (schleicher, 2019). the score obtained in 2015 was 389 while in 2018 it was 379 (schleicher, 2019). there are various causes of the decline and low pisa scores, including the lack of facilities in the form of textbooks provided in solving mathematical problems with the real world (jannah et al., 2019; novita et al., 2012; zulkardi et al., 2021). therefore, there is a lack of fulfillment in terms of providing textbooks, so it is better to develop various kinds of pisa questions and activities that can be used during the learning process which is believed to improve mathematical literacy for students (munayati et al., 2015) another demand that must be met by educators is to make learning integrated with the surrounding environment and daily life. this is related to learning that uses contexts that are close to students (magen-nagar, 2016). problems that can be used as the closest context at this time include the case of covid-19. the covid-19 case was caused by the corona virus that originated in the wuhan area and spread to various parts of the world (irfan et al., 2020; pertiwi et al., 2021; zulkardi et al., 2021). understanding students' concepts is very important to be considered as one of the requirements in solving various kinds of math problems related to everyday life (edo et al., 2015). therefore, understanding important concepts is made meaningfully in the learning process (magen-nagar, 2016). this study aims to develop questions and activities with the pisa framework for shape and space content using the context of social distancing during a pandemic. from the background of the problems that have been stated, it is known that the formulation of the problem in this study are: what are the characteristics of pisa type questions and activities with shape and space content? is social distancing valid, practical and has potential effects. the purpose of this research is to find outthe characteristics of pisa type questions and activities in the shape and space context of social distancing during the pandemic are valid, practical and have potential effects. in previous studies, there have been studies on the development of pisa questions including the use of shape and space content in the context of soft tennis and volleyball, and so on (efriani et al., 2019; jannah et al., 2019; kohar et al., 2019; meryansumayeka et al., 2020). however, until now no one has made the development of questions from the context of covid-19 to be studied. 2. method this research is a type of development study which has two main stages including the preliminary and formative evaluation stages (nieveen & folmer, 2013; tessmer, 1993). this study aims to obtain pisa type questions and activities that are valid, practical and have potential effects. validation in this study was held on activitiesexpert review with master's colleagues (gravemeijer et al., 2017), as well as junior high school mathematics teachers and led by experts in the development of pisa questions. in addition, there is a one to one validation stage for heterogeneous students with each of them having low, medium and high abilities who are not included in the research subject. then it was tested on 12 junior high school students who have high, medium and low abilities online (via zoom meeting) so that the practicality of the questions obtained can be obtained. after that, a trial was conducted on students in one particular class as many as 20 students in the field test to see the potential volume 12, no 1, february 2023, pp. 1-12 3 effect on pisa type questions and activities with shape and space content using the context of social distancing during a pandemic (bakker & wagner, 2020). in this study, using walkthrough, observation and test data collection techniques. data collection techniques used are walkthrough, observation and test techniques. the walkthrough technique is used to see whether or not the questions and activities made by researchers against experts are based on the content, constructs and language used from suggestions and comments obtained during expert reviews and fgds. however, at the same time as validating from the experts, the researchers also conducted one to one on students who had been selected to be tested in later development. observation in the study aims to observe and know the characteristics and needs of these students as well as when the trial took place. other than that, tests are also carried out which are useful to see the practicality of the questions that have been made by the researchers that will be done by the students. at this stage of the test or trial, 12 junior high school students with heterogeneous abilities (high, medium and low). furthermore, the results of the student's answers were analyzed qualitatively in order to see the practicality of the questions being worked on. 3. result and discussion 3.1. result during the preliminary design stage, researchers made observations first to junior high schools in palembang city to look for various kinds of information needed in sorting research subjects, time and knowing the flow of learning and teaching and learning activities in the classroom as well as taking care of permits as administrative requirements in carrying out research. at the school concerned and analyze various kinds of pisa questions and activities and then develop questions and activities using the context of social distancing during the pandemic. during formative evaluation, the first thing to do is self-evaluation. at this stage, the researcher evaluates the questions that have been developed. the questions are made in the form of questions and activities with a total of two types of activities (sharing tasks and jumping tasks) with the pisa type of shape and space content with the context of keeping a distance during the pandemic. the pisa questions developed were taken from the pisa questions in 2006. figure 1 is a picture of the 2012 pisa questions taken as an example for the development of the questions in this study. figure 1. original pisa questions in 2006 with the context of “rock concert” sistyawati et al., development of pisa types of questions and activities content shape and space … 4 table 1 shows the development of questions that have been developed by researchers. this problem is then used as prototype 1 (see figure 2). figure 2. development of pisa prototype 1 questions translated into english: during the covid-19 pandemic, tickets for the bumi sriwijaya stadium art show were sold out. in this case, to prevent transmission of the virus, a 1.5 meters distance rule is applied for each visitor. to answer these questions, follow the steps in the following activity! question 1: look at the pictures of the stadium! what shape do you think it is? question 2: after knowing the shape, do you still remember how to find the area? please write down the formula! question 3: if you already know the formula, determine the area of the stadium based on the description of the image! question 4: please take a look above and see the illustration of the following picture! after you know the area, what is the distance between the audience that must be obeyed? question 5: if you already know the extent and distance between one and the other in each person, then to find out the number of visitors based on what operations can you do? give the reason! question 6: you have set the type of calculated operation to be used, then next is that you have to perform the operation on the area and information distance between individuals. so, how many visitors are there? volume 12, no 1, february 2023, pp. 1-12 5 you have set the type of calculated operation to be used, then next is that you have to perform the operation on the area and information distance between individuals. so, how many visitors are there? after obtaining questions with prototype 1, then validation tests were carried out by experts and fgds as well as one to one which aims to get valid questions in terms of constructs, content and language. in the characteristics seen by researchers for the shape and space content in pisa with the current independent emergency curriculum. furthermore, in terms of constructs to see the suitability of the level of students' abilities, especially in the ability to examine problems that are in accordance with reality for students. furthermore, in terms of language, several aspects that are seen include the suitability of writing questions with eyd rules, using sentences that are easy to understand and not experiencing problems. ambiguity in the meaning of the developed questions. in line with fgd and expert review, the researcher also did one to one to see the opinions of students in working on the problem and to see comments and suggestions in the development of this prototype 1 question. after doing this stage, valid pisa prototype 2 type questions and development activities are obtained based on comments and suggestions obtained on prototype 1 questions. table 1. comments / suggestions and revision decisions no. comments and suggestions revision decision 1 in the picture, it is better to use an image that is more relevant to the flat shape that students want to ask the image has been corrected and selected based on the reference on the problem to be relevant 2 in questions, it is better to use more than one picture for every three questions in the form of one illustration. so the total image becomes two images have been added to the problem so that not just one picture 3 the context of the questions that were made, some of them used ineffective sentences, so it was suspected that it would make students confused when working on them sentences have been revised so that the sentences used are more effective the next stage is to conduct direct trials of students with small groups to see the practicality of prototype 2 questions for students of smp n 13 palembang. trial questions in small groups are carried out face-to-face. in the implementation, students are given 20 minutes to solve the problem. after doing the small group, the researcher then revises the questions and activities that have been made so that they become prototype 3 for the students of smp n 13 palembang. then, there are several kinds of inputs and suggestions for use as prototype 3. here are some kinds of input given to students for the implementation of small group prototype 2 trials (see table 2). sistyawati et al., development of pisa types of questions and activities content shape and space … 6 table 2. comments / suggestions for the small group no. comments and suggestions revision decision 1 it's good to change the word wake up space to wake up flat already repaired student answer analysis the questions developed are pisa type questions and activities with the context of online shopping during a pandemic which consists of one item. the problem is solved by students within a period of 20 minutes through one-to-one. in these questions, students are given several kinds of information about the field, the size of the field, the distance between the people in it and the steps in solving these problems and activities. these questions are included in level 4 by solving problems based on sequential procedures. the material used is the area and circumference of a flat figure. figure 3. one-to-one answers of student 1 figure 3 shows that the student's answer can be answered based on the activity in the problem. these activities are based on the context of keeping a distance in the pandemic period. based on these answers, students answer questions correctly but there is a mistake in calculating the results of the operation. the result of 110 x 90 is 9900. but the student was translated into english: answer 1: rectangle answer 2: length times width answer 3: 110 x 90 = 900 answer 4: 1,5 meters answer 5: 900/1,5=600 answer 6: 600 visitors volume 12, no 1, february 2023, pp. 1-12 7 wrong in answering 900. the student can answer the question correctly but cannot describe it. figure 4. one-to-one answers of student 2 figure 4 show that the student's answer is still incorrect. the actual formula of the rectangle area is p x l and the roving formula is 2length + 2width. but the student wrote it the formula of the two in reverse. for the one-to-one stages, it can be seen in the image above. in the answers of students 1 and 2 above, it can be seen that students understand the information about the questions well. however, student 1 can answer the question correctly but does not describe the desired answer. in addition, for student 1 there was an error in calculating the questions and activities. then for student 2, it is known that they can answer the questions well and can describe the answers as desired. however, student 2 has misperceptions about the area and perimeter formulas. so the use of area and perimeter formulas is used in reverse. translated into english: answer 1: rectangle answer 2: area = 2 × length+width answer 3: l = 2 × length + width = 2×90+110=2×200=400m2 answer 4: distance per person 1.5 meters answer 5: to find out the number of visitors, the calculated operation used is division, because the previous question was already known widely and the distance between one person and another. answer 6: 𝐿. 𝑠𝑡𝑎𝑑𝑖𝑜𝑛 = 400𝑚2 𝑎𝑛𝑑 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 1,5𝑚2 400𝑚2 1,5𝑚 = 40.000𝑐𝑚2 150𝑐𝑚 = 26.000 𝑝𝑒𝑟𝑠𝑜𝑛 sistyawati et al., development of pisa types of questions and activities content shape and space … 8 figure 5. small group student answers figure 5 shows that in this case students can highlight the activity problem as in figure 2 properly and correctly. in addition, in this case students can also describe the answers they choose. but in this case, for problem number 3, students' understanding of the unit concept of broad operations is still so lacking. after the questions and activities have been revised, the next step is to conduct a small group trial on prototype 2. the picture shows students who can know that students have understood well the questions asked. however, for the elaboration of question number 5, it can be seen that the student still has not answered what he wants from the question. the student only answered using the division operation without explaining the reason why he chose the division operation in solving the problems and activities. in addition, for question number two, students should clearly write down the units in the problem to be solved in the completion step. however, the student only wrote down the units in the result of the solution. translated into english: answer 1: rectangle answer 2: because the rectangle has a length and width, then write the formula area = width x length answer 3: area = width x length = 110 x 90 = 9900m answer 4: the distance between units of people with others is 1.5 meters. answer 5: to find out the number of spectators is by operating the count (division) answer 6: area : distance = 9900 : 1,5 = 6600 audience volume 12, no 1, february 2023, pp. 1-12 9 figure 6. student field test answers figure 6 shows the results of students' answers at the time of the field test. at this trial stage, there is one student who becomes the reference for the analysis at this stage. almost all students have the same answer. from these answers, it can be analyzed that the student can work on the questions in accordance with the directions from the questions and activities given. for question number 3, the student can explain the reasons why he chose the arithmetic division operation in the process of analyzing questions and activities. 3.2. discussion after conducting trials from one-to-one, small group to field tests, the results obtained from this research are whether the questions and activities developed can be classified as valid and practical questions and developments or not. based on the results of the data analysis above, it can be seen that the literacy skills of students in this case are classified as good. this can be seen from the one-to-one process, small group to field tests where it can be seen that the student can answer questions according to the desired direction. in the process of assessing their mathematical literacy skills, there are several aspects that are used as a reference in assessing student's mathematical literacy include (1) communication, (2) mathematization, (3) restating, (4) reasoning and giving reasons, (5) using problem solving strategies, (6) use symbols, formal language and techniques and (7) using mathematical tools (hesse et al., 2015; lin & tai, 2015; nusantara & putri, 2018; putri & zulkardi, 2018). from these seven aspects, it can be seen that most of the students who are the subjects of this research have good mathematical literacy skills. this can be seen from the way students answer various questions on the questions and activities that have been given. there are some students who are not able to explain the reasons when choosing a particular operation path as stated by question number 5. however, there are also some of them who can describe it well. translated into english: answer 1: rectangle answer 2: length x width answer 3: area = length x width = 90 cm x 110 cm = 9900 answer 4: 1,5 meters answer 5: division, the reason is because by way of division can get a smaller value answer 6: during pandemic = 1920 distance: 1,5 meters 1920 x 1,5 = 2880 area = 2880 sistyawati et al., development of pisa types of questions and activities content shape and space … 10 4. conclusion this research produces questions and develops pisa types with shape and space types using social distancing during a pandemic that is valid, practical and has potential effects. in this case, the question can be said to be valid based on fgd activities, expert validation and one-to-one trials. meanwhile, to see whether the question is practical or not, it can be seen from the results of the small group that has been held for several students. then to see the potential effects based on the field test trials. based on the results of the answers obtained from these students, it can be classified that the questions are included in the type of questions and practical activities because they can be solved easily by students and can be interpreted well with various kinds of student responses in answering and adjusted to the level of difficulty of class students ix. after this research has been carried out with valid, practical and potential effects and questions, it is hoped that future researchers can develop other research on pisa questions with much more diverse content and other contexts. acknowledgements this study was financially supported by hibah profesi universitas sriwijaya with contract number: 0014/un9/sk.lp2m.pt/2021. the researcher would like to thank the principal of smp negeri 13 palembang who has allowed the researcher to collect data at the school and the class ix students who have participated in this research. as well as to ma’am kania sitisyarah who has helped us during the research. 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(2021). designing pisa-like task on uncertainty and data using covid-19 context. journal of physics: conference series, 1722(1), 012102. https://doi.org/10.1088/1742-6596/1722/1/012102 https://doi.org/10.22342/jme.10.2.5243.277-288 https://doi.org/10.1007/978-3-319-17187-6_43 https://doi.org/10.1007/978-3-319-10121-7_1 https://doi.org/10.4324/9780203061978 https://doi.org/10.1088/1742-6596/1722/1/012102 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p121-136 121 pre-service mathematics teachers’ conception of higher-order thinking level in bloom's taxonomy damianus d. samo nusa cendana university, jalan adisucipto penfui, kupang, indonesia demm_samo@yahoo.com received: april 06, 2017 ; accepted: may 01, 2017 abstract the purpose of this study is to explore pre-service mathematics teachers' conception of higher-order thinking in bloom's taxonomy, to explore pre-service mathematics teachers' ability in categorizing six cognitive levels of bloom's taxonomy as lower-order thinking and higher-order thinking, and preservice mathematics teachers' ability in identifying some questionable items as lower-order and higher-order thinking. this research is a descriptive quantitative research. the participants are 50 third-year students of mathematics education department at universitas nusa cendana. the results showed: (1) pre-service mathematics teachers' conception of lower-order and higher-order thinking more emphasis on the different between the easy and difficult problem, calculation problem and verification problem, conceptual and contextual, and elementary and high-level problem; (2) preservice mathematics teachers categorized six cognitive levels at the lower-order and higher-order thinking level correctly except at the applying level, preservice mathematics teachers placed it at the higher-order thinking level; (3) pre-service mathematics teacher tend to made the wrong identification of the test questions that were included in the lower-order and higher-order thinking. keywords: bloom’s taxonomy, higher-order thinking. abstrak tujuan penelitian ini adalah untuk mengeksplorasi konsepsi calon guru matematika tentang higher order thinking dalam taksonomi bloom, mengeksplorasi kemampuan calon guru matematika dalam mengkategorikan enam level kognitif dalam taxonomi bloom sebagai lower order thinking dan higher order thinking, dan kemampuan calon guru matematika dalam mengidentifikasi item test sebagai lower order thinking dan higher order thinking. penelitian ini adalah penelitian deskripstif kuantitatif. subjek penelitian adalah 50 mahasiswa tingkat tiga di jurusan pendidikan matematika, fakultas keguruan dan ilmu pendidikan, universitas nusa cendana. hasil penelitian menunjukan bahwa: (1) konsepsi calon guru matematika tentang lower-order thinking dan higher-order thinking lebih menekankan pada perbedaan tingkatan masalah yang sulit dan mudah, masalah perhitungan dan pembuktian, konseptual dan kontekstual, tingkat berpikir elementer dan tingkat berpikir lanjut; (2) calon guru matematika mengkategorisasikan enam level kognitif pada level lower-order thinking dan higher-order thinking secara tepat kecuali pada level aplikasi, banyak calon guru matematika yang menempatkannya pada level higher-order thinking; (3) calon guru matematika cenderung salah mengidentifikasi soal test yang termasuk dalam lower order thinking dan higher-order thinking. kata kunci: higher order thinking, taksonomi bloom how to cite: samo, d. d. (2017). pre-service mathematics teachers’ conception of higherorder thinking level in bloom's taxonomy. infinity, 6 (2), 121-136. doi:10.22460/infinity.v6i2.p121-136 samo, pre-service mathematics teachers’ conception of … 122 introduction the national educational goal is to develop the potential of students' to become a man of faith and fear of god almighty and noble, healthy, knowledgeable, skilled, creative, independent, and become citizens of a democratic and responsible. this national educational goal contains two important aspects that are cognitive and attitude aspects. cognitive aspect emphasizes proficient and creative. proficient in the indonesian big dictionary is 1) capable of doing something; 2) clever, proficient; 3) have the ability and skill to grind. the definition shows that proficient is more than just remembering. being able to do something means having the knowledge, understanding, and skill enough in doing the job. this condition shows that proficient is not enough based on the usual way of thinking but requires a higher way of thinking to work in all situations or problems. while creative means 1) have created; can create; 2) contains creativity, work that requires intelligence and imagination. creative is the highest level in the taxonomy of education or a part of a higher-order thinking level. therefore, the national educational goal requires that learning should be developed and designed to develop higher-order thinking ability. bloom, engelhart, hill, furst, & krathwhol (1956) divided the taxonomy of educational into six levels: (1) knowledge; (2) comprehension; (3) application; (4) analysis; (5) synthesis and (6) evaluation. the sixth level is grouped into two major parts, namely, lower order thinking (knowledge, comprehension, and application) and higher-order thinking (analysis, synthesis, and evaluation). bloom’s taxonomy is revised by anderson & kratwohl (2001) and brings new educational taxonomy into (1) remembering; (2) understanding; (3) applying; (4) analyzing; (5) evaluating, and (6) creating. the aspects of higher-order thinking in the taxonomy are revised: (1) analyzing; (2) evaluating; and (3) creating. the last three levels which are defined as higher-order thinking have many attributes or characteristics that distinguish one another; allows the use as a part in learning activities in both the process and its evaluation. analyzing is associated with cognitive processes by giving, attributing, organizing, integrating and validating. evaluating includes checking, critiquing, hypothesizing, and experimenting. creating includes generating, designing, producing, and devising. some experts define higher-order thinking by referring to bloom's taxonomy revisions to mention higher-order thinking as the ability to think analytic, evaluative and creative (pegg, 2010; thompson, 2000). thompson (2000) defined higher-order thinking involves solving tasks where an algorithm has not been taught or using known algorithms while working in unfamiliar contexts or situations. the definitions stated above imply higher-order thinking as problem-solving activities. aside from being a problem-solving activity, some authors present some additional attributes: the ability to think critically and creatively (brookhart, 2010; miri, david, & uri, 2007). the addition attributes critical and creative thinking skills showed an understanding that aspect of problem-solving has a broad scope so it can be analyzed into small parts that show the other thinking skills that can be developed. moreover, some experts also define higher-order thinking by showing many of the attributes that is, as the ability to solve problems that involve critical thinking, logical, reflective, metacognition, reasoning, and creative (lewis & smith, 1993; murray, 2011). higher-order thinking in this research is a non-algorithms thinking include analytical, evaluative and creative thinking which involves metacognition. the importance of developing higher-order thinking has some reasons: (1) to organize knowledge learned into long-term memory. organizing raises enough information retention volume 6, no. 2, september 2017 pp 121-136 123 longer than if stored in short-term memory that is characteristic of lower order thinking. for example, students who learn to memorize tend to quickly forget what is memorized than students who learned how to discover. memorization process will push that knowledge into short-term memory, while the process of discovering will push that knowledge into long-term memory. knowledge stored in long-term memory is easily accessed and is used in various situations that tend to change: (2) to develop adaptability to a variety of new problems that is found in life, exercises to develop a higher order thinking ability in formal education will develop an attitude and a way of creative thinking to get out of life's problems are complex, (3) to encourage the creation of quality human resources that can compete with other nations. in connection with the importance of developing the higher-order thinking ability, then the teacher as a major subject in learning should be able to design learning that accommodates strategy of developing higher-order thinking ability. the strategies of developing higher-order thinking are using the contextual problem, using the test items of higher-order thinking, asking and discuss the critical and analytical problem, and encourage students to develop ideas and think out of context. teacher's ability in designing the learning activities should be based on an understanding of higher-order thinking itself. a complete understanding of higher-order thinking in the bloom taxonomy aims to support preservice teachers in the development of problem-solving and mathematics ability. delima (2017) reveals there is an influence of problem-solving ability to students mathematical thinking. furthermore, a good understanding of bloom taxonomy will support preservice mathematics teachers in constructing questions to access students' thinking ability. asking a question requires good communication skills so that the message is a question of the mastery of good grammar, that of the material, and communicative skills (hendriana, 2017). questions expressed in addition to clear must be analytical, critical, and can be imaginative expressions. higher order thinking in the bloom’s taxonomy is a skill that can be trained to secondary school students and university students. higher-order thinking at the university level is better known as advanced mathematical thinking because of the substance of the material at the advanced level. advanced mathematical thinking includes ability in representing, abstracting, creative thinking, and mathematical proving (herlina, 2015). so here, there are many impacts of a good conception of the higher-order thinking and lower order thinking of bloom taxonomy. a good understanding should be given to pre-service mathematics teachers as teachers in the future. the understanding can be given in the course of learning evaluation of learning and in other courses through the introduction of problem according to the cognitive level of bloom’s taxonomy and discussion in the learning activities. therefore, in this study, i will explore pre-service mathematics teachers' conceptions of higher-order thinking in bloom's taxonomy and the results of exploration can be a reference in preparing pre-service teachers to become a teacher of the future. exploration is done on final year students who have been equipped with learning evaluation courses and subjects supporting pedagogy. so that the problem in this research are; 1) how do pre-service mathematics teacher's conceptions of lower-order and higher-order thinking in bloom's taxonomy?, 2) how do pre-service mathematics teachers categorize six cognitive levels of bloom's taxonomy as lower-order and higher-order thinking, 3) how do pre-service mathematics teachers identify the test as lower-order and higher-order thinking?. the benefit got with this exploration is to inform pre-service teacher mathematics educators in the samo, pre-service mathematics teachers’ conception of … 124 decision to equip pre-service mathematics teachers with the knowledge and understanding of the bloom's taxonomy completely in classroom learning implementation. method this research is a descriptive quantitative study. the data were analyzed and visualized by percentages and diagrams. the participants are 50 third-year students of mathematics education department at universitas nusa cendana. the instruments used in this study were questions items and interview guideline. the question items are: (1) the preservice mathematics teachers' conception of lower-order and higher-order thinking in bloom's taxonomy, (2) the cognitive domain classification of six major categories, and (3) the identification of lower-order and higher-order thinking questions. the third question items presented several linear equations and functions questions of 7th and 8th-grade mathematics which have all six categories of bloom's taxonomy. based on the synthesis of experts', lowerorder and higher-order thinking definition are; 1) lower-order thinking is the type of algorithm thinking or simple thinking that follows existing pattern or procedures. in bloom’s taxonomy, lower-order thinking includes remembering, understanding, and applying. the remembering level question is using the information retrieval in one simple step and wrote down directly what it is. the understanding level question involves recognizing or remembering information in one step and well explained. the applying level question involves using acquired knowledge in one step and applying the information to solve the problem with correct procedures. 2) higher-order thinking is the type of non-algorithm thinking which include analytic, evaluative and creative thinking that involves metacognition. in this context, problemsolving question belongs to the higher-order thinking question which can be reflected in analyzing, evaluating, and creating. the analyzing level is problem-solving question whereby solution does not use the information (formulas, rules or procedures) directly but instead use other additional supporting information. the evaluating level question is problem-solving question containing elements of decision making. in mathematics context, the decision can be considered as the correct or incorrect of given information or statement, or the examination procedures of results of problem-solving. the creating level question is problem-solving question containing orders or commands to create something new as clue or guidance to solve the next problem. results and discussion results the following are the research results and are organized according to the research question: pre-service mathematics teachers’ conception of higher-order thinking subjects wrote different conceptions of lower-order thinking and higher-order thinking based on their understanding. most of the subjects wrote that lower-order thinking is thinking ability on easy problems, whereas higher-order thinking is thinking about difficult problems. a subject wrote that lower-order thinking is the ability to remember the concepts while higherorder thinking is problem-solving ability level. another subject wrote that lower-order thinking is elementary school problems and higher-order thinking is high school problems. some conception of higher order thinking and lower order thinking is shown in the following figure: volume 6, no. 2, september 2017 pp 121-136 125 figure 1. subjects’ conception the summary of subjects’ conceptions can be shown as follows: table 1. the summary of subjects’ conception description lower-order thinking higher-order thinking easy level problems difficult level problems ability to remember the concepts using existing concepts to solve problems use understanding in problem-solving developing an existing understanding in solve the complex problem. not making a mathematical model making mathematical model simple thinking level that uses a part of the brain complex thinking level that uses all the brains elementary school problem thinking high school problems thinking problem-solving with little step procedures problem-solving with many step procedures calculation problem thinking verification problem thinking solve concept problems solve word or contextual problems the summary of subjects’ conceptions of lower-order thinking and higher-order thinking is more emphasis on the different between the level of easy and difficult problem, calculation problem and verification problem, conceptual and contextual, and elementary and high level. subjects’ conceptions have touched on lower-order thinking and higher-order thinking description yet as two different cognitive processes level at procedural and nonprocedural or algorithm and nonalgorithmic. some subjects construe the difference in making mathematical models and not. mathematical modeling is a characteristic of the understanding level. on the other hand, misconceptions arise at the distinction in calculation and verification. verification is a part of higher-order thinking in analyzing level; however, the calculation is at higherorder thinking and lower-order thinking level. misconceptions also seem at the distinction in lower-order thinking and higher-order thinking characteristic which at the conceptual and contextual problem. contextual problems are considered as higher-order thinking problems even though those problems apply the formula which is known directly. subjects' misconceptions impact on the formulation of higher-order thinking mathematics problems. in order to strengthen the conception of lower order thinking and higher order thinking bloom’s taxonomy students’ written answer, then interviewed several subjects. the results of interviews on the written answers of the two subjects above are presented as follows: r : you write down the characteristics of lower order thinking and higher order thinking based on different levels of easy and difficult questions. what is the reason? s1 : i understand that lower order thinking is a simple type of thinking that certainly in doing math problems does not require long and difficult process while higher order thinking otherwise. r : could you explanation it more concrete? s1 : i mean, the lower order thinking problem is asked to solve the math problem directly, for example, find the roots of quadratic equation, this is easy and not a long process. compare with the matter of quadratic equations but in story form. samo, pre-service mathematics teachers’ conception of … 126 --------------------------------------------------------------------------------------------------------------------------------------- r : lower order thinking and higher order thinking characteristics you write that is conceptual and contextual problem solving. why is that? s2 : lower order thinking works on the realm of facts or concepts. this means we're only asked to solve the problem by mentioning concepts we remember from what we've learned before. r : what about higher order thinking? s2 : higher order thinking is more complex, we are asked to use the concepts we remember or mention it to solve the contextual problems we encounter. pre-service mathematics teachers’ conception of six cognitive levels the second research question is how preservice mathematics teachers categorize six cognitive levels of bloom's taxonomy as a lower-order thinking and higher-order thinking. subjects provided six cognitive levels of bloom's taxonomy and identified at lower-order thinking and higher-order thinking level. some categorization of higher order thinking and lower order thinking is shown in the following figure: figure 2. subjects’ categorization those results are presented as follow: table 2: the percentage of subjects’ categorization results cognitive level percentage lower-order thinking higher-order thinking remembering 50 (100%) 0 (0%) understanding 38 (76%) 12 (24%) applying 16 (32%) 34 (68%) analyzing 10 (20%) 40 (80%) evaluating 8 (16%) 42 (84%) creating 0 (0%) 50 (100%) volume 6, no. 2, september 2017 pp 121-136 127 figure 3. subjects’ categorization results based on the table, all of the students categorize the remembering level in lower order thinking and creating in higher order thinking. in understanding level, 76% subjects categorize in lower order thinking and 12% categorize in higher order thinking. in analyzing level, 20% subjects categorize in lower order thinking and 80% subjects categorize in higher order thinking. in evaluating level, 16% subjects categorize in lower order thinking and 84% subjects categorize in higher order thinking. the subjects’ categorization of lower-order thinking and higher-order thinking on above table showed that subjects made a mistake when they placed applying level at higher-order thinking. these conditions accordance with the exploration results at the first research question where many subjects mentioned that higherorder thinking as a type of solving contextual or word problem. in quantitative terms, application level is interpreted as higher-order thinking level, qualitative exploration to obtain any information why subjects categorize applying level at higher-order thinking would be explained on qualitative research later. to explore the results of lower order thinking and higher order thinking bloom’s taxonomy conception then conducted interviews on several subjects. the results of interviews on the written answers of two subjects above are presented as follows: r : you put remembering and understanding in lower order thinking while others are in higher order thinking. what is your explaination? s1 : actually, i forgot about dividing the taxonomy level of bloom but based on my intuition, remembering and understanding it requires a simple process. as i said before that the lower order thinking is a simple process, not complicated and not long. r : what about the other four levels you categorized in higher order thinking? s1 : because higher order thinking need a long and complicated process then the matter of applying level and the other exactly categorized in higher order thinking. this is because, at the level of this problem we work on the problem with a long work process, using many concepts and difficult. ---------------------------------------------------------------------------------------------------------------- r : you put remembering, understanding and applying in lower order thinking level while analyzing, evaluating and creating in higher order thinking. what is your explanation? s2 : in bloom’s revision taxonomy, distinguished in two levels of lower order thinking and higher order thinking with the cognitive level i wrote this. in my opinion, the first three levels as lower order thinking require simple cognitive work, whereas higher order thinking requires more complex cognitive work because it involves many related concepts in contextual problem solving. rememberi ng understand ing applying analizing evaluating creating low 50 38 16 10 8 0 hot 0 12 34 40 42 50 50 38 16 10 8 0 0 12 34 40 42 50 0 10 20 30 40 50 60 low hot samo, pre-service mathematics teachers’ conception of … 128 r : many of your friends put applying into higher order thinking. what is your opinion about that? s2 : the applying is related to the use of formulas or rules directly so it does not require complex cognitive work. r : can you give an example to prove your answer? s2 : in example we are asked to find the length of one side of the right triangle if we know the length of two other sides then we use the formula phytagoras directly. this is easy. so this is an applying level. pre-service mathematics teachers’ identification of lower-order and higher-order thinking questions. the third research question is how pre-service mathematics teachers identify tests items as the lower-order thinking and higher-order thinking in one-variable linear equations and function. the tests items are shown as follows: section i (one-variable linear equations) 1. the sum of three consecutive even numbers is 48. what are these numbers? 2. what is an open sentence? 3. write the one-variable linear equation form of following sentence: “the sum of twice of y and (-5) is 21” 4. find the solution set of 5x = x – 40 5. find the mistake in solving one-variable linear equations 2x = 11x + 45 2x – 11x = 11x-11x + 45 9x = 45 9 45 9 9  x x = 5 6. write the real-world phrase as a variable expression: 2x + 5 = 6 7. write an equation with the solution x=2, the equation should have the variable on both sides 8. a car and a motorcycle set off from the same point to travel the same journey. the car has a start of four minutes before the motorcycle sets off. if the car travels at 80km/h and the motorcycle travels at 90km/h, how many kilometers will be traveled when the two vehicles are level? some identification result of cognitive level is shown in the following figure: volume 6, no. 2, september 2017 pp 121-136 129 figure 4. subjects’ identification identification results that are presented as follow: table 3. levels categorization results question items cognitive level of bloom’s taxonomy c1 c2 c3 c4 c5 c6 1 2 8 22 12 5 1 2 30 16 3 1 0 0 3 11 20 17 2 0 0 4 9 18 15 8 0 0 5 0 1 12 15 12 10 6 1 12 2 13 5 17 7 0 8 11 15 0 16 8 0 6 35 7 2 0 total 53 89 117 73 24 44 figure 5. subjects’ categorization results remem bering unders tanding applyi ng analizi ng evaluat ing creatin g categorization 53 89 117 73 24 44 53 89 117 73 24 44 0 100 200 categorization categorization samo, pre-service mathematics teachers’ conception of … 130 the first section presented higher-order thinking question at number 5, 6, 7 and 8. the fifth questions are identified differently by subjects. question number one is applying level, 44% subjects identified as applying level and 24% subjects identified as analyzing level. question number 5 is evaluating level, 74% subjects identified as higher-order thinking level, 30% subjects identified as analyzing level and 24% subjects identified as evaluating level. question number 6 and 7 are creating level, 34% subject identified as creating level at number 6 dan 32% subject identified as creating level at number 7 whereas, there are still many subjects that categorize at analyzing level. what is interesting is at the question number 8, about 70% subjects identified as applying level. in general, eight questions that are presented above, the majority of subjects categorized them in the applying level. if we review the group higher-order thinking and lower-order thinking then students categorize 64.75% in lowerorder thinking category. this condition in accordance with the subject's conception which subjects’described higher-order thinking as the word or contextual problem. section 2 (function) 1. given a function g(x) = ax + 7. the value of g(x) for x = -2 is 1. find the value of g(x) for x = 5. find the equation of g(x). describe your ways. 2. given a function f(x) = 2x – 3 with the domain of f(x) is a = {7, 9, 11, 13} and a function dan fungsi g(x) = 3x + 4 with the domain p = {x x  3, x r }. compare the two graphs the functions, what can you conclude? 3. what are relation and function? 4. two graphs of function are parallel if the comparison of variable x and the comparison of variable y are equal. whereas two graphs of function coincide if the comparison of variable x, the comparison of variable y and if the comparison of coefficient is equal. whether the information is correct? 5. draw an arrow diagram to represent the function that related from a set of a to b. if r is a relation 'is greater than' from a to b. 6. given a function f(x) = -x + 3 with the domain k = {-3, -1, 1, 3, 5, 7} repersent f(x) diagramatically find the value of f(x) for x = -3, x = 5 find the range of f(x). 7. given a = {p, q, r} and b = {2, 3, 4}. draw all the possible mapping from set a to set b with arrows diagram some identification result of cognitive level is shown in the following figure: volume 6, no. 2, september 2017 pp 121-136 131 figure 6. subjects’ identification the result of identification are presented below: table 4. levels categorization results question items cognitive level of bloom’s taxonomy c1 c2 c3 c4 c5 c6 1 0 3 16 18 12 1 2 0 1 19 23 7 0 3 27 20 2 1 0 0 4 0 1 5 23 20 1 5 0 13 19 12 5 1 6 0 1 20 15 6 8 7 1 1 15 11 10 12 total 28 40 96 103 60 23 figure 7. subjects’ categorization results the second section presented higher-order thinking question at number 1, 2, 4, 5 dan 7. the fifth questions are identified differently by subjects. question number 1 and 2 are analyzing remem bering underst anding applyin g analizi ng evaluat ing creatin g categorizaton 28 40 96 103 60 23 28 40 96 103 60 23 0 100 200 categorizaton categorizaton samo, pre-service mathematics teachers’ conception of … 132 level. from both questions, more than 90% subjects identified as higher-order thinking level, 36%, and 46% subjects answered correctly as analyzing level. question number 4 is evaluating level, 46% subjects identified as higher-order thinking at analyzing level and 40% at evaluating level. question number 5 is creating level, 36% subjects identified as higherorder thinking with details 24% categorize at analyzing level, 2% subjects located at creating level, while 38% identified as applying level. question number 7 is creating level, 24% subjects identified as creating level whereas 30% identified as applying level. in general, nine questions that are presented above, majority of subjects categorized them in the analyzing level. if we review the group higher-order thinking and lower-order thinking then students categorize 64.75% in higher-order thinking category. discussion the purpose of this study is to explore pre-service mathematics teachers' conception of higher-order thinking in bloom's taxonomy, categorization six cognitive levels of bloom's taxonomy as lower-order and higher-order thinking, and identification some question items as lower-order and higher-order thinking. this research reveals preservice mathematics teachers’ conceptions of higher-order thinking in bloom’s taxonomy that inclined erroneous. pre-service mathematics teachers' conception of lower-order and higher-order thinking more emphasis on the different between the easy and difficult problem, calculation problem and verification problem, conceptual and contextual, and elementary and high-level problem. pre-service mathematics teachers categorized six cognitive levels at the lower-order and higher-order thinking level correctly except at the applying level, preservice mathematics teachers placed it at the higher-order thinking level. subjects categorize remembering and understanding level in lower order thinking whereas applying until creating in higher order thinking. these categorization base on their intuition and learning experience that mention the word problem as application question and the application question is problem-solving question. word problem question and problemsolving question are two different things. word problem can be categorized in problemsolving question and applying level question. the problem-solving question is not merely the word problem. in other sections, pre-service mathematics teacher tend to made the wrong identification of the test questions that were included in the lower-order and higher-order thinking. bloom's taxonomy is something that is very familiar to the preservice mathematics teachers but not understood deeply. according to coffman (2013), although the preservice teachers have heard of bloom’s taxonomy, it does not necessarily come to mind when discussing higher order thinking, until they are prompted. pre-service mathematics teachers described higher-order thinking that is different from lower-order thinking on the different at elementary thinking and advances thinking, the level of difficulties, conceptual and contextual, and the type of calculation and verification. in accordance with these research findings, thompson (2008) states that teachers misinterpret higher-order thinking level of bloom's taxonomy. teachers define higher-order thinking as problem-solving, finding a pattern; interpret information and conceptual understanding. in contrast, most of the teachers define higherorder thinking based on characteristics such as; a) a number of measures "necessary" to solve tasks, (b) level of difficulty, or (c) algebra as subjects that create higher order thinking. cognitive levels of bloom’s taxonomy are used in learning activity objectives. the misconceptions in the formulation of learning objectives provide the competencies achievement that is wrong direction anyway. the opinions of coffman (2013) and thompson volume 6, no. 2, september 2017 pp 121-136 133 (2008) supports the results of this study reveal that blooms’ taxonomy often heard even allow potential teachers but not understood properly. pre-service mathematics teachers identify the higher-order thinking test in accordance with their conception which is described above. it seems that the first and the second section of test items are identified by preservice mathematics teachers at applying level. this is consistent with harpster (1999), which revealed that mathematics teachers have a mistake perception about higher-order thinking as lower-order thinking. the mistaken understanding impact of the preservice teachers of higher-order thinking and lower-order thinking is that the ability to create the problem will be low. saeed & naseem (2016) reveal question papers were largely assessing students’ lower cognitive abilities (knowledge and comprehension); a few items were assessing higher cognitive abilities (application and analysis). moreover, hamafyelto, hamman-tukur, hamafyelto (2015) in their research found there were significant relationships between teachers of commerce competence and content validity, the areas of teachers’ competence in constructing examination questions was low. it was found that teachers concentrated on the lower levels of the cognitive domain (remembering, understanding applying). both of these opinions provide the same idea that many teachers are still inclined to make problems at the low levels. the fact that teachers are only able to make a problem with low levels can be seen from the ability of preservice teachers who also have the wrong conception of higher-order thinking and lower-order thinking. the higher-order thinking processes that occur in the process of solving mathematical problems are characterized by the application of multiple criteria, which may not be known in advance (rubin & rajakaruna, 2015). as a mathematical process with many criteria, higher order thinking requires linking ideas, concepts, and disciplinary content is an underused yet effective educational strategy (richland & begolli, 2016). it is certainly different from the conception of preservice mathematics teacher in this study. the misconceptions of higher order thinking in bloom's taxonomy impact on the development of mathematical thinking ability. the development of mathematical thinking ability is the purpose of the current mathematics education (kaya & aydin, 2016; katagiri, 2004). the development of mathematical thinking ability is formulated in the objectives of learning activities. pre-service mathematics teachers use bloom's taxonomy to develop learning indicators but often not aligned with the formulation of test items. sometimes, tests items are not aligned with learning indicator. pre-service mathematics teachers just introduced the similar types of problems that are more conditioned the cognitive structure to calculation not thinking. in addition, educators have limited knowledge and creativity in developing test items that vary according to the cognitive level of bloom's taxonomy. test items which are tested also based on a textbook that allows students to learn with the existing standard procedures. kocakayaa & kotluka (2016) explained the agreement in interpretation of standards has to increase, especially for teachers, to allow all students a fair chance to attain the same standards regardless of teachers and schools. the low levels of inter-and intra-judge consistency for the teachers may have negative effects on students’ learning, on fairness in assessments. ryan & mccrae (2006) described the errors and misconceptions made by pre-service teachers were used here to inform either personal development or collective treatment during preservice teacher education. teacher errors deserve attention not least to avoid transfer to children in schools. errors provide opportunities for pre-service teachers to examine the basis of their own understanding so that knowledge can be reorganized and strengthened. samo, pre-service mathematics teachers’ conception of … 134 teachers’ misconceptions are caused they have a limited understanding of bloom's taxonomy. in education, standards have to be interpreted, for planning of teaching, for development of assessments and for alignment analysis (näsström, 2009). here, the understanding of the blooms’taxonomy correctly become very important in the development of students' thinking skills. conclusion based on results and discussion above, it can be concluded that: (1) preservice mathematics teachers have wrong conceptions of higher-order thinking. the preservice mathematics teachers identified higher-order thinking questions type according to elementary and advanced level thinking conception, the level of difficulties, conceptual and contextual, as well as the type of calculation and verification. this conception did not follow the description of lower-order and higher-order thinking as the different cognitive level in procedural or nonprocedural, algorithm or nonalgorithm problem solving; (2) most of the preservice mathematics teachers categorize six cognitive levels at the lower-order and higher-order thinking level correctly except at the applying level, preservice mathematics teachers placed it at the higher-order thinking level; (3) preservice mathematics teacher tend to made the wrong identification of the test questions that were included in the lower-order and higher-order thinking. the wrong conception of lower-order and higher-order thinking lead the preservice teachers to identification results misconception. according to conclusion described above, there are suggestions and recommendations as follow; (1) preservice mathematics teachers should be provided with better understanding of cognitive levels of bloom's taxonomy, especially the characteristics of each level, attributes attached to each level and math skills, to be part of the higher-order and lower-order thinking level, and other skills relevant to math; (2) preservice mathematics teachers should be familiarized of higher-order thinking questions start from their first-year of study. the purpose of this attempt is to inform pre-service teachers the type of higher-order thinking, to distinguish and develop insight to think outside the context, divergent, critically and creatively; (3) the preservice mathematics teachers should be trained to develop higher -order thinking level questions. thus in the future, they will be able to creatively develop nonroutine test items. acknowledgments i would like to thank the third-year students of mathematics education department at nusa cendana university, who have taken the time been the subject of research and fill this instrument, without their help this work would never have been possible. references anderson, l. w., & krathwohl, d. r. (2001). a taxonomy for learning, teaching, and assessing: a revision of bloom’s taxonomy of educational objectives. new york: longman publishing. bloom, b. s., engelhart, m. d., hill, h. h., furst, e. j., & krathwohl, d. r. (1956). the taxonomy of educational objectives, the classification of education goals, handbook i: cognitive domain. new york: david mckay company. volume 6, no. 2, september 2017 pp 121-136 135 bookhart, s. m. (2010). how to assess higher-order thinking skills in your classroom. alexandria: ascd. coffman, d. m. (2013). thinking about thinking: an exploration of preservice teachers’ views about higher order thinking skills. university of kansas: unpublished doctoral dissertation. delima, n. (2017). a relationship between problem solving ability and students’ mathematical thinking. infinity, 6(1), 21-28. hamafyelto, r. s., hamman-tukur, a., & hamafyelto, s. s. 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(2008). mathematics teachers’ interpretation of higher-order thinking in bloom’s taxonomy. international electronic journal of mathematics education, 3(2), 97-109. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p235-258 235 investigation of students’ behavior in mathematical problem solving yulyanti harisman*1, muchamad subali noto2, wahyu hidayat3 1universitas negeri padang, indonesia 2universitas swadaya gunung jati, indonesia 3institut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received may 28, 2021 revised july 1, 2021 accepted july 2, 2021 this study is a qualitative research by using the descriptive method that aims to examine the behaviour of eighteen students in bandung, indonesia. six issues related to geometry were given to eighteen of second-grade junior high school students with heterogeneous abilities. the problems given to the students contained all of the problem-solving strategies such as guessing and checking, make a picture, make a list, make a table, working backwards, looking patterns, and using a logical reason, solving simple problems and making questions. data collection was conducted through mathematical problem-solving tests, recording students’ presentations, and interviewing among researchers and students after doing the problems. the result of recording was a video during the presentation process, and the interview would explore their understanding of the given problems to see the behaviour used by subjects of the research. the data in this research showed that many students’ behaviour identified; in the relevant literature, there are terms of the behaviour of problem-solving naive, routine, and sophisticated. however, the category "naïve," "routine," and "sophisticated" did not fully draw various behaviours observed, it was obtained additional category termed behavioural problem solver "naïve," "routine," "semi-sophisticated" and "sophisticated". it was due to the category of regular students can be divided into two, some students can be directed, and some of them cannot be directed to sophisticated behaviour. thus, the routine category can be classified into two categories: routine and semi-sophisticated. keywords: behavior, naïve, problem solving, routine, semi-sophisticated, sophisticated copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: yulyanti harisman, departement of mathematics education, universitas negeri padang jl. prof. dr. hamka, air tawar, padang city, west sumatra 25171, indonesia email: yulyanti_h@fmipa.unp.ac.id how to cite: harisman, y., noto, m. s., & hidayat, w. (2021). investigation of students’ behavior in mathematical problem solving. infinity, 10(2), 235-258. 1. introduction the curriculum of mathematics education in indonesia has changed several times. the changes were enacted curriculum starting from 1947 until 2013. the 2013 curriculum applies nowadays, insists on the importance of thinking skills, communication, and a deep understanding of students in solving problems encountered in the learning process of mathematics, therefore it is necessary to change the curriculum. https://doi.org/10.22460/infinity.v10i2.p235-258 harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 236 regulation of the minister of education and culture (permendikbud) no. 22 the year, 2016 about the standard of primary and secondary education, stated that the skills acquired through the activity of "observing, asking, trying, reasoning, presenting, and creating." characteristics of competence and the difference of acquisition affect the process standard characteristics (mecri, 2016). to encourage the ability of learners to generate contextual work, either individually or in groups, are strongly suggested to use learning approaches that produce project-based learning. it appears that the standard process of primary and secondary education in the curriculum also emphasizes on solving mathematical problems (chew, shahrill, & li, 2019; olivares, lupiáñez, & segovia, 2021; stacey, 2005). the studies conducted by experts mostly focus on the effective implementation of teachers to improve the abilities and behaviour of students in solving problems. a study conducted by hidayat (2017) examine the problem-solving behaviour of senior high school students on non-routine problems associated with the concept of calculus material. another study also examined the implementation of teachers conducted by sulak (2010), which saw the effects of problem-solving strategies towards the achievement of problem-solving in primary school students. furthermore, cai (2003) did an international study that documented the problemsolving approach, which covers most of the students, describing the successful implementation problem-solving approach in learning mathematics. besides the studies evaluating the implementation of teachers’ strategies to improve students' problem-solving, several studies also investigated how the approach used by students in solving problems. the research had done by de hoyos, gray, and samson (2002) that observed the processes carried out by two students in solving the problem. muir, beswick, and williamson (2008), "although there have been studies that see the approach of the students, there is no effort of researchers to verify whether the selected students are consistent in any issue." muir et al. (2008) conducted a study that saw the consistency of behaviour that he selected 20 elementary school students from 5 different schools in solving problems. this study has six questions which contain various strategies proposed by polya (1957). a research conducted by muir et al. (2008) was grounded on research from schoenfeld (1982), which examined how a problem-solving expert and novice choose behaviours that were used in solving the problem. according to schoenfeld (1982), behaviours selected by an expert in solving the problem are: the expert on problem-solving tend to recognize the patterns of problems, tends to change strategy when a strategy is not working, and an expert on problem-solving can generate its strategy in solving problems. furthermore, some of the behaviour displayed by novice problem solvers in solving the problem is: only recognize the problem of the surface of it, tend to manipulate numbers in solving problems, and unable to move strategy if a strategy does not work (harisman et al., 2019) the research findings of muir et al. (2008) complemented the research done by schoenfeld (1982). if schoenfeld categorized the behaviour of students when solving problems in two categories, namely expert and novice, then muir et al. (2008) categorized it into three categories: naive, routine, and sophisticated. naive behaviour-oriented on problem solver behaviour associated only with manipulating numbers that exist in the problem. routine behaviour-oriented on structured behaviour and sophisticated problem solvers oriented on problem solver who can generate their strategies when they faced problems (harisman et al., 2020). volume 10, no 2, september 2021, pp. 235-258 237 1.1. the process of problem solving talking about solving mathematical problems certainly will not be released from the core problem itself. problems solving questions are usually non-routine problems. problemsolving also related to other terms such as reasoning, critical thinking, creative thinking, decision-making is a subset of problem-solving (ekawati et al., 2020; hendriana & fadhillah, 2019; hutajulu et al., 2019; kariadinata, 2021; maulidia et al., 2019; praekhaow et al., 2021). through problem-solving, it will broaden a person’s thought, from which he does not know transform into he knows when carrying out the process of solving the problem. according to elia, van den heuvel-panhuizen and kolovou (2009), in answering the questions of problem-solving, we can use flexible strategies, and they can be modified and changed if the strategies do not work. besides, using problem-solving strategies require several abilities include interpreting information, planning and working methods, checking the results, and trying to look for an alternative strategy (muir et al., 2008). heuristic planning most recommended by researchers (e.g., schoenfeld, 1982) to facilitate the resolution of the problem is coming from polya (1957) and requires the problem solver to understand the problem, set a plan, implement the plan, and check back the solution which has been obtained, polya (1957) further explained in how to solve it as outlines shows four main steps in solving the problem, they are: understanding the problem, devising a plan, carrying out the plan, and looking back. heuristic and strategies identified are most widely used in problem-solving, guessing and checking, making an image, creating a list, tables, working backwards, seeing patterns, and using a logic reason, solving simple problems and making questions (muir et al., 2008). based on the indicators outlined in the polya (1957), it created a rubric that will be used to check the students’ work in solving the problems (see table 1). table 1. problem solving ability scoring rubric (muir et al., 2008) scores problem solving ability rubric understanding the problem devising a plan carrying out the plan 0 unable to identify the information that exists in the problem not using one of the problem-solving strategies loading errors in every step of the completion 1 able to identify the information contained in the problem using problem-solving strategies but contains wrong steps there are errors in several steps to resolve 2 understanding the meaning of every information in the problem using problem-solving strategies with the right steps there are no errors in every step of completion based on these rubrics, students’ work will be analyzed, and during problem-solving, it will be seen the consistency of students' problem-solving behaviour. many things can affect the performance of solving the problem; one of them is effective. several dimensions of affective that affect problem-solving are emotion, the level of awareness, and control of students (mcleod, 1988). fitzpatrick (1994) examined the relationship of various cognitive factors, attribution, and gender in finding the solution of a mathematical problem solving harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 238 with 100 smu. based on these other aspects need to be examined outside the problemsolving process mentioned by polya (1957), which becomes the main reference in this article. things that affect performance problem solving outlined in problem-solving behaviour. besides, students' ability to solve their behaviour problem, it also needs to see the weaknesses of them in solving problems. below are descriptions of behaviours that indicate to students in solving problems through observation of research experts’ findings. 1.2. the behaviour of problem solving behaviour is a response to the stimulus given to the students. the behaviour can be measured and observed through observation. if people are in a situation of containing the problem, then the individual will show a course of conduct to solve the problem. the mathematics learning process often exposes students to the issues of problem-solving. to solve these problems, of course, the student will show a set of behaviours. most of the researchers focus only on tests of mathematical problem solving, but this approach has been criticized because it is not sufficient, because the tests do not provide information about how students think (fitzpatrick, 1994). in order for researchers not only focus on the test and see things other than the investigator should observe the behaviour of students in solving the problem. there have been studies on mathematics education that examined students’ behaviour in facing problems solving. schoenfeld (1982) researched how problem-solving expert and novice problem solvers behave in solving problems with student research subjects in colleges. through these studies, schoenfeld found that the novice problem solvers tend to exhibit behaviour in recognizing the problem only from the surface, while the expert problem solvers show their behaviour in recognizing the pattern of these problems. the novice problem solver tends to manipulate the numbers on the issue, read quickly and at a glance, do not give consideration to the context of the problem, not diligent in resolving the problem if the chosen strategy does not work, do not access other problem-solving strategies, do not show thinking metacognitive. the experts tend to exhibit their different behaviour to manage the problem better, motivated intrinsic and high curiosity, like the challenge, always work actively, always recheck the solution that has been obtained, explore more to find alternative solutions muir et al. (2008) also viewed behaviour that indicated 20 elementary school students in solving problems. according to muir et al. (2008), experts and new problem solvers do not accommodate the behaviour intended to show. the rubric of problem-solving behaviour after twenty students worked on six problems solving and conducted interviews with them (see table 2). table 2. overview of students’ behaviour based on a range (muir et al., 2008) factors category of behaviors naïve routine sophisticated knowledge ownership making mistakes on the four-step problemsolving there is no attempt to verify the solution (making mistakes at some problemsolving steps) a high score on every step of problem-solving volume 10, no 2, september 2021, pp. 235-258 239 factors category of behaviors naïve routine sophisticated unable to use the earlier problems that have been resolved able to identify similar issues, but not on the mathematical structure identifying similar problems in the structure of the mathematics frequent use the same way to solve all problems focus on one way to solve a particular problem. identifying other ways to solve the problem written communication and verbal are not sufficient verbal communication is usually clear written communication and verbal are adequate controls thinking metacognitive does not appear, either in written communication and verbal thinking metacognitive looked verbally thinking metacognitive is clear in response to written and verbal confidence doing almost the same strategies implementing strategies in a systematic way generates its strategy relying on one or two strategies relying on more than two strategies, but can not move to another strategy when a strategy can not work willing to use a combination of several strategies affective confidence is in line with a quick achievement answers often express lack of confidence in the ability to solve problems reveals confidence in the ability to solve problems the rubric (see table 2) based on things that affect the behaviour presented by lester and kroll (1993). according to lester and kroll, problem-solving performance is affected by five factors: acquisition knowledge and utilization, control, confidence, affective, and socio-cultural context. 1.3. statement of the problem the implications of the research findings from muir et al. (2008) state that "a decision of this behaviour that has an impact on studying and learning of solving the problem". after analyzing the behaviours that are shown by muir et al. (2008), the authors interested in conducting the study in secondary schools in the city, bandung. particularly, it addresses the following qustions: (1) what behaviours are shown by some secondary high school students in the city of bandung, indonesia? (2) how far is the consistency carried out by each student? the selected subject is geometry based on the experience of junior high school teachers who explained that students have difficulty in understanding the material. harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 240 2. method 2.1. description of study this study was conducted to see the behaviour of eighteen from three different junior high school students in the city of bandung in solving problems. the study was based on a study done by (muir et al., 2008) where it will be seen how all knowledge of ownership, control, confidence, and affective of students when they were given problem-solving questions. muir et al. (2008) saw it on elementary students in tasmania. the researcher would like to try to do this on a junior high school in bandung city. 2.2. participants subjects in this study were eighteen students in three junior high schools in the city. students were selected with varying capabilities (high, medium, and low). the ability of students viewed from the midterms that were given by teachers. the students were chosen because they considered in the transition of the period from children to teenagers. it was expected that the behaviour they appear would be more expressive and spontaneous. it was expected that strategies would be more informal displayed by students. junior high school students are considered students who will be at the level that will bring a model of solving problems that will show models for the solution of such problems. 2.3. assessment instrument the task given to the students was the problem solving that contained all the problem-solving strategies. the problem was designed to be seven questions. seven problems raise a wide variety of strategies that will have students like to make a list, a portrait of the diagram, see patterns, and so on. this matter has also been used in studies of observing gesture gender with different students in solving problems (harisman et al., 2017). selected topics in this study was the topic of geometry. this topic was chosen according to the junior high school teaching experience. the students often have difficulties when they are faced with the problems of geometry (see table 3). table 3. description of the problems no description of the problems 1 mr hok guan, the owner of the pipe shop, wants to tie his pipes with rope, each bond is six pipes. he is confused about how to tie them (r = 10 cm). is the model bonding a rectangular or triangular model? because the rope used to tie the pipes is limited, mr hok guan expects more efficient use of the rope. help mr hok guan to solve the problem. which binding is the most economical? volume 10, no 2, september 2021, pp. 235-258 241 no description of the problems 2 here are cube nets that are numbered 1 through 6. what is the largest of three numbers on the sides, which together form the cube-corner point? 3 it is known that a swimming pool with a pool size of 60 meters long, 15 meters wide, depth of 1 meter left hand and kept right ramps up to a depth of 5 meters. calculate the volume of water needed to fill the pool to the brim. 4 pay attention to the unit cube pile below. if the unit cube pile pattern continues, what is the number of the unit cube on stacks to-5? 5 it gives a square abcd with a side length of 2 units. the sides of the square are the radius of the circle. calculate the area shaded? 6 mr ahmad wants to include several books of the same size, namely: length 20 cm, width 15 cm, and 2 cm thick in a box-shaped beam with a size of 30 cm x 25 cm x 25 cm. what is the maximum number of books that can be inserted into the box? 7 the volume of a cube is equal to the volume of a beam that is. it is known that the long beam twice as long as high-beam cube and half times the width of the beam. determine the whole surface area of the beam! after the eighteen students asking to answer seven problems above, they were interviewed about the answers to the questions submitted. students were asked to respond to these questions while the interviewee was recording and making observations notes of the interview process. the interview approach is called the semi-structural approaches interviewed by burns (muir et al., 2008). the students' answers were analyzed by using rubric existing sections in table 4. harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 242 table 4. problem solving capabilities scoring rubric (muir et al., 2008) scores problem solving ability rubric understanding the problem explaining the plan of problem solving checking back the problem solving 0 unable to identify the information that exists in the problem not using one of the problem-solving strategies loading errors in every step of the completion 1 able to identify the information contained in the problem using problem-solving strategies but contains wrong steps there are errors in several steps to resolve 2 understanding the meaning of every information in the problem using problem-solving strategies with the right steps there are no errors in every step of completion data were analyzed with the mentally correct answer and ranked based on the scoring rubric listed in table 4. tally also conducted on how many strategies are chosen by students in completing the seven questions. then, the data were analyzed to see the students’ behaviour through the analysis of the results of the interviews. 3. results and discussion 3.1. behavior problem solving orientation students’ behaviour of three junior high schools in bandung is presented in this section. six students were selected from each school with different abilities. two students have a low capability, two students have the middle ability, and two students have a high ability. the students were recommended by school teachers based on mathematical ability. students were labelled a to high-ability students, s for middle students, and r for lower-ability students. furthermore, in labelling schools, s-1 is for the school one, s-2 for the school two, and s-3 is for school three. the following explanation is in table 5 the names and abilities of students in each school. table 5. name and ability of students at each school no student's school origin school one school two school three 1 lutfi (r) s-1 dwi nanda (r) s-2 fauzan (r) s-3 2 divy (r) s-1 aulia fauza (r) s-2 rizky (r) s-3 3 fikri (s) s-1 najla (s) s-2 mesya(s) s-3 4 dhea (s) s-1 febrina (s) s-2 dita (s) s-3 5 annisa (t) s-1 zara (t) s-2 yani (t) s-3 6 alvaro (t) s-1 salma (t) s-2 nadhira (t) s-3 volume 10, no 2, september 2021, pp. 235-258 243 labelling is done to facilitate in grouping the students based on the behaviour category at the end grain. rubric behaviour muir et al. (2008), which presented in the previous section (table 2), used as a reference early in categorizing the students’ behaviour. the description of data is only presented to students who are at the same school. for other schools also analyzed in the same way, but it is not described in the discussion. recapitulation of the analysis results for each student in each school is included at the end of the description. the behaviour of mathematical problem-solving in each indicator of students at school (s-1) will be described — the names of the students: lutfi, divy, fikri, dhea, annisa, and alvaro. the following is an overview of students’ behaviour in every aspect of mathematical problem-solving behaviour. 3.2. knowledge ownership 3.2.1. applying heuristics polya step in mathematical problem solving students’ apathetic behaviour made a mistake on the four-step problem-solving. this is indicated by fikri answer in solving two problems in figure 1 at the following tasks in table 3. figure 1. representation of fikri's mathematical problems two fikri did not understand the problem because fikri did not know the purpose of the three sides that would form the vertices, fikri has not been able to choose the right strategy and implemented it. fikri supposedly tried-piloted three sides which will form the corner points then choose which of the numbers that are on the three sides if it is added together, it will result in big numbers. problem two was designed with one strategy solution, they are guessing and check, but because fikri did not understand the problems of the strategy selected, so the finding was not right, that only by choosing three numbers on the sides nets of the cube that when the nets were added together, it would produce the largest number, and fikri has not made the resulting checking to the answer. the unreached comprehension, the wrong strategy selection, and implementation and has not been checked back into the solutions are also demonstrated through interviews with the following fikri. interviewer : what is going with this? fikri : the biggest number is 4, 5 and 6 interviewer : it is, but the problems are known here, nets of a cube. what are the cube nets? fikri : i do not know interviewer : nets of a cube are a series of squares that belong to a two-dimensional figure if they are strung together back, they will form a geometrical cube, remember? fikri : yes interviewer : well, l the things looked for is the biggest of three numbers, but which together form the corner points, now the question is whether the 6, 4, and 5 form a vertex? harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 244 fikri : i do not know interviewer : try, for example; the base is two, then four, then six, then the roof is five, is it going to form a corner? try to imagine! fikri : i do not know interviewer : (taking a cube). take a look, how is the pedestal? fikri : two interviewer : left? firi: : six interviewer : right? fikri : four interviewer : the roof? fikri : lima interviewer : now, are six, four, and five forms a corner? fikri : i do not know interviewer : not form, which forms a corner that is part of this, this and this (pointing to three different sides that form the vertex). so, although the 6, 4, and 5 are the three largest numbers they do not form a corner point, it should be tried first one-on-one, can we imagine or not? fikri : yes although fikri has been directed to understand the problem and led to determine which three sides that will form the vertices, fikri always replies with answer directives do not know. furthermore, students who behave routine behaviour shown by the representation and dhea’s answer to problem number two, respectively, are as follows in figure 2 and figure 3. figure 2. representation of mathematical dyvi for problem two volume 10, no 2, september 2021, pp. 235-258 245 figure 3. mathematical representation of dhea for problem two based on the answers and interviews, dyvi and dhea understood the existed problems, in which they understood where the three numbers on the sides of the cube that would shape vertices. they were assumed that they already chose the right strategy, guess and check, three digits on the side that would form the corner points, but still had several steps to resolve the error and not presenting the correct answers yet. dyvi was right in choosing three numbers that would form the corner points: 2, 3, 6 and 2, 3, 4, but she was still wrong in choosing the numbers 1, 2, 3, and not presenting the correct answer which was not determined yet where the three biggest numbers that would form the corner points. dhea was also right in choosing three numbers that would form the corner points: 2, 3, 4, and 1, 4, 5, but she was still wrong in choosing the numbers 1, 2, 3, and 6, 2, 4, and dhea also did not present the correct answer that was not yet determined where the three largest numbers that would form the vertices. the following interview on dyvi and dhea is presented below. interviewer : in question number two, what story is this? what is this? dyvi : cube, oh yes nets interviewer : what is the net? dyvi : that is they are; it is hard to explain interviewer : is it hard to explain it? so if the nets are folded they are going to form a cube dyvi : yeah, like that interviewer : well, what would be asked now? dyvi : what are the three largest number sides together that will form a cube corner point? based on the result of the interview, dyvi understood what was meant in the questions, but did not have enough knowledge to verify the solution. dyvi assumed that what she did was correct. when she was directed that the three numbers written by dyvi were wrong, dyvi also did not attempt or desire to seek out the truth of the response. the following interview shows this: interviewer : (take wake cubes), what number is in the base? dyvi : two interviewer : how many are these? (pointing to the left side) dyvi : six interviewer : how many are these? (pointing to the front side) dyvi : three harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 246 interviewer : how much are these? (referring to the right side) dyvi : four interviewer : how many are these? (referring to the rear side) dyvi : one interviewer : how many are the roof? dyvi : lima interviewer : okay, now what was just asked? dyvi : the biggest sum of three numbers interviewer : what three numbers did that mean? dyvi : the numbers that will form the corner; it was just correct; it was explained one by one. based on the interview above, dyvi felt confident with her answers, and she had no attempt to verify the answers. the way of dyvi’s thinking in planning strategies have been passed fikri that is still in the orientation behaviour of apathy, dyvi did not only focuses on the numbers that exist in the questions, but had planned to try-piloted figures that would form the corner points, but there was still mistaken in several steps of problem-solving when the interviewer directed her, dyvi did not show her effort to verify the mistaken answers. the different things were shown by dhea that was indicated by an effort to verify the answers. it can bee seen in the description of the interview below. interviewer : why did you choose 6 + 2 + 4 ? dhea : that will make the corner point interviewer : what is the name of this? dhea : nets interviewer : what are they? dhea : ooh, what are they? interviewer : for example, the nets of the cube are folded, if they are folded, what are they going to be? dhea : becoming a cube dhea firstly was sure with her answers that the 6, 2, and 4 sides would form the corner point, but, if it were re-check of the three sides, it would not form a corner point. dhea already understood what the cube nets are, she planned the strategies well, but she still did a mistaken in executing the plan. when the interviewer directed her, dhea showed her effort in verifying solutions on several steps of problem-solving that were still confusing. it can be seen from dhea’s responses below. interviewer : interviewer point on one of the bases. by helping (geometry that forms a cube) the base is two, so what about the above? dhea : five interviewer : well, now which side is formed this corner point? (pointing to one of the cubes) dhea : left, front, and above left, front, top interviewer : yes, that is correct, what is the number? dhea : six, five, and four interviewer : why four? try to see it again. dhea : oh six, five, and three interviewer : yes, is there anything bigger? 6, 4, and 5 are the largest numbers, aren’t they? volume 10, no 2, september 2021, pp. 235-258 247 dhea : this six, five, and four have no corner points (pointing to the three sides on the cube) interviewer : so, why not six, four, and five? dhea : because not form a corner point when the researcher was directing that the sides chosen by dhea would not form on the cube. dhea could verify the incorrect answers she gave. furthermore, when she was given another problem that is related to another one, dhea was also able to explain it well. it can be seen from her answers that showed she did not choose the six, five, and four sides because the combination of the sides would not make a corner point. she showed an effort in verifying solution even though there was still found mistakes on steps of problem-solving. according to the case, dhea passed the problem-solving behaviour category that was shown by dyvi (routine) but cannot be categorized yet on sophisticated behaviour category (muir et al., 2008). the orientation of sophisticated behaviour category that was shown by alvaro can be categorized as sophisticated behaviour because alvaro already had a high score on every step of problem-solving. it can bee seen on the result of alvaro’s answer sheet for problemsolving number 2 in figure 4. figure 4. representation of alvaro’s mathematic for problem 2 alvaro already understood the three biggest numbers that formed a corner point chose a right strategy by describing the nets of cube and put all of the numbers on the sides of the cube, and ran the strategy by choosing three biggest numbers which would form corner point, i.e. 5, 3, and 6, and also rechecked his answers. in the interview, alvaro showed his well understanding of the problem. it can be seen in the description of the interview of alvaro below. interviewer : number two was correct, o.k? why did you choose 5, 3, and 6 on this? alvaro : imagine! if this base is two, so the roof is five, and this one is three, this one is six on the left, and that is the biggest one. imagine this picture in your mind interviewer : yes, that is correct based on alvaro’s answer, it shows that he understands the problem has a good strategy in doing it. the next is the description of students’ behaviour category on indicators to two aspects of knowledge ownership as follow. 3.2.2. use previous knowledge for mathematics problems in using the previous knowledge to solve a mathematical problem, two terms should be understood. first, students can only connect the problem by seeing the figure, for example, only with remembering the formula or concept that are used in the problem.; harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 248 second, students can connect the problem to see the mathematical structure, such by seeing the principles on formula or concept which is connected to the problem that the students face. on the aspect of using the previous knowledge to solve mathematical problems, the apathetic students will show a behaviour that they cannot use the previous problem, which was over to solve a new problem. fikri showed it. almost all over the problems he always answered do not know and not remember when he was asked about whether he was ever given the same question before or not. here is the answer of fikri in an interview in solving problem one in figure 5, figure 6, and figure 7. figure 5. representation of fikri mathematics to solve problem 1 interviewer : have you ever been given the same question like this before? fikri : i think i have interviewer : then, what have you still remember? fikri : forget interviewer : how did the question look like? fikri : not remember interviewer : the, can you solve this problem, how? fikri : remember the formula that was given on the test only interviewer : see! you ever got a question like this before in a course. how did the question look like? fikri : forget again figure 6. representation of dhea mathematics for problem one volume 10, no 2, september 2021, pp. 235-258 249 figure 7. representation of annisa mathematics for problem one interviewer : have you ever been given the same question like this before? dhea : yes, i have interviewer : what have you remembered then? dhea : at the moment if i am not mistaken that there were 4 circles given interviewer : how did the ropes look like? dhea : like a square, but there are six ropes in this question. at that time, the teacher gave the question where there 4 circles interviewer : so, how to do this? what was connected at that time? dhea : the formula n x d +, circumference may be at that time there were four circles, so the formula was 4 x d, now there are six circles; thus, the formula changes by 6 x d. the principle is the same depending on how many ropes there are. interviewer : do you know or not why it is like that when the ropes are connected on the pipes? can you show me what the six ropes are? dhea : which one? i do not know interviewer : please point which ones the ropes are! dhea : i don’t know, i want to use this formula interviewer : all right here is the description of annisa interview interviewer : had you been given a problem like this before? annisa : yes, i had interviewer : what did you remember? annisa : at that time the circle was less, there were four if i was not wrong interviewer : what kind of shape was its bond? annisa : like a first, there were four circle interviewer : so could you finish it, how? were you linked with that time or not? annisa : stay replaced n x d at that time being 4 x d, and now replaced 6 x d, the principle was like this. interviewer : do know what the reason was why it could be changed? annisa : i did not know, and it depended on many circles, didn't it? if the circle was 4, n was replaced 4, if there were six, just replaced 6. because in what book teacher once said it, n it was the number of circles. it was not so, was it? interviewer : yes, like that if we remember the formula, just changed. however, understanding the principle why can be like that? trying to connect to its strap harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 250 annisa : was it a picture of four? (trying to redraw a bond form with four-pipe) interviewer : yes, what was the relationship with the diameter? annisa : oh, this meant, it was a rope which was here a diameter or two fingers. this one, this one, and this one (while tracing with the pen). so, there is four. if there were six diameters (trace again by using a pen) interviewer : yes, good this documentation was presented when annisa tried to analyze the relationship between previous knowledge with new problems not only based on external features but also more to the mathematical structure through the direction of the interviewer. figure 8. annisa described rope with four paralon and six paralon figure 9. annisa representation in figure initially, both of students could identify the same problems that had been resolved, but not on the mathematical structure, but only on the external features by remembering the formula and associating it with the number of circles. the difference was that annisa could be directed to relate by looking at its mathematical structure while dhea insisted on using volume 10, no 2, september 2021, pp. 235-258 251 the previously thought formulas. based on figure 8 and figure 9, annisa tried to redefine and identify the mathematical structure based on the interviewer's direction. based on the explanation, dhea had exceeded the behavioural categories shown by fikri. fikri could not use previously problems that had resolved, whereas dhea could identify similar problems, but not its mathematical structure (only on external features) and could not be directed to identify similar problems according to its mathematical structure, while annisa initially could only identify similar problems with out-of-trouble features, but could be directed to identify similar problems according to their mathematical structure. furthermore, alvaro could identify a similar problem but not remember the formula and immediately looked to the diameter and radius on the pipe binding strap. alvaro identified a similar problem according to the mathematical structure, not on external features only. here were answers (see figure 10) and interviews to alvaro in a row. figure 10. alvaro’s mathematical representation for problem one interviewer : had you ever been given this question before? alvaro : i thought i had interviewer : so, what did you remember? alvaro : it was simple then; these were four circles, the same principle. this was the shorter string, here was only one diameter in length, if six into two diameters were here (pointing at the picture) interviewer : did not use a form to complete it? alvaro : i did not remember the formula. just the principle, it was just counting this string and see how much diameter. based on the interview, alvaro did not remember and did not use the formula. alvaro connected the length of the strap on the pipe with the diameter of the circle. it can be inferred that alvaro had identified a similar problem based on his mathematical structure, not merely identifying based on external features only. 3.2.3. many ways were used in mathematical problem solving based on the interview, alvaro did not remember and did not use the formula. however, connected the length of the strap on the pipe with the diameter of the circle. it can be inferred that alvaro had identified a similar problem based on his mathematical structure, not merely identifying based on external features only. furthermore, fikri and annisa have shown several variations in how to solve all problems given fikri has used various strategies in solving all problems but has not wanted to find another way or the right way to solve the problem. therefore, fikri is categorized who have a routine orientation to the variety of strategies in problem-solving. annisa differently shows this; annisa also has a variety of ways in solving the seven problems given. harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 252 annisa has shown variations in how to solve problems such as drawing and using logic. unlike fikri annisa, she wants to find a variety of other ways to solve problems. students categorized as sophisticated classified by muir et al. (2008), which have an orientation to behaviour able to identify other ways of solving all problems shown by alvaro's behaviour. for each problem given, from several presentations, alvaro tends to have other ways of solving problems from other students. alvaro has his strategy in problemsolving, without remembering systematic formulas as demonstrated by the previous behaviour of fikri and annisa. 3.2.4. written and verbal communication in mathematical problem solving students who behave apathetic tend to show inadequate verbal and written communication. dhea has shown this behaviour from the answers given that it appears that dhea's written communication is inadequate. when interviewed about the answers obtained by dhea also cannot explain these answers. furthermore, the characteristics displayed by routine students are unclearly written communication but can be clarified through verbal communication. annisa's behavior shows this, but annisa does not want to clarify the written language she has made even though the interviewer has directed it. alvaro showed the same thing. alvaro shows behaviour when communicating written language is unclear and can be explained through verbal language, but has a difference with annisa, that is when directed by the interviewer, alvaro has a desire to clarify the language of his writing. when confirmed about the answer given, alvaro can explain it well. based on the analysis of the three students, it can be concluded that: dhea showed unclearness in both written and verbal language, anissa showed written language that was unclear but could clarify verbal language, and there was no desire to clarify the language of his writing, alvaro in resolving problem four showed lack of clarity in the written language but can clarify the verbal language and have an effort (desire) to clarify the written language, furthermore for the problem of the two written and verbal languages alvaro is adequate and clear. 3.3. control things that are considered in the aspect of self-control (control) is how students think metacognitive looks or not in written or verbal communication. the characteristics of student behaviour oriented toward apathy are metacognitive thinking does not appear both in writing and verbally. fikri's behaviour indicates this in solving problem two. fikri's metacognitive thinking does not appear in writing, because fikri's answers have neat (smooth) characteristics, without scribbles, and are wrong. when answering fikri also often answers fast but wrong. the orientation of dhea behavior shows different things. dhea has a metacognitive thinking orientation behaviour that does not appear in writing, but metacognitive appears verbally but cannot be directed to perform metacognitive processes on written answers. furthermore, annisa also shows metacognitive thinking in writing and shown in verbal language, and metacognitive thinking can be directed to the writing or answers made. alvaro shows the orientation of sophisticated behavior. alvaro shows the metacognitive processes appear in written and verbal. alvaro has also shown the process of obtaining the beam elements correctly, so it can be said that alvaro's written answers are neat, clear and true meaning the metacognitive process appears in writing. from the results of the interview, alvaro was very careful and rethought what he had done. alvaro said he volume 10, no 2, september 2021, pp. 235-258 253 had calculated many times. alvaro is always rethinking about the way he had thought before. so, alvaro can already be categorized in the advanced category in the aspect of self-control (control). 3.4. confidence 3.4.1. confidence in how to implement strategies in mathematical problem solving in this indicator, muir et al. (2008) categorizes students behaving apathetic by doing the same strategy because they lack confidence in their ability to solve the problem strategies used, students, regularly behaving routine implementing strategies and students who sophisticated can produce their strategies because they have high confidence in the use of problem-solving strategies. furthermore, students with sophisticated categories produce their strategies. alvaro did not use a systematic formula in solving problems as dhea and annisa did, but alvaro found his strategy in solving the given problem. 3.4.2. confidence in the variety of strategies used in solving mathematical problems based on the findings of this indicator, there are four behavioural category orientations. students in the apathetic category lean on one strategy. apathetic students tend always to use strategies to manipulate numbers in the problem. fikri often shows this behaviour although, in some indicators, they are classified as routine categories. when interviewed, they often answered that they did not understand the problem if directed, they tended to answer with answers that did not know. in students who are categorized routinely, tend to rely on more than two strategies, when one strategy does not work, he does not turn to another strategy. this behaviour is often shown, lutfi answered knowing more than one strategy, but when a solution was not found, he did not turn to another strategy. lutfi, when directed by the interviewer to use strategies that lead to solutions, tend to reject and can not do it. furthermore, dhea leaned and knew more than two strategies; when one strategy did not work, he did not switch to another strategy but could be directed to switch strategies. alvaro has combined several strategies, as in the case of two alvaro has already described, then imagined and used a guess and check strategy. 3.5. affective in the affective aspects of the indicator seen is how students' confidence in solving problems. confidence is observed when students answer questions from the interviewer. during interviews, students who behave apathetically often do not think in advance what they say. apathetic students' confidence is in line with the quick answers they express. dyvi and fikri are students who exhibit this behaviour. in other cases, dhea and annisa often show a lack of confidence in answering interviewer questions. although both have the behaviour of a lack of confidence in solving problems, there are differences between dhea and annisa. dhea often expressed a lack of confidence in solving problems but did not want to explore further ways of solving them. on the other hand, annisa showed the same lack of confidence as dhea but wanted to explore further the solution. in another case, alvaro showed confidence in his ability to solve problems. almost all problems alvaro answers confidently and correctly. alvaro can be categorized as sophisticated behaviour in the aspect of self-confidence, although alvaro cannot solve all the problems given correctly. harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 254 based on the description of the findings, muir et al. (2008) had categorized student behaviour into three categories: naïf, routine, and sophisticated that was not yet fully drawing the behaviour of the eighteen observed students. category of routine students, there were two different orientations: there were students who could be directed, and there were students who could not be directed to the orientation of students behave sophisticated. the conclusion of the analysis of student orientation findings on routine categories could be classified into two categories: routine and semi-advanced. the orientation of these behaviours will be described in table 6. table 6. description of student behavior at the junior high school level no factor indicator category of behavior apathetic routine semi sophisticated sophisticated 1 knowledge ownership the application of polya's heuristic steps in mathematic al problem solving made a mistake on the four solving steps no effort to verify the solution (make a mistake on some troubleshoot ing steps) there is an attempt at verifying the solution but still making mistakes on some troubleshooting steps a high score on each troubleshooting step the use of prior knowledge for mathematic al problem solving cannot use previously resolved issues can identify a similar problem, but not on the mathematic al structure (can identify similar problems only on external features only) can identify a similar problem, but not on the mathematical structure, but can be directed to look at problems in the mathematical structure identify similar problems according to the mathematical structure his many ways are used in mathematic al problem solving often use the same way to solve all problems because of the limitations of knowledge possessed focus on one way with the knowledge you have to solve a particular problem, but have no desire to generate the right way to solve the problem focus on one way with the knowledge you have to solve a particular problem, but have a desire to generate the right way to solve the problem identify other ways of troubleshooting volume 10, no 2, september 2021, pp. 235-258 255 no factor indicator category of behavior apathetic routine semi sophisticated sophisticated written and verbal communicat ion in mathematic al problem solving written and verbal communicatio n is inadequate written communicat ion is not clear, but can clarify through verbal communicat ion written communication is not clear, but can clarify through verbal communication and can be directed to clarify his written communication written and verbal communication is sufficient (obviously) 2 self control (control) thinking metacogniti on in mathematic al problem solving communicat ion thinking metacognition is invisible, both in written and verbal metacogniti on does not appear in writing, but metacogniti on appears verbally metacognition is not without writing, but metacognition appears verbally and can be directed to perform metacognition processes in written answers thinking metacognition appears in written and verbal responses 3 confidence confidence in how to implement the strategy in problemsolving doing exactly the same strategy implementi ng the strategy in a systematic way implementing the strategy in a systematic, yet expandable way to generate its strategy strategy mathematics generate your own confidence in the variety of strategies used in mathematic al problem solving strategy rely on one strategy relying on more than two strategies, when one strategy does not work does not switch to another relying on more than two strategies, when one strategy does not work, does not switch to another strategy, and can be directed to move strategy desiring to combine several strategies 4 affective often declare lack of confidence in solving problems, but desire to explore further how to solve them with confidence confidence in solving mathematical problems confidence is in line with the quick answer often declare lack of confidence in solving problems appears confident in problem-solving skills harisman, noto, & hidayat, investigation of students’ behavior in mathematical problem … 256 the results of this study try to discover the behaviour presented by eighteen students of junior high school. after analyzing the work and constructing the video of each student's interview results, it was obtained four students' behaviour orientations in problem-solving. muir et al. (2008) had studied the consistency of students' behaviour in previous problemsolving, which classified three student orientations in problem-solving, including naive, routine, and sophisticated. this paper completes the behavioural orientation found by muir et al. (2008) while solving the problems, in which these three orientations were not considered to cover the whole behaviour shown by the eighteen students of junior high school. two types of routine problem-solvers were found in this study in which there are both routine troubleshooters who have a high curiosity about solutions of the problem solving and routine troubleshooters who were not interested in exploring further issues. category of problem solvers in this paper is grouped into four groups, namely: ignorant, routine, semi-sophisticated and sophisticated. the results of this behavioural analysis can be used as a reference to teachers to make decisions in implementing what instruction decisions are appropriate for the students to behave sophistically in solving problems. this research also aims to investigate the behaviour to make some assumptions to what extent students can solve the problems. according to malloy and jones (1998), the assessment of the behaviour of children could see the irregularities and consider the investigation for the assumptions we have towards students. he conducted and investigated the problem-solving skills of 24 african students, and he thought that his research results did not make the reader agree to lower expectations for african students. according to molloy, the results of this study could be used for teachers to take the next step to improve classroom learning practices. 4. conclusion category of problem solvers in this paper is grouped into four groups, namely: ignorant, routine, semi-sophisticated and sophisticated. this article can also be used for a reference to reflect, investigate, and diagnose to what extent the children behave and contribute to problem-solving questions so that teachers should determine appropriate leading strategies and methods. in this study, it also shows that aspects affecting the students’ success in problem-solving are not only aspects of knowledge ownership but also aspects of metacognitive, control, self-confidence. his research showed that students in certain contexts “learners with good metacognitive ability” exhibited better academic achievement. revealed smith (2013). this article describes how children who have metacognitive abilities and care about errors in problem-solving adding their reflexes to the solutions they produce. in the other hand, this also aims to communicate and collect the most important arguments espousing problem-solving in applying for math, especially on space geometry lesson to junior high school’ students. for further research, it is expected to look at the consistency of problem-solving behaviour by using plentiful junior high schools as a sample and can be performed in diverse topics and materials. references cai, j. 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(2010). effect of problem solving strategies on problem solving achievement in primary school mathematics. procedia-social and behavioral sciences, 9, 468-472. https://doi.org/10.1016/j.sbspro.2010.12.182 https://doi.org/10.5951/jresematheduc.29.2.0143 https://doi.org/10.22460/infinity.v8i1.p1-10 https://doi.org/10.5951/jresematheduc.19.2.0134 https://doi.org/10.1016/j.jmathb.2008.04.003 https://doi.org/10.1080/0020739x.2020.1738579 https://doi.org/10.22460/infinity.v10i1.p121-132 https://doi.org/10.5951/jresematheduc.13.1.0031 https://doi.org/10.1016/j.jmathb.2005.09.004 https://doi.org/10.1016/j.sbspro.2010.12.182 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p177-182 177 the relation between self-efficacy toward math with the math communication competence sylvia rahmi 1 , rifka nadia 2 , bibih hasibah 3 , wahyu hidayat 4 1 smp negeri 25 pekanbaru, jl. kartama, maharatu, marpoyan damai, pekanbaru, riau, indonesia 2 smp negeri 10 banda aceh, jl. poteumereuhom, kuta alam, banda aceh, indonesia 3 smp negeri 1 soreang, jl. ciloa no. 3, pamekaran, soreang, bandung, indonesia 4 stkip siliwangi, jl. terusan jenderal sudirman, cimahi, indonesia 1 sylviarahmi.sr@gmail.com, 2 rifkanadia28@yahoo.com, 3 bibih_hasibah@yahoo.com, 4 wahyu@stkipsiliwangi.ac.id received: june 08, 2017 ; accepted: september 11, 2017 abstract the aim of this research is to analyze the relationship between self-efficacy toward mathematics with mathematical communication competence. the design of this research is survey and correlation technique. the research instruments used in this research are math communication competence test and attitude scale. the instruments used are 5 questions about math communication competenca teat and 28 statements about self-efficacy. the research population is all student of smp negeri 1 soreang. with samples of this research were 70 students of 7th grade student chosen by cluster random sampling. the data analyzed quantitatively done through math communication data and self-efficacy attitude scale. spss 20 used in this research. the result shows that student self-efficacy influences students math communication competence. keywords: communication skills mathematics, self-efficacy. abstrak penelitian ini bertujuan untuk menganalisis hubungan antara self-efficacy terhadap matematika dengan kemampuan komunikasi matematik. desain penelitian ini adalah survey dengan teknik korelasi. instrumen penelitian yang digunakan dalam penelitian ini berupa tes kemampuan komunikasi matematik dan skala sikap self-efficacy siswa. intrumen yang digunakan yaitu 5 soal kemampan komunikasi matematik dan 28 pernyataan mengenai self-efficacy. populasi dalam penelitian ini adalah seluruh siswa smp negeri 1 soreang dengan sampel penelitian ini adalah 70 siswa kelas vii sebanyak dua kelas yang dipilih secara cluster random sampling. analisis data dilakukan secara kuantitatif yang dilakukan terhadap data kemampuan komunikasi matematik dan skala sikap self-efficacy. dalam perhitungan statistik menggunakan spss 20, hasil penelitian menunjukkan bahwa self-efficacy siswa mempengaruhi kemampuan komunikasi matematik siswa. kata kunci: kemampuan komunikasi matematik, self-efficacy. how to cite: rahmi, s., nadia, r., hasibah, b., & hidayat, w. (2017). the relation between self-efficacy toward math with the math communication competence. infinity, 6 (2), 177182. doi:10.22460/infinity.v6i2.p177-182 mailto:bibih_hasibah@yahoo.com rahmi, nadia, hasibah, & hidayat, the relation between self-efficacy toward math … 178 introduction education is one important aspects that will determine the quality of life of a person and a nation. in formal education, one of the subjects in school that can be used to build the students' way of thinking is mathematics. therefore, math lessons at schools not only emphasize the giving formulas but also teach students to be able to communicate ideas related to everyday life. one of the mathematical skills that must be possessed by the students is the ability of mathematical communications. mathematical communication can be interpreted as a student's ability to convey something he knows through dialogue events or interrelationships that occurr in the classroom environment, where there is a transfer of the message. displaced messages contain mathematical materials that learn, such a concepts, formulas, or a problemsolving strategies. parties involved in communication events in the classroom are teacher and student. the way the message can be transmitted can be smoken or written. in order for student’s mathematical communication skills to develop well, then in the process of learning mathematics, teachers need to provide opportunities for students to be able to improve their ability to communicating mathematical ideas. pimm (lindawati, 2013) states that children are given the opportunity to work in a groups in collecting and presenting data, show good progress as they listen to one anothers’s ideas, discuss it together and then draw up the conclusions that the group views. apparently they learn most from communicating and constructing their own knowledge. according anggraeni (2013) summarizes the opinions of some experts and nctm and identifies some mathematical communication skills such as: a) stating a situation, into drawings, diagrams, language, symbols, expressions or mathematical models; b) state image, diagrams, language, symbols, expressions or mathematical models in their own language; c) listening, discussing, writing mathematics; d) read a mathematical presentation with understanding; e) revisit a mathematical description in its own language; and f) compile questions about mathematics.through mathematical communication students exchange and explain their ideas or understanding to their friend. the communication process helps students construct the meaning of a series of mathematical processes and make generalizations. in an effort to explore and develop students’ mathematical communication skills, teachers should expose students to a variety of contextual issues and invite them to communicate their respective ideas (hidayat, 2017). in the 2013 curriculum also said that in mathematics learning hard skills and soft skills including mathematics education values in the culture and character that should be developed simultaneously and balanced through learning scientific approach. one of the soft skills that mathematics is self-efficacy. self-efficacy leads to a person's beliefs about its ability to organize and carry out a series of actions to achieve results (bandura, 1997). it can be concluded self-efficacy is the belief that students need to have in order to succeed in the learning process. pintrich and de groot (khaerunisak, kartono, hidayah, and fahmi, 2017) found that students who believed they could perform academic tasks using cognitive and metacognitive strategies were more and still doing better than unbelieving students. self-efficacy makes a difference in the way people act, as a follow-up of feelings and thoughts. people who believe they can do something that has the potential to transform environmental events are more likely to act and more likely to succeed than those with low self-efficacy. behavior is influenced by the extent to which a person believer can perform the actions required by the particular situation. volume 6, no. 2, september 2017 pp 177-182 179 method this research design is a survey by using correlation technique, where writer take two class as sample of research. this research conducted in smp negeri 1 soreang. the development of self-efficacy variables students about math starting with the 28-point declaration preparation is complete with 4 choices. the scale used is a likert scale. with a choice of answers ss (strongly agree), s (agree), ts (disagree), and sts (strongly disagree). for determination calculations using score ss = 4, c = 3, ts = 2, and sts = 1 for statements favorable (positive), whereas a score ss = 1, s = 2, ts = 3, and sts = 4 for statements unfavorable (negative). the instrument was then consulted with the supervisor in order to have the validity of the content. meanwhile, in order to have the empirical validity of the instrument is then tested to determine the validity, reliability, distinguishing power and distress index. math test used was a test of mathematical communication skills. the test of mathematical communication ability is arranged in the form of description. the reason for preparation of the test in the form of description because it is tailored to the purpose of this study which prioritizesthe process of the results. tests in the form of description is not much to give a chance to speculate, it can even encourage students to dare to express their opinions in their own and language. the steps taken in the preparation of this instrument is (1) make a grid of test based indicators matematic communication skills, (2) make the scoring guidelines, (3) preparing test problems; (4) assess conformity between material, indicators and test questions. results and discussion results to see how strong the relationship between self-efficacy and mathematical communication skills, then the correlation test pearson with and the hypothesis is h0: ρ = 0 h1: ρ ≠ 0 with the criteria: if sig> 0.05 then h0 accepted table1. correlation guildfrord’s criteria range criteria 0.01 to 0.20 very weak 0.20 to 0.40 weak 0.40 to 0.70 strong enough 0.70 to 0.90 strong 0.90 to 1.00 very strong rahmi, nadia, hasibah, & hidayat, the relation between self-efficacy toward math … 180 table 2. results of self-efficacy correlation and mathematical communication skills self efficacy posttest pearson correlation self-efficacy 1,000 0,424 postes 0,424 1,000 sig. (one-tailed) self-efficacy 1,000 0,000 postes 0,000 1,000 n self efficacy 70 70 postes 70 70 based on table 2, the result of correlation between self-efficacy dam mathematical communication ability of students is 0.424 and the significance value (sig) of 0.000. correlation values (r) obtained was 0.424, which means is quite powerful. due to the significant value of 0,000 is smaller than = 0.05, h0 is rejected, meaning that there is a relationship between self-efficacy toward mathematics with mathematical communication skills. to determine the influence ofanatar self-efficacy with mathematical communication skills then tested using a regression coefficient of linear regression analysis. this analysis was conducted to see the direct influence of self-efficacy of students' mathematical communication abilities of students. the hypothesis tested were: h0 : self-efficacy students about math does not affect students' mathematical communication skills h1 : self-efficacy affects students about math mathematical communication skills of students with criteria: if sig> 0.05 then h0 accepted results of the analysis are shown in table 3 table 3. regression analysis self-efficacy with communication capabilities of mathematical model unstandardized coefficients standardized coefficients t sig. b std. error beta 1 (constant) 72,888 2,935 24,832 0,000 postes 0,190 0,049 0,424 3,865 0,000 based on table 3, it can be seen the regression equation y = 72.888 + 0,190x which means, the greater the value of self-efficacy,the greater the students' mathematical communication skills of students, and vice versa. because the significance value of 0.000 is smaller than = 0.05, it can be concluded under h0 is rejected it means significantly self-efficacy affects students toward math mathematical communication skills of students. discussion based on the analysis of data to test the hypothesis, the conclusions from the findings made by that self-efficacy of students towards mathematics in general affect students' mathematical communication skills. self-efficacy affects the ability of mathematical communications volume 6, no. 2, september 2017 pp 177-182 181 because a higher level of confidence in one's self to the higher mathematics mathematical communication skills. this is shown by the significant value of 0.024, which means smaller than = 0.05, which means self-efficacy affect the ability of mathematical communications. the correlation coefficient of 0.380 and a positive value indicating that a positive relationship and have the power relationships within the category of being between two variables. this supports the hypothesis that there is evidence of the relationship between self-efficacy toward mathematics with mathematical communication skills. self-efficacy is one of the factors that influence the adjustment to the ability of the student (sumarmo, hidayat, zulkarnaen, hamidah, & sariningsih, 2012; irfan & suprapti, 2014; haji & abdullah, 2016; hendriana, 2017). in addition, the results are also in line with those proposed by hendriana, rohaeti & hidayat (2016) that mathematical communication ability is also influenced by various factors, including self-efficacy factor. conclusion based on the analysis the conclusion is: (1) there is a relationship between self-efficacy toward mathematics with mathematical communication skills. (2) self-efficacy influences students toward math mathematical communication skills. references anggraeni, d. (2013). meningkatkan kemampuan pemahaman dan komunikasi matematik siswa smk melalui pendekatan kontekstual dan strategi formulate-share-listen-create (fslc). infinity journal, 2(1), 1-12. bandura (1997). self-efficacy: the excercise of control. new york: w.h freeman and company. hendriana, h., rohaeti, e. e., & hidayat, w. (2016). metaphorical thinking learning and junior high school teachers’ mathematical questioning ability. journal on mathematics education, 8(1), 55-64. hendriana, h. (2017). senior high school teachers’ mathematical questioning ability and metaphorical thinking learning. infinity journal, 6(1), 51-58. hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. irfan, m., & suprapti, v. (2014). hubungan self-efficacy dengan penyesuaian diri terhadap perguruan tinggi pada mahasiswa baru fakultas psikologi universitas airlangga. jurnal psikologi pendidikan dan perkembangan, 3(3), 172-178. khaerunisak, k., kartono, k., hidayah, i., & fahmi, a. y. (2017). the analysis of diagnostic assesment result in pisa mathematical literacy based on students selfefficacy in rme learning. infinity journal, 6(1), 77-94. lindawati, s. (2013). pembelajaran matematika dengan pendekatan inkuiri terbimbing untuk meningkatkan kemampuan pemahaman dan komunikasi matematis siswa sekolah menengah pertama. jurnal pendidikan, 2(2), 16-29. haji, s., & abdullah, m. i. (2016). peningkatan kemampuan komunikasi matematik melalui pembelajaran matematika realistik. infinity journal, 5(1), 42-49. doi:http://dx.doi.org/10.22460/infinity.v5i1.190 http://dx.doi.org/10.22460/infinity.v5i1.190 rahmi, nadia, hasibah, & hidayat, the relation between self-efficacy toward math … 182 sumarmo, u., hidayat, w., zukarnaen, r., hamidah, m., & sariningsih, r. (2012). kemampuan dan disposisi berpikir logis, kritis, dan kreatif matematik (eksperimen terhadap siswa sma menggunakan pembelajaran berbasis masalah dan strategi thinktalk-write). jurnal pengajaran mipa, 17(1), 17-33. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p273-284 273 infinity mathematical self-esteem ability of junior high school students in project-based learning dahlia fisher1*, jarnawi afgani dahlan2, beni yusepa ginanjar putra1 1universitas pasundan, indonesia 2universitas pendidikan indonesia, indonesia article info abstract article history: received dec 28, 2021 revised sep 21, 2022 accepted sep 22, 2022 the main objective of this research is to study the achievement of their mathematical self-esteem, as a result of project-based learning (pbl) and conventional learning (cl). this research is conducted mathematical selfesteem to be possessed by students, but in reality shows their self-esteem are still low. this research used quasi experimental methods. the population of this study is students of state junior high schools in bandung city, west java province. the sample comprises the students of grade viii. one class as an experimental group that received pbl learning and another class as a control group that received cl learning. the instruments used in this study are prior mathematical knowledge (pmk) worksheet and mathematical self-esteem scale. the results indicate that: (1) the achievement of mathematical selfesteem of the students who received pbl learning is better than the students who received cl learning; (2) there is no interaction effect between learning model and pmk on the achievement of students' mathematical self-esteem. keywords: conventional learning, mathematical self-esteem, prior mathematical knowledge, project-based learning this is an open access article under the cc by-sa license. corresponding author: dahlia fisher, department of mathematics education, universitas pasundan jl. tamansari no.6-8, bandung city, west java 40116, indonesia. email: dahliafisherpmat@unpas.ac.id how to cite: fisher, d., dahlan, j. a., & putra, b. y g. (2022). mathematical self-esteem ability of junior high school students in project-based learning. infinity, 11(2), 273-284. 1. introduction the demands of life in the 21st century force human resources not only to increase their knowledge abilities. knowledge ability must be balanced with attitude and skills in managing the ongoing comprehensive self-potential development. the driving factor that is no less important is the 2013 curriculum which demands multidimensional competence on students. the competencies that must be achieved are not only limited to the cognitive domain, but also to attitude and psychomotor competencies. students' mathematical thinking abilities are influenced by their internal elements (pamungkas et al., 2017). young and hoffmann (2004) suggested that the success of students in school is strongly influenced by the self-admiration or self-esteem of these https://doi.org/10.22460/infinity.v11i2.p273-284 https://creativecommons.org/licenses/by-sa/4.0/ fisher, dahlan, & putra, mathematical self-esteem ability of junior high school students … 274 students for their ability to solve problems. this is in line with the opinion of fisher and kusumah (2018), the findings are that there is a positive correlation between student achievement and mathematical self-esteem. based on several arguments about the importance of self-esteem, it is increasingly clear that mathematical self-esteem has very important implications for education (casino-garcía et al., 2021). students need to have a sense of worth, deserve and useful when involved in learning. such feelings are termed selfesteem (fauzan & herman, 2011). rosenberg (2015) defines self-esteem as a person's overall positive or negative assessment of himself or in other words self-esteem is the overall attitude that a person holds about himself, ranging from negative to positive. the essence of the notion of self-esteem raises the growing feeling that “i am capable and i am worthy". the current fact regarding students' mathematical self-esteem is that less than 40% of students have self-admiration in solving mathematical problems (fitriah & aripin, 2019). in the previous year, fisher and kusumah (2018) conducted a study involving 140 junior high school students. the results of the study revealed that the average score of mathematical self-esteem obtained by male and female students was 63.84 and 63.27 respectively, while the ideal maximum score was 100. paying attention to the facts that occur in the field, it is very necessary to implement learning models that can improve students' mathematical abilities, attitudes and skills. research results from nielsen et al. (2010), struyven et al. (2010), and la nani et al. (2020), revealed that a learning model that can improve students' learning abilities and achievement is a project-based learning model (pbl). through pbl, students are prepared cognitively and emotionally to solve complex challenges collaboratively. constructivism perspective views learning as an experiential process. learning becomes more meaningful and produces quality values. through pbl, the learning atmosphere becomes active, collaborative, increasing students' self-confidence so that students feel worthy, precious, capable and useful for others. according to dias and brantley-dias (2017), the goal of pbl is to help students increase their knowledge and understanding of the subject as well as success skills (i.e., 21st century skills). they highlighted that students were expected to put their subject-matter knowledge into practice while working on the project. thus, through understanding and applying a variety of abilities, including critical thinking, problem solving, collaboration, and self-management, students encourage their deep learning of the subject. pbl is a learning model that is based on project development, imagination, planning, design, and is a student-centered teaching method that allows students to build interdisciplinary relationships as they work by bringing real-life environments to the classroom (fisher et al., 2020, 2021; kalayci, 2008). klein et al. (2009) explained that pbl is a learning model that empowers students to gain new knowledge and understanding based on their experiences through various presentations. pbl contains project-based complex tasks based on very challenging questions and problems, and requires students to design, solve problems, make decisions, carry out investigative activities, and provide opportunities for students to work independently (nasution et al., 2021). the characteristics pbl show that students can choose topics and/or presentation projects/products, produce final products such as presentations, recommendations for solving problems related to the real world, involve various disciplines, vary in duration of time, feature teachers in facilitator role (hernández-ramos & de la paz, 2009; kamdi, 2015). this study tries to dig deeper into the mathematical self-esteem of students who learn to use pbl. some of the literature is well studied to enrich the novelty in this research, so that it can be seen more comprehensively how pbl can provide a new paradigm in learning mathematics. based on these considerations and problems, the researcher took the title “mathematical self-esteem of junior high school students in project-based learning”. volume 11, no 2, september 2022, pp. 273-284 275 infinity 2. method 2.1. research design the method used in this study was a quasi-experimental method. there were two groups of students. as the experimental group was students who acquire teaching mathematics under project-based learning (pbl) model, while the control group were students who acquaire teaching mathematics under conventional learning (cl). this study implemented postest only for both groups of students. the research design involved two factors, namely learning models and student broup based on factors prior mathematical knowledge (pmk). the first factor consisted of pbl model and direct instructions. the second factor consisted of a group of students based on pmk (high, middle, and low). this research design could be described as the relationship between the factors as presented in table 1. table 1. relationship of mathematical self-esteem ability pmk teaching model pbl cl high (hi) hipbl hicl middle (mi) mipbl micl low (lo) lopbl locl 2.2. sample the population of this study is students of state junior high schools in bandung city, west java province. the sample comprises the students of grade viii at junior high school. whereas the sample was 130 students (66 students as an experiment group and 66 as a control group). 2.3. research procedure research activities initiated by determining the study sample. after the sample was set, each student was given a pmk test. the test is intended to classify students based on pmk (high, middle, and low). after the experimental and the control groups were formed, the students were given the treatment using pbl and cl. after the treatment, posttest on mathematical self-esteem scale. this scale consists of 25 statements items arranged with four answer choices (responses), and the scale consist of positive statements (favorabel) and negative statements (unfavorable). for data analysis, researchers used ibm spss statistics 25. 2.4. data analysis there were two main hypotheses to be tested. the first one was related to test two independent samples with the interval ratio of measurement. the data was analyzed by t test, t’ test. in the second hypothesis, data could be tested using two ways anova if the conditions were available. if the conditions were not available, the interaction effect would be add the count using aligned rank transformation (art). fisher, dahlan, & putra, mathematical self-esteem ability of junior high school students … 276 2.5. instrument 2.5.1. test of prior mathematical knowledge (pmk) all of the instruments were developed by the researchers in this study and through doing the try outto fulfill the requirements of qualified validity, reliability. test of pmk became required to measure the students’ mathematical prior knowledge about the substances of mathematics which have been studied before, after they have been at grade vii. the materials assist in mastering the core of discussion which became discussed throughout this research. researchers choose test questions from the national examination (un) during 2010 to 2017 junior high school mathematics class vii material. the selection of un questions is assumed to have met national standards as a good measuring tool. the question is in the form of multiple choice and each item has four answer choices. the purpose of the initial mathematical ability test is to place students on the basis of their initial mathematical ability. the indicators are as follows: (1) students are able to perform arithmetic operations with rational numbers; (2) students can use the principles, division, addition and subtraction in integer operations; (3) students are able to perform arithmetic operations on problems related to comparisons; (4) students are able to determine the solution of a one-variable linear equation; (5) students are able to present data; (6) using the concept of circumference of a flat shape in daily life. the pmk categories are presented in table 2. table 2. category of pmk pmk category 𝑃𝑀𝐾 ≥ �̅� + 𝑠 high �̅� − 𝑠 ≤ 𝑃𝑀𝐾 < �̅� + 𝑠 middle 𝑃𝑀𝐾 < �̅� − 𝑠 low 3. result and discussion 3.1. result descriptive statistical analysis of the result of mathematical self-esteem ability was presented in table 3. table 3. description statistics of students’ mathematical self-esteem ability prior mathematical knowledge (pmk) stat. experiment control mixed posttest n posttest n posttest n high �̅� 86.98 7 80.73 8 83.86 15 𝑠 3.62 3.90 3.76 middle �̅� 73.47 48 67.99 49 70.73 97 𝑠 7.35 5.30 6.32 low �̅� 48.54 9 48.26 9 48.40 18 𝑠 8.17 7.51 7.84 mixed �̅� 69.66 64 65.66 66 67.66 130 𝑠 6.38 5.57 5.97 table 3 show that the achievement of mathematical self-esteem ability that students acquire teaching under pbl was relatively higher than students who acquire teaching under cl, the well-viewed as a whole and viewed based on the level of pmk. the percentage of volume 11, no 2, september 2022, pp. 273-284 277 infinity achievement in students' mathematical self-esteem based on learning, pmk (high, medium, low), and overall can be seen more clearly in the bar chart in figure 1. figure 1. the percentage of achievement mathematical self esteem ability inferential statistical analysis of the results of students’ mathematical self-esteem ability to experimental and control groups were presented in table 4. table 4. difference of students’ mathematical self-esteem ability variable group difference test se mixed experiment t-test 0.028 different control se high experiment t-test 0.030 different control se middle experiment t’-test 0.000 different control se low experiment t-test 0.930 not differrent control table 4 show that there was difference in achievement mathematical self-esteem ability significantly between students who attained teaching under pbl (experimental group) and students who attaind teaching under conventional learning (cl), the well-viewed as whole (mixed) and viewed based on the prior mathematical knowledge (high and middle). if these results were associated with the results in table 3 and figure 1. it can be concluded that the achievement of students’ mathematical self esteem ability who attained teaching under pbl were higher than students who attained teaching under cl. the interaction effect between model learning and pmk toward achievement of students’ mathematical self-esteem ability would be tested by using two ways anova. before using two ways anova, it was necessary to be viewed whether the data of each facor was distrbuted normally and homogeneity test. the result of distributution normality was presented in table 5. fisher, dahlan, & putra, mathematical self-esteem ability of junior high school students … 278 table 5. normality distribution on mathematical self-esteem ability grup ability distribution normality test implications sig. conclusions experiment 0.200 normal two ways anova is used control 0.200 normal experiment high 0.137 normal control 0.200 normal experiment middle 0.200 normal control 0.218 normal experiment low 0.254 normal control 0.096 normal furthermore, the homogeneity test of the variance of the data on the achievement of students' mathematical self-esteem based on learning and pmk was carried out using the levene statistical test. the result of the homogeneity test was presented in table 6. table 6. the homogeneity test on mathematical self-esteem ability statistik levene (f) df1 df2 sig. 𝐇𝟎 2.845 5 126 0.018 not homogeneous from the table 6, it could be stated that there was not homogeneity the variance of mathematical self-esteem ability significantly between model learning and pmk. because the data is not homogeneous, the test uses the adjusted rank transformation test (art test) (aubuchon & hettmansperger, 1984; conover & iman, 1981; higgins et al., 1990; leys & schumann, 2010; sawilowsky, 1990). the first step is to calculate the average score of the observations contained in each row and column according to the variables studied. furthermore, each observation score in each row and column is reduced by the average. art test is presented in table 7. table 7. adjusted rank transformation test b1 b2 score rank score rank a1 �̅�𝐴1.𝐵1.1 − �̅�𝐴1 − �̅�𝐵1 �̅�𝐴1.𝐵1.2 − �̅�𝐴1 − �̅�𝐵1 �̅�𝐴1.𝐵1.3 − �̅�𝐴1 − �̅�𝐵1 ⋮ �̅�𝐴1.𝐵1.𝑖 − �̅�𝐴1 − �̅�𝐵1 𝑌𝐴1.𝐵1.1 𝑌𝐴1.𝐵1.2 𝑌𝐴1.𝐵1.3 ⋮ 𝑌𝐴1.𝐵1.𝑖 �̅�𝐴1.𝐵2.1 − �̅�𝐴1 − �̅�𝐵2 �̅�𝐴1.𝐵2.2 − �̅�𝐴1 − �̅�𝐵2 �̅�𝐴1.𝐵2.3 − �̅�𝐴1 − �̅�𝐵2 ⋮ �̅�𝐴1.𝐵2.𝑖 − �̅�𝐴1 − �̅�𝐵2 𝑌𝐴1.𝐵2.1 𝑌𝐴1.𝐵2.2 𝑌𝐴1.𝐵2.3 ⋮ 𝑌𝐴1.𝐵2.𝑖 a2 �̅�𝐴2.𝐵1.1 − �̅�𝐴2 − �̅�𝐵1 �̅�𝐴2.𝐵1.2 − �̅�𝐴2 − �̅�𝐵1 �̅�𝐴2.𝐵1.3 − �̅�𝐴2 − �̅�𝐵1 ⋮ �̅�𝐴2.𝐵1.𝑖 − �̅�𝐴2 − �̅�𝐵1 𝑌𝐴2.𝐵1.1 𝑌𝐴2.𝐵1.2 𝑌𝐴2.𝐵1.3 ⋮ 𝑌𝐴2.𝐵1.𝑖 �̅�𝐴2.𝐵2.1 − �̅�𝐴2 − �̅�𝐵2 �̅�𝐴2.𝐵2.2 − �̅�𝐴2 − �̅�𝐵2 �̅�𝐴2.𝐵2.3 − �̅�𝐴2 − �̅�𝐵2 ⋮ �̅�𝐴2.𝐵2.𝑖 − �̅�𝐴2 − �̅�𝐵2 𝑌𝐴2.𝐵2.1 𝑌𝐴2.𝐵2.2 𝑌𝐴2.𝐵2.3 ⋮ 𝑌𝐴2.𝐵2.𝑖 volume 11, no 2, september 2022, pp. 273-284 279 infinity for data art, researchers used ibm spss statistics 25. the results are then arranged sequentially (rank) and only then can the two-ways anova method as usual. two way anova test was presented in table 8 and figure 2. table 8. two ways anova on mathematical self-esteem ability source type iii sum of squares df mean square f sig. partial eta squared corrected model 128239.563a 5 25647.913 50.963 .000 .669 intercept 327171.983 1 327171.983 650.098 .000 .838 teaching model 3793.111 1 3793.111 7.537 .007 .056 pmk 108406.702 2 54203.351 107.703 .000 .631 teaching model* pmk 1513.955 2 756.977 1.504 .226 .023 error 63411.437 126 503.265 total 775388.000 132 corrected total 191651.000 131 a. r squared = .669 (adjusted r squared = .656) figure 2. the interaction effect teaching model and prior mathematical knowledge from the table 8 and figure 2, it could be stated that there was no interaction effect between teaching models and prior mathematical knowledge toward achievement of student’s mathematical self-esteem ability. 3.2. discussion overall, the average achievement of mathematical self-esteem of students who received pbl was higher than students who received cl. the average mathematical selfesteem achievement of students who received pbl was 69.66 from the ideal maximum score fisher, dahlan, & putra, mathematical self-esteem ability of junior high school students … 280 which was classified as moderate, and students who received cl of 65.66 were also classified as moderate. unlike the research conducted by fadillah (2012), which used learning with an open-ended approach, this study found that students' self-esteem in mathematics who received learning with an open-ended approach was not better than those who received ordinary learning, in terms of overall students. in pbl in a group, someone with high self-esteem tends to be more courageous and critical of the group. although it does not affect directly, self-esteem can affect a person's leadership traits (baumeister, 2013). in line with lawrence (2006), students with high self-esteem will maintain a natural curiosity in learning and have enthusiasm and enthusiasm when facing new challenges. these two opinions further strengthen the findings in this study, that the average achievement of mathematical self-esteem of students who receive pbl is higher than students who receive cl. although the average self-esteem achievement of students who received pbl and students who received cl were both classified as moderate, the results of statistical tests confirmed that the overall mathematical self-esteem achievement of students who received pbl was significantly better than students who received cl. this means that overall, pbl learning is higher than students who receive cl. these results are possible, because as suggested by fatah et al. (2016), that one of the efforts to increase students' self-esteem is by giving responsibility to students. in pbl, students learn to be responsible for solving a given situation or problem, starting from making a project implementation schedule to reporting the results of project work to express perceptions of the project problems faced according to the results of their respective thoughts. then individually, students are responsible for being able to explain their work to their friends. the teacher's role as a facilitator in pbl emphasizes more on the efforts made by students, not on the results. no matter how simple the results of student thinking, teachers still appreciate and give appreciation. when students make mistakes, the teacher emphasizes to students that mistakes are part of the learning process. it is not a failure. thus, students become more confident to be actively involved in learning, feel that their existence is valued, feel that they are needed by others, and in the end students can respect themselves. based on the pmk category, the average achievement of mathematical self-esteem of high pmk students who received pbl was 86.98 from the ideal maximum score of 100 (classified as high). meanwhile, the average achievement of mathematical self-esteem of high pmk students who received cl had an average achievement of 80.73 from the ideal maximum score (high). the statistical test concluded that the achievement and increase in self-esteem of high pmk students who received pbl learning were better than those who received cl. based on the pmk category, the average achievement of mathematical self-esteem of moderate pmk students who received pbl was 73.47 from the ideal maximum score (classified as moderate). the average mathematical self-esteem achievement of pmk students who received cl had an average achievement of 67.99 from the ideal maximum score (classified as moderate) based on the results of statistical tests, the achievement of mathematical self-esteem of middle pmk students who received pjbl learning was more than students who received cl. like the research conducted by pamungkas et al. (2017), which used the inquiry cooperation learning model, this study found that for each category of initial mathematical ability (high, middle, and low) and school ranking (high and middle), achievement and self-improvement the mathematical self-esteem of students who received pbl learning was better than students who received cl. based on the pmk category, the average mathematical self-esteem achievement of low pmk students who received pbl learning was 48.54 from the ideal maximum score (classified as low). the average achievement of mathematical self-esteem of low pmk volume 11, no 2, september 2022, pp. 273-284 281 infinity students who received cl had an average achievement of 48.26 from the ideal maximum score (classified as low). the results of statistical testing concluded that the achievement of mathematical self-esteem of low pmk students who received pbl was not better than students who studied conventionally. the results of statistical tests for each category of pmk (high, middle, low), concluded that the achievement and improvement of mathematical self-esteem of students who received pbl were better than students who received cl. this shows that in the three pmk categories (high, middle, low), pbl learning is better than cl. therefore, pbl is more appropriate for students who have high and middle initial abilities than cl. based on the results of the study, it was found that there was no significant interaction between learning (pbl, cl) and pmk (high, middle, low) on the achievement of students' mathematical self-esteem. it means, the interaction between learning and pmk does not have a significant effect on differences in students' self-esteem achievement. differences in the achievement of students' self-esteem are only caused by differences in learning factors. in other words, the learning factor does not depend on the pmk factor. in any pmk category, the achievement of self-esteem of students who received pbl was better than students who received cl. even though at high and middle pmk, the achievement of mathematical selfesteem of students who received pbl was better than students who received cl, but when it seen from the difference in average achievement, high and middle pmk students who benefited much more from learning pbl compared to low pmk. 4. conclusion the achievement of self-esteem of students who received pbl learning was better than students who received conventional learning. the percentage of students' achievement of mathematical self-esteem who received pbl learning and those who received learning were both relatively high. there is no interaction effect between learning model (pbl-cl) and pmk (high, middle, and low). acknowledgements i would like to thank faculty of teacher training and education universitas pasundan who has given full support so that this paper can be realized. references aubuchon, j. c., & hettmansperger, t. p. 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(2004). self-esteem in children: strategies for parents and educators. national association of school psychologists. retrieved from: https://vresvalhallaschools.enschool.org/ourpages/auto/2009/2/5/57593823/selfesteem_young. pdf https://doi.org/10.33387/j.edu.v18i2.2119 https://doi.org/10.1016/j.jesp.2010.02.007 https://doi.org/10.22460/infinity.v10i1.p109-120 https://doi.org/10.7227/ijeee.47.2.7 https://doi.org/10.3102/00346543060001091 https://doi.org/10.1080/02619760903457818 https://vres-valhallaschools.enschool.org/ourpages/auto/2009/2/5/57593823/selfesteem_young.pdf https://vres-valhallaschools.enschool.org/ourpages/auto/2009/2/5/57593823/selfesteem_young.pdf https://vres-valhallaschools.enschool.org/ourpages/auto/2009/2/5/57593823/selfesteem_young.pdf fisher, dahlan, & putra, mathematical self-esteem ability of junior high school students … 284 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p177-192 177 ketupat eid tradition of the north coast of java as an alternative mathematics learning media wikan budi utami1, fikri aulia2, dian permatasari3, muhammad taqiyuddin4, sri adi widodo5* 1universitas pancasakti tegal, indonesia 2universitas negeri malang, indonesia 3universitas islam negeri sunan kalijaga, indonesia 4university of auckland, new zealand 5universitas sarjanawiyata tamansiswa, indonesia article info abstract article history: received aug 24, 2021 revised jan 13, 2022 accepted feb 18, 2022 learning mathematics requires real situations that students often face. tradition as local wisdom from the surrounding community can be used for learning mathematics so that students can reach the context of abstract mathematics. in this article, the researcher explains how teachers can take advantage of local culture and traditions in the form of ketupat eid on the north coast of java to be used in learning mathematics. this research was conducted on the northern coast of java, where every seventh day after eid mubarak, the area performs the ketupat eid tradition. researchers tried to connect some foods that only exist in the ketupat eid tradition in the form of ketupat and lepet with mathematics in school. the results showed that diamonds could be used for rhombuses, prisms, and beams, and lepet can be used as a tube context in geometry learning. keywords: ethnomathematics, ketupat eid tradition, ketupat, lepet, mathematics learning this is an open access article under the cc by-sa license. corresponding author: sri adi widodo, department of mathematics education, universitas sarjanawiyata tamansiswa jl. batikan uh iii/1043, tuntungan, tahunan, yogyakarta 55167, indonesia. email: sriadi@ustjogja.ac.id how to cite: utami, w. b., aulia, f., permatasari, d., taqiyuddin, m., & widodo, s. a. (2022). ketupat eid tradition of the north coast of java as an alternative mathematics learning media. infinity, 11(1), 177-192. 1. introduction mathematics learning that is currently developing still tends to be less flexible, less applicable, very theoretical, and less contextual. mathematics learning in secondary schools is less varied, thus affecting students' interest in further learning mathematics. teaching mathematics in schools is too formal, so the mathematics that children find in everyday life is very different from what they find in school. in other words, most of the teachers in learning mathematics have provided abstract mathematical concepts, without relating them to the context of the surrounding environment, so that students consider mathematics to be the most difficult subject for them. https://doi.org/10.22460/infinity.v11i1.p177-192 https://creativecommons.org/licenses/by-sa/4.0/ utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 178 the effect of learning mathematics in indonesia which tends to be less flexible, less applicable, very theoretical and less contextual can be seen from the ability of students in mathematics to have a tendency to be below the international average score of 500. the results of a survey conducted by trends in international mathematics and science study in 2015 obtained a score of 397 (mullis et al., 2015), as well as the results of the program for international student assessment (pisa) which measures reading, math, and science literacy skills, in 2015 the average score was successively the participants were 397, 386, and 403 which were still below the international average score of 500 (oecd, 2016; wijaya, 2016). these results are in line with the results of previous studies which showed that students' ability to solve mathematical problems was very weak. mathematics is recognized as developing along with the development of human civilization, while human civilization always produces culture. this means that mathematics is indirectly very closely related to the culture of an area. in other words, mathematics is closely related to every aspect of social life (d'ambrosio, 2001; irfan et al., 2019; rosa & orey, 2016; rosa & shirley, 2016). this is what causes the development of the mathematical context in each region to be different. for example, the context of the day in the javanese calendar uses modulo 5, while the context of the day in the christian calendar which is used internationally uses modulo 7. even in several regions in indonesia, architecture. various studies that study ethnomathematics, including researching the architecture of mosques in several areas have been identified for learning mathematics (hardiarti, 2017; lusiana et al., 2019; putra et al., 2020), regional musical instruments for learning mathematics (andarini et al., 2019; marina & izzati, 2019), batik for mathematics learning (fatkhurohman et al., 2021; sudirman et al., 2018; wahyudi et al., 2021), including traditional regional cultures such as in mandailing natal and sleman areas can be used for learning (dewita et al., 2019; irfan et al., 2019). integrating local culture or traditions or traditions in the environment around students with the context of mathematics is often called ethnomathematics (d'ambrosio, 2001; rosa & shirley, 2016). ethnomathematics is the way various cultural groups carry out mathematics in their activities. ethnomathematics is perceived as a lens for viewing and understanding mathematics as a cultural product (rosa & orey, 2013, 2016), so that ethnomathematics can be used as a culture-based approach in learning school mathematics so that mathematics can be understood well by students. the process of integrating the surrounding environment such as tradition or culture in learning mathematics is in line with the paradigm of meaningful learning for students (reiser & gagné, 1982; widodo et al., 2018). but the implementation in the field, mathematics learning in schools has not fully used the local cultural context (aini et al., 2019; ayuningtyas & setiana, 2019; sudirman et al., 2018), mathematics learning in schools is still focused on abstract mathematical material and cannot be observed directly by students. each region has a tradition that is a distinctive feature to distinguish culture from other regions. as in the northern coast of java, one of the eid mubarak traditions that are still being preserved is the bodo kupat (ketupat eid) tradition. in this tradition, two types of food that must be present to commemorate this are ketupat and lepet. apart from the vegetable menu, such as rendang and opor as a complement to the eid mubarak tradition, they are also made as a companion to eating ketupat. there has not been much research on the ketupat eid tradition as an alternative medium for learning mathematics. most researchers reveal the ketupat eid tradition from the aspect of character education values such as the adhesive rope of friendship between humans or for muslims it is often called hablum minannas (arif & lasantu, 2019), cultural acculturation such as the formation of a javanese muslim family mindset to carry out the ketupat eid tradition (including the volume 11, no 1, february 2022, pp. 177-192 179 tradition of eating ketupat) in various areas occupied by javanese muslim families (misbah, 2019). for this reason, this study seeks to identify typical foods in the ketupat eid tradition that can be used as an alternative medium for learning mathematics. learning media involves the use of activities that require mental processes in learning. so that mental activity in learning mathematics can be generated by using systematic manipulation of instructional events. in this connection, the learning media used are ketupat and lepet which exist in the ketupat eid tradition on the north coast of java. it is hoped that by using the ketupat eid tradition, mathematics learning becomes more meaningful because abstract mathematics content can be brought into real mathematics learning. 2. method 2.1. research design this is qualitative and descriptive research with an ethnographic approach (creswell, 2012). descriptive data is collected in the form of words and pictures (fraenkel et al., 2012; mohajan, 2018). meanwhile, the ethnographic method is used to describe, explain and analyze the cultural elements of a society or ethnic group using a more contemporary language (bass & milosevic, 2018; dobbert, 1982). ethnography means writing about a cultural group (bass & milosevic, 2018; dobbert, 1982; naidoo, 2012). 2.2. data collection authors are irreplaceable when used as a human instrument (brisola & cury, 2016; wa-mbaleka, 2020). therefore, the data collection techniques in this research were obtained through documentation and in-depth literature research. literature searches are carried out in the form of primary and secondary reference sources such as journals, research reports, thesis, dissertations, proceedings papers, books, and internet sources. 2.3. analyzing of data the data analysis technique used includes four main processes, namely: (1) data collection through literature and documentation, (2) data reduction, (3 ) data presentation in the form of narrative text, and (4) conclusion (creswell & creswell, 2017; fraenkel et al., 2012). the data collected were validated using a triangulation technique (carter et al., 2014; renz et al., 2018). this technique checks data through several relevant sources (informants). data verification is a step used to confirm the conclusions from the collected data (carter et al., 2014; creswell & creswell, 2017). verification is carried out by reviewing literature researches, interviews results, documentation, and placing a copy of the determined data using the validity technique (fraenkel et al., 2012). 3. result and discussion 3.1. result this study focuses on the food served in the ketupat lebaran tradition. this tradition in the islamic calendar occurs on the 7th of shawwal. different from the eid tradition in general which occurs on the 1st of shawwal, the level of crowds in the ketupat eid tradition is higher than the eid tradition. this is because on the 3th to 6th of shawwal, muslims still carry out the fasting worship of shawwal. utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 180 in general, there are two foods that must be present in the ketupat lebaran tradition. the two foods are ketupat and lepet (see figure 1). although some muslim families provide opor menus as a complement to ketupat, not all muslim families provide opor menus. figure 1. on the figure left is a ketupat food, while the right is a lepet food 3.2. discussion there are two discussions that will be written in this article. the first discusses the philosophical foundations of the ketupat eid tradition, along with the ketupat food and lepet foods. the second is talking about mathematics learning media using ketupat and lepet food. 3.2.1. historical of ketupat eid eid mubarak is a muslim holiday that occurs after the fasting worship of ramadan. in some areas in java island, such as the northern coastal region, two-holiday momentums occur in the month of shawwal, namely eid mubarok and ketupat eid. ketupat eid, also known as riyoyo ketupat or bakda kupat in javanese tradition, occurs on the 8th of shawwal after the one-week sunnah fasting. muslims carry out the sunnah of shawwal fasting because the reward erases one's sins for a year. however, sunan kalijaga or raden mas said introduced the bakda ketupat on the 8th of shawwal, in acculturating the javanese language bakda (after) becoming bodo, to obtain bodo kupat. in this regard, ketupat eid, which occurs on the 8th of shawwal, is calculated based on the intuition that the 1st of shawwal is the feast of eid mubarak carried out in one week (7 days). however, muslims are not allowed to fast on the 1st of shawwal because it is a day of tasrik. in some areas on the java island, such as regencies of jepara, pati, rembang, kudus, blora, and grobogan, bodo kupat was carried out on the 7th of shawwal. the difference in the bodo kupat is due to variation in time calculation between the javanese and general calendar. the calculation of time on the javanese calendar starts after asar (3 pm), leading to a bodo kupat celebrated on the 7th of shawwal. the javanese northern coastal java community usually prepares special foods to celebrate the bodo kupat tradition. the typical food is ketupat and lepet, in addition to making opor or gulai as a companion to enjoy kupat. ketupat is a food made with the basic ingredients of rice wrapped in woven yellow young coconut leaves, as shown in figure 2. volume 11, no 1, february 2022, pp. 177-192 181 figure 2. the philosophy of ketupat ketupat, in javanese, also known as kupat is an acronym for ngaku lepat and laku papat, which means admitting a mistake and four-step, respectively. ngaku lepat in javanese tradition, specifically in the syawalan and sungkeman, is used by children to admit for mistakes. in this tradition, children are taught to respect their elders by apologizing and asking for guidance and blessing. the symbol of sungkeman tradition is the process of using ketupat as a treat to ask for forgiveness. the door is automatically opened and all mistakes that occurred between the two will be erased once the guest eats ketupat. therefore, the use of this symbol to acknowledge mistakes by humans (hablu minnanas) and ask for forgiveness are in accordance with allah swt. the words laku papat philosophically come from 4 terms, namely lebaran (eid), luberan (overflow), leburan (melted), and laburan. lebaran (eid) means the end of the fasting month of ramadan and preparing to welcome the day of the victory of eid mubarak (return to holy). luberan (overflow) means to melt and overflow due to significant volume. it is morally associated with the culture of sharing with the poor by paying zakat to achieve sacredness. leburan means exhausted or united, with the eid moment used to eradicate sins against one another by apologizing and forgiving. laburan comes from the word labur or lime, which means using a white dye to purify liquids. this means that humans need to maintain their inner and outer purity. each element in ketupat has a philosophical meaning. for instance, kupat means purity of heart after apologizing for mistakes made to others. janur is a young and yellow coconut leaf which means jatining nur or conscience. according to some preliminary researches, the leaves used as wrappers are taken from the arabic java nur, which means the light has come. rice as the main ingredient for making ketupat symbolizes human lust. therefore, the kupat made from rice wrapped in coconut leaves represents lust limited by conscience. this means that humans need to be able to restrain the lust of the world with their conscience. the ketupat weaving has intricate details, meaning that human life is also full of twists and turns and is expected to strengthen each other physically and spiritually. utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 182 the rectangular shape of the ketupat also symbolizes the four lusts of the world, namely, anger, hunger, the desire to have something beautiful, and the desire to force oneself. those who eat ketupat are likened to be able to control these four passions during fasting. in addition, the rectangular shape of the diamond depicts kiblat papat lima pancer, which means four cardinal directions and one center, namely the direction of human life where the center is allah swt. lepet is a snack made with sticky rice and grated coconut wrapped in corn husks, coconut, or banana leaves, as shown in figure 3. lepet lexically comes from the word silep kang rapet (close tight), which means closing the past mistakes committed by our brothers and sisters and forgiving them. mistakes are forgiven when admitted upon (lepat), with the promise for not repeating it, thereby making brotherhood closer to such sticky rice in a lepet. figure 3. lepet the existence of ketupat and lepet food during the celebration of ketupat eid in the javanese islamic tradition is a form of reaffirmation. furthermore, apologizing and forgiving after fasting is manifested by real actions, not just lip-smacking and saying sorry during holidays with words such as minal aidzin wal faidzin. the word sorry needs to be born from the inner heart and students do not need to fight each other again. 3.2.2. media of learning learning media is a tool used to convey messages for students to ensure that they obtain its objectives (trisniawati et al., 2019; widodo et al., 2019). in this regard, the use of media in learning needs to consider the objectives adjusted to the level of student development and teachers ability in accordance with the characteristics of the material and support facilities (balaji & chakrabarti, 2010; kerres & witt, 2003; scardamalia & bereiter, 1991). volume 11, no 1, february 2022, pp. 177-192 183 the use of appropriate learning media generates new desires and interests, motivates, and stimulates learning activities to improve their understanding, facilitate data interpretation, and condense information (malaini et al., 2021; sedkaoui & khelfaoui, 2019; yusandra, 2021). hence, learning media is a very important factor used by the teacher (yusandra, 2021). this is because it is closely related to the learning experience and the meaningfulness of student outcomes (utami, 2019; widodo, 2018). the characteristics of abstract mathematics make students unable to understand mathematical concepts (scandura & wells, 1967; swanson & williams, 2014). therefore, media is needed to provide the right concepts concretized to understand mathematical material. previous researches stated that the use of media in learning improves students' cognitive abilities (irfan et al., 2019; widodo et al., 2018; widodo et al., 2021), activities (jonassen et al., 1994), and creativities in solving problems. however, this research indicates that the media used in learning promotes students ability to understand the material presented by the teacher during learning. several researches have been carried out to examine the use of media for learning, emphasizing technology-based learning media, specifically computer-based. although the curriculum in indonesia has emphasized the introduction of computer and information technology since the elementary school level, the limited facilities do not support the widespread application of computer game-based learning. therefore, this led to the use of learning media in the surrounding environment as an alternative solution. one aspect that exists in the environment and can be used as a learning medium is culture or tradition. mathematics is born from the activities of the cultural environment (bishop, 2013; dominikus et al., 2020; gerdes, 2013). ethnomathematics is the cultural anthropology of mathematics, which uses broad mathematical concepts (d'ambrosio, 1985, 1989, 2001; d’ambrosio, 2006; rosa & shirley, 2016). therefore, culture in learning mathematics is possible, specifically in indonesia, where each region maintains heritage and respects ancestors. one of the traditions or cultures still maintained is bodo kupat and lepet. therefore, using these 2 foods as learning media help students understand abstract mathematical concepts. figure 4. rhombus a rhombus is a parallelogram with 4 equal right-angled triangles, as shown in figure 4. a rhombus has the properties that all four sides are the same length, with the two diagonals utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 184 perpendicular to each other, while the opposite angles are equal with 2 axes of symmetry. its area and circumference are determined using 𝐿 = 𝑑1 𝑥 𝑑2 2 and 𝑘 = 4 𝑥 𝑠, respectively. the shape of the diamond, when viewed in two dimensions, is similar to a rhombus, as shown in figure 5. the diamond found in bodo kupat tradition is used as a medium for learning rhombus. figure 5. ketupat that resembles a rhombus ketupat is used as a learning medium for rhombuses and prisms, a three-dimensional shape bounded by an identical base and covered in an n-sided and the upright sides in the form of a square or rectangle. in other words, a prism is a shape with a cross-section of equal size. figure 6 shows that the shape of ketupat resembles a rectangular prism. figure 6. ketupat resembles a rectangular prism rectangular prisms are often referred to as beams due to their similar characteristics. these include: (1) having a rectangular base and roof that are congruent, (2) 6 side planes, (3) 4 vertical side planes, (4) 12 ribs, and (5) 8 corner points. a rectangular prism is called a cube when its base and roof are congruent with the 4 vertical sides. the volume of a rectangular prism or block is determined as 𝑉 = 𝑝 𝑥 𝑙 𝑥 𝑡, where 𝑝 𝑥 𝑙 denotes formula from wide, therefore, 𝑣 = 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 𝑥 𝑡. in addition to ketupat, the lepet in ketupat eid tradition are used as a medium for tube learning. a lepet is made such as rice cake resembles a tube, as shown in figure 7. the characteristics of the tube are found in the lepet shape. it includes: (1) the circular base and bag area and (2) a rectangular blanket area. the concept of volume used to determine the volume of the tube is 𝑉 = 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 𝑥 𝑡𝑎𝑙𝑙. since the base of the cylinder is a circle, the area of the base is 𝜋 𝑥 𝑟2, therefore, the volume of the cylinder is 𝑉 = 𝜋 𝑥 𝑟2𝑥 𝑡. volume 11, no 1, february 2022, pp. 177-192 185 figure 7. the picture of a slime that resembles a tube from the previous explanation, ketupat and lepet foods that exist in ketupat eid tradition can be used as a medium for learning geometry, such as rhombuses, rectangular prisms, rhombus prisms, and blocks. lepet can be used as a learning medium for tubes in connection with the formation of rhombus and other mathematical shapes. in addition, ketupat and lepet food when viewed from the manufacturing process can be used to formulate mathematical problems with diamond context such as the following. if 10 ketupat is used to commemorate eid mubarak with an average thickness and side of 4 cm and 10.5 cm. determine the average area of the leaf used to wrap ketupat and its volume! from this problem, it is known that the average thickness and side of the diamond are 4 cm and 10.5 cm. terms of the area of the leaves needed to wrap ketupat can be interpreted by the surface area. therefore, 𝐿𝑘𝑒𝑡𝑢𝑝𝑎𝑡 = 10.5(10.5 + 2(4)) = 10.5(18.5) = 194.25 𝑐𝑚2. the volume of ketupat is obtained from 𝑉𝑘𝑒𝑡𝑢𝑝𝑎𝑡 = (10.5)(10.5)(4) = 63 𝑐𝑚2. in addition to use ketupat context, the process of making diamonds can also be used to arrange questions. suppose every 500 grams of rice produces 12 diamonds, therefore, each ketupat is only filled by 2/3 of the volume which is not too thick pera (harsh) and mushy. unlike ketupat, lepet is made using a mixture of sticky rice and grated young coconut. it takes 1 kg of sticky rice mixed with 2.5 young coconuts grated to 400gr to make 40 lepets at a boiling time of 4 to 5 hours. from the process of making ketupat and lepet, a teacher can use the context of ketupat and lepet to make math problems such as the following: mrs. retno is a seller of lepet and ketupat. during eid mubarak she had 100 leaves with 1 leaf used to make ketupat, while half is used to produce lepet. in addition, to make 12 ketupat it takes 500 grams of rice, therefore, the capital needed to make 12 ketupat is rp. 6,000.00. to make 40 lepet 1 kg of sticky rice and 2.5 coconuts are needed. the price of 1 kg of sticky rice is idr 25,000.00, while 1 coconut is idr 5,000.00. hence, the capital needed to make 40 lepet is idr 37,500 at a total of idr 100,000.00 with a selling price of rp. 3,000.00 and rp. 2,000 for ketupat and lepet, respectively. suppose 𝑥 = number of ketupat production, and 𝑦 = number of sticky productions, then to determine the number of ketupat and lepet made from 100 leaves using 𝑥 + 1 2 𝑦 = 200. conversely, 500 grams of rice is used to make to make 12 ketupat. therefore, the utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 186 capital is rp. 6,000.00. to make 40 lepet requires 1 kg of sticky rice and 2.5 coconuts. the price of 1 kg of sticky rice is idr 25,000.00, while 1 coconut is idr 5,000.00. therefore, the capital needed to make 40 lepet is idr 37,500 with a total of idr 100,000.00 6000 12 𝑥 + 37,500 40 𝑦 = 100000. due to the number of available leaves and limited capital, the appropriate sign of inequality is “≤" from table 1. table 1. estimation of ketupat and lepet modeling variable sum of ketupat (𝒙) sum of lepet (𝒚) symbol stock sum of janur 1 ½ ≤ 100 startup capital 6,000 12 = 500 37,500 40 = 937.5 ≤ 100,000 the following constraint functions were obtained from table 1: a. 𝑥 + 1 2 𝑦 = 100 or 2𝑥 + 𝑦 = 200. b. 6,000 12 𝑥 + 37,500 40 𝑦 = 100,000 or 500𝑥 + 937.5𝑦 = 100,000. c. 𝑥 and 𝑦 are positive integers therefore 𝑥 ≥ 0 and 𝑦 ≥ 0. the objective function is obtained from the statement “if the selling price of ketupat is rp. 3,000.00 and lepet is rp. 2,000.00” then the objective function is 𝑍 = 3000𝑥 + 2000𝑦. furthermore, use inequalities to determine the area of the set of solutions. for instance, 2𝑥 + 𝑦 = 200 is used to determine the point of intersection of the coordinates (0,200) and (100,0). meanwhile, for inequality 500𝑥 + 937,5𝑦 = 100000 the point of intersection for the coordinate are (0,106.67) and (200,0). therefore, a solution set for inequalities is formulated from these points, as shown in figure 8. figure 8. ketupat and lepet problem modeling volume 11, no 1, february 2022, pp. 177-192 187 figure 8 shows that the solution set is limited by points (0,0), (100,0), (106.67,0) and (63.64, 72.73). point (63.64, 72.73) is used to obtain the intersection of lines 2𝑥 + 𝑦 = 200 and 500𝑥 + 937.5𝑦 = 100000 by eliminating or substituting the two equations. then, compare the value of the objective function at each corner point and substitute it into the objective function (see table 2). it is found that the number of ketupat and lepet required to obtain the highest sales result of rp. 479,000.00 are 63 ketupat and 72 lepet. table 2. value of the objective function for each boundary points point 𝒁 = 𝟑𝟎𝟎𝟎𝒙 + 𝟐𝟎𝟎𝟎𝒚 (0,0) 𝑍 = 3000(0) + 2000(0) = 0 (100,0) 𝑍 = 3000(100) + 2000(0) = 300.000 (0,106) 𝑍 = 3000(0) + 2000(106) = 212.000 (63,72) 𝑍 = 3000(127) + 200(145) = 479.000 based on the two examples of cases of using ketupat and lepet for learning mathematics that have been previously described, namely the area and perimeter of several geometric shapes, and linear programming problems. does not rule out the possibility that the context of ketupat and lepet can be used for mathematics learning media on other materials. 4. conclusion learning mathematics requires the integration of real objects that are close to students and school life. therefore, the use of ketupat and lepet in kupat eid tradition is often practiced by muslims in some areas on the north coast of java as learning media to build rhombuses, rectangular prisms, rhombus prisms, and blocks help in making this process easier for students. similarly, for linear programming material, ketupat and lepet contexts are used to formulate mathematical problems. the research results that have been obtained, it is very possible to learn mathematics using the context of ketupat and lepet. both of these foods are very exposed to residents on the north coast of java, so that teachers in these areas can use the context of ketupat and lepet in learning mathematics. acknowledgements the authors would like to thank the universitas pancasakti tegal, universitas negeri malang, universitas islam negeri sunan kalijaga, university of auckland, and universitas sarjanawiyata tamansiswa, which has provided 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(2019). understand, develop and enhance the learning process with big data. information discovery and delivery, 47(1), 2-16. https://doi.org/10.1108/idd-09-2018-0043 sudirman, s., son, a. l., & rosyadi, r. (2018). penggunaan etnomatematika pada batik paoman dalam pembelajaran geomteri bidang di sekolah dasar [the use of ethnomathematics in paoman batik in learning geometry in elementary schools]. indomath: indonesia mathematics education, 1(1), 27-34. swanson, d., & williams, j. (2014). making abstract mathematics concrete in and out of school. educational studies in mathematics, 86(2), 193-209. https://doi.org/10.1007/s10649-014-9536-4 trisniawati, t., muanifah, m. t., widodo, s. a., & ardiyaningrum, m. (2019). effect of edmodo towards interests in mathematics learning. journal of physics: conference series, 1188, 012103. https://doi.org/10.1088/1742-6596/1188/1/012103 https://doi.org/10.26740/jrpipm.v4n1.p10-22 https://doi.org/10.3102/00346543052004499 https://doi.org/10.1177/1049732317753586 https://doi.org/10.1007/978-3-319-30120-4_3 https://doi.org/10.1007/978-3-319-30120-4_4 https://doi.org/10.3102/00028312004003295 https://doi.org/10.1207/s15327809jls0101_3 https://doi.org/10.1108/idd-09-2018-0043 https://doi.org/10.1007/s10649-014-9536-4 https://doi.org/10.1088/1742-6596/1188/1/012103 volume 11, no 1, february 2022, pp. 177-192 191 utami, w. b. (2019). student experience about higher order thinking skill with contextual learning based ethnomathematics using learning media and math props. international journal of recent technology and engineering (ijrte), 8(ic2), 719-721. wa-mbaleka, s. (2020). the researcher as an instrument. in a. p. costa, l. p. reis, & a. moreira, in computer supported qualitative research, cham. https://doi.org/10.1007/978-3-030-31787-4_3 wahyudi, h., widodo, s. a., setiana, d. s., & irfan, m. (2021). etnomathematics: batik activities in tancep batik. journal of medives: journal of mathematics education ikip veteran semarang, 5(2), 305-315. https://doi.org/10.31331/medivesveteran.v5i2.1699 widodo, s., irfan, m., leonard, l., fitriyani, h., perbowo, k., & trisniawati, t. (2019). visual media in team accelerated instruction to improve mathematical problemsolving skill. in proceedings of the 1st international conference on science and technology for an internet of things. european alliance for innovation (eai). https://doi.org/10.4108/eai.19-10-2018.2281297 widodo, s. a. (2018). selection of learning media mathematics for junior school students. turkish online journal of educational technology-tojet, 17(1), 154-160. widodo, s. a., darhim, d., & ikhwanudin, t. (2018). improving mathematical problem solving skills through visual media. journal of physics: conference series, 948, 012004. https://doi.org/10.1088/1742-6596/948/1/012004 widodo, s. a., prihatiningsih, a., & taufiq, i. (2021). single subject research: use of interactive video in children with developmental disabilities with dyscalculia to introduce natural numbers. participatory educational research, 8(2), 94-108. https://doi.org/10.17275/per.21.31.8.2 wijaya, a. (2016). students' information literacy: a perspective from mathematical literacy. journal on mathematics education, 7(2), 73-82. https://doi.org/10.22342/jme.7.2.3532.73-82 yusandra, t. f. (2021). utilization of audio visual media as an online learning solution during the covid-19 pandemic. hispisi: himpunan sarjana ilmu-ilmu pengetahuan sosial indonesia, 1(1), 281-289. https://doi.org/10.1007/978-3-030-31787-4_3 https://doi.org/10.31331/medivesveteran.v5i2.1699 https://doi.org/10.4108/eai.19-10-2018.2281297 https://doi.org/10.1088/1742-6596/948/1/012004 https://doi.org/10.17275/per.21.31.8.2 https://doi.org/10.22342/jme.7.2.3532.73-82 utami, aulia, permatasari, taqiyuddin, & widodo, ketupat eid tradition … 192 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p33-54 33 infinity integrating sundanese ethnomathematics into mathematics curriculum and teaching: a systematic review from 2013 to 2020 dindin abdul muiz lidinillah1*, rahman1, wahyudin1, sani aryanto1,2 1universitas pendidikan indonesia, indonesia 2universitas bhayangkara jakarta raya, indonesia article info abstract article history: received jan 20, 2021 revised aug 1, 2021 accepted aug 8, 2021 as a country with the most ethnic, cultural, and linguistic backgrounds globally, indonesia has great potential for the development of ethnomathematics studies. among those ethnic groups, sundanese ethnic group and culture is the second-largest after the javanese ethnic group and culture, making it interesting to study because it has rich cultural elements that can be integrated into the mathematics curriculum in schools. this article explores the development of research on ethnomathematics based on sundanese culture, including those which are based on the study's scope, the integration of sundanese ethnomathematics into the school curriculum, and mathematics learning. based on the investigation, there has been no systematic literature review on sundanese ethnomatematics. the research was conducted using the systematic literature review method with the prisma protocol. the results showed that there were various kinds of research on sundanese ethnomathematics. there were various kinds of study on sundanese ethnomathematics integration into mathematics curriculum and teaching using five model categories. the results also showed that rme, ctl and pbl were the most widely used teaching approaches for learning sundanese ethnomathematics. however, most sundanese ethnomathematics teaching uses a special approach that varies, developing teaching material, teaching media, and using the context of traditional games. keywords: curriculum, ethnomathematics, sundanese, systematic review this is an open access article under the cc by-sa license. corresponding author: dindin abdul muiz lidinillah, departement of mathematics education, graduate school of the universitas pendidikan indonesia jl. dr. setiabudi no.229, bandung city, west java 40154, indonesia email: dindin_a_muiz@upi.edu how to cite: lidinillah, d. a. m., rahman, r., wahyudin, w., & aryanto, s. (2022). integrating sundanese ethnomathematics into mathematics curriculum and teaching: a systematic review from 2013 to 2020. infinity, 11(1), 33-54. 1. introduction research on ethnomathematics has developed rapidly since ubiratan d'ambrosio delivered his opening speech in 1984 at the fifth international congress on mathematics education in adelaide, australia, on the relationship between mathematics, culture and https://doi.org/10.22460/infinity.v11i1.p33-54 https://creativecommons.org/licenses/by-sa/4.0/ lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 34 society (rosa & orey, 2016). d'ambrosio (1985) has provided a foundation for the study of ethnomatematics and its relationship to history and pedagogy. meanwhile borba (1990) places ethnomathematics as an epistemological approach to mathematics and presents the basis for studying the relationship between ethnomathematics and education. since 1984, six international conferences on ethnomathematics (icem) have been held every four years, attended by representatives of more than 20 countries (rosa & orey, 2016) and most recently held in colombia in 2018. this shows that the study of ethnomatematics has so far developed into an interesting field for research. ethnomathematics was defined as an anthropological study of mathematical thinking and practice about the relationship between mathematics and culture (d'ambrosio, 1985; eglash, 2000) and mathematical modeling, which can help students translate various mathematical ideas and practices found from a cultural element (rosa & orey, 2016). ethnomathematics is a human creation that directs attention to the cultural roots and social history of mathematics, expressed in the language code of certain sociocultural groups (cimen, 2014). ethnomathematics is practiced by cultural groups, including not only indigenous peoples, but also groups of workers, professional classes, and children from specific age groups (d'ambrosio, 1985). therefore, ethnomathematics is closely related to the mathematical way of thinking of a society that is related to its culture, and can be integrated in the school curriculum. ethnomathematics is a field of research that includes the process of transmitting, disseminating, and institutionalizing mathematical knowledge (ideas, processes, and practices) that originates from the diversity of cultural contexts in history (rosa & orey, 2016). the international study group on ethnomathematics (isge) identifies four areas of general interest in ethnomathematics: (a) field research to collect data on mathematics in culture; (b) mathematical work in cross-cultural situations; (c) the application of ethnomathematics in the classroom; and (d) theoretical, sociological, and ethnomathematics policy studies (rosa & shirley, 2016). meanwhile, there are six essential dimensions of ethnomathematics: cognitive, conceptual, educational, epistemological, historical, and political, that can be used to analyze the socio-cultural roots of mathematics (rosa & orey, 2016). both the areas of general interest and the essential dimensions of ethnomathematics provide consideration in choosing a research focus on ethnomathematics even though they present different areas of study which are substantially the same. the ethnomathematics theoretical framework was regarding how to create a more culturally responsive mathematics classroom (brandt & chernoff, 2015; maksimova, 1967). teachers must design mathematics teaching based on a diversity of practices that are placed historically, culturally, socially, and politically like other human activities (gay, 2000; nam et al., 2018). a pedagogical approach based on ethnomathematical studies can develop mathematics teaching based on student perspectives (ricardo & mafra, 2020) or mathematics teaching that considers the daily mathematics practices of culturally diverse students (françois, 2010) to make meaningful relationships and deepen their understanding of mathematics (d'ambrosio, 2001). this framework can be an ethnomatematic based learning approach in schools. there are five ethnomathematics curriculum models that integrate aspects of student culture into a holistic learning environment: mathematical epistemology, its content, classroom culture, and mathematics teaching approaches (rowlands & carson, 2002), namely: (a) presenting ethnomathematics as a context which is meaningful for the development of thinking skills; (b) presenting ethnomathematics as a specific cultural content distinct from universal mathematical concepts; (c) as a stage in the development of mathematical thinking that the child goes through; (d) creating a classroom containing a volume 11, no 1, february 2022, pp. 33-54 35 infinity cultural context in the form of values, beliefs and learning theory; and (e) presenting mathematical concepts and practices derived from the culture of students. teaching mathematics relevant to culture and personal experiences help students know more about reality, culture, society, environmental problems, and themselves (orey & rosa, 2006) ethnomathematics provides enrichment and new topics that students have never seen before that can be found in cultural practices worldwide (rosa & shirley, 2016). the arguments for using ethnomathematical examples in the classroom are (a) to show students that their own culture contributes to mathematical thinking and (b) to show students different cultures from around the world to build respect for others and contribute to global education (rosa & shirley, 2016). this perspective emphasizes students to have respect for a variety of cultural backgrounds in addition to developing mathematical thinking with cultural contexts.although ethnomatematics has developed since 1984 to cover various areas and dimensions of research, research on ethnomatematics in indonesia began in 2011 based on tracing results in the indonesian journal portal, namely http://garuda.ristekbrin.go.id and the indonesian repository portal http://rama.ristekbrin.go.id. several scientific articles published in international journals and proceedings were published after that year. the study of culture-based education in the indonesian context is fascinating to conduct. it is because based on the 2010 indonesian central statistics agency census results (na’im & syaputra, 2010), there are 1340 ethnic groups in indonesia, which are grouped into 31 clusters of ethnic groups . the javanese dominate and rank first, which is around 40,22% of the total population, followed by the sundanese of around 15,5%, and the third and fourth ranks are the batak (3,58%) and sulawesi ethnic groups (3,22%). moreover, based on the mapping and verification results, there are 652 local languages in indonesia. this number does not include dialects and sub-dialects this data illustrates the challenges of studying ethnomathematics in indonesia, which has a high diversity of cultures and ethnicities. the diversity of indonesian cultures inspires research activities on ethnomathematics in every ethnicity and culture in indonesia. as the second largest ethnic group in indonesia, research on sundanese culture from the perspective of ethnomathematics is very interesting to do, as well as research on other diverse ethnic cultures in indonesia, especially javanese ethnomathematics as the largest ethnic group.it should be known that many ethnomathematics studies have been carried out involving other ethnic groups and cultures in indonesia, which gives color to indonesia's ethnomathematics diversity. mathematics has existed since the time of the sundanese people's ancestors and is still part of daily life, especially in rural communities. the challenge is how teachers in schools can integrate ethnomathematics into mathematics teaching that is more dynamic and exciting, fun, and easier to learn (abdullah, 2017). sundanese ethnomathematics based teaching as a curriculum unit can be developed based on five ethnomathematics curriculum models (rowlands & carson, 2002). the development of learning units that are based on an ethnomathematics curriculum can refer to an ethnomathematics based teaching approach framework (adam, 2004) (see figure 1). http://garuda.ristekbrin.go.id/ http://rama.ristekbrin.go.id/ lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 36 figure 1. the framework for the ethnomathematics curriculum unit (adam, 2004) this model (see figure 1) can be developed as a learning phase that enables students (adam, 2004): (a) to become aware of the mathematical activities that exist in their culture; (b) to understand and experience these cultural activities from a mathematical perspective; (c) to relate their mathematical knowledge to other experiences or cultures that use mathematical thinking; (d) to study the use of mathematical systems, notations, and techniques in response to human needs; and (e) to understand conventional mathematics better so as to contribute to understanding the principles of culture-based mathematics. ethnomathematics curriculum models and ethnomathematics based teaching approach framework are used as frameworks for analyzing the model of integration of ethnomathematics into the school curriculum and ethonomathematics-based learning syntax contained in articles and scientific papers on sundanese ethnomathematics. this study is expected to provide a reference as a preliminary study describing the framework and projection of sundanese ethnomathematics studies which are specifically related to how to integrate sundanese ethnomathematics into the curriculum and teaching mathematics and what mathematics learning goals can be developed with sundanese ethnomathematics based teaching. to explore more deeply about how ethnomathematics has been carried out in various studies, especially those related to how sundanese ethnomathematics is integrated into the mathematics curriculum and teaching in schools, the following are the questions raised in this research which uses a systematic review method. the third research question related to the objectives of learning mathematics completes the framework for integrating ethnomathematics into the school curriculum that refers to the standards developed by joyner and reys (2000) and national reasearch council (kilpatrick, 2001), a) what are the models for integrating sundanese ethnomathematics into the school mathematics curriculum? b) what are the approaches to teaching mathematics based on sundanese ethnomathematics? c) what are the objectives of learning mathematics with a sundanese ethnomathematics approach? 2. method researchers used a systematic review with the prisma protocol (preferred reporting items for systematic review and meta-analysis protocol). the role of the prisma protocol is a procedure for examining and selecting all appropriate empirical volume 11, no 1, february 2022, pp. 33-54 37 infinity evidence used to answer the identified research questions. the empirical evidence referred to in this case is scientific research results in the form of journal articles, seminar proceedings and research reports. this method minimizes bias and provides guidelines and a structure for reporting. the prisma protocol consists of 27 checklist items and a four-phase flow chart. the prisma protocol is an evidence-based approach to accurately and reliably report findings from articles for systematic review (moher et al., 2009). the focus of scoring articles for this systematic review was based on items from the checklist, including title, author, year, research question, cultural setting, method, sample/subject, and findings. items were selected primarily those of relevance to the research questions and supported additional analysis. the target documents that were traced were journal articles (local, international) and scopus-indexed proceedings, undergraduate theses, master theses, dissertations. the search was mainly focused on journal portals and repository portals in indonesia (http://garuda.ristekbrin.go.id and http://rama.ristekbrin.go.id). subsequently, document searches were carried out on google scholar and the publish and perish app to find articles that did not appear on both search portals. several international journal articles and scopus-indexed proceedings were obtained. the keywords used to search for articles on the garuda protal a portal are ethnomathematics, “etnomatematika”, “ethnomatematics”, and mathematics with culture, local and traditional. meanwhile, searches in international journals and proceedings used the keywords ethnomathematics and indonesia.the following is a detailed description of the stages of selecting study sources, namely: (a) identifying journal articles and student research results in the form of undergraduate theses, master theses, dissertations; (b) multiple document screening assisted by mendeley; (c) removing incomplete documents or obtaining disclaimer notes; (d) selecting documents relevant to sundanese ethnomathematics studies; (e) checking the quality of documents relevant to the research question; (f) recapping the relevant documents for the next analysis needs. descriptive statistics were used to present the data extracted and tabulated. the findings related to the research questions were then presented in the narrative summary. the following is the sample selection process flowchart as a source of the article review in accordance with the prisma protocol for systematic review (see figure 2). http://garuda.ristekbrin.go.id/ http://rama.ristekbrin.go.id/ lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 38 figure 2. prisma protocol flowchart to demonstrate the document selection process in detail, the following is the number of sources generated through the selection process at each stage of prisma. table 1. selection of documents for research types of document search engine number of documents identification screening eligibility included local journals garuda 311 311 47 29 international journals publish & perish 13 13 7 5 scopus-proceedings publish & perish 62 62 14 7 undergraduate theses rama 74 66 38 14 master theses rama 13 13 2 0 dissertations rama 4 4 2 1 total 477 469 110 56 source documents obtained before the final selection stage described the scope of the sundanese ethnomathematics study obtained from these source documents. then, specifically, the presentation and discussion of data were focused on research questions. id e n ti fi c a ti o n record identified through garuda and rama portal (n=402) in c lu d e d e li g ib il it y s c re e n in g record identified through google scholar and publish and perish (n=75) record after duplicates and disclaimer removed (n=469) record screened (n=469) record exluded: not sundanese ethnomatematics (n=359) full-text excluded (not relevant with research question) (n=54) sundanese ethnomatematics study included in review (n=56) full-text assessed for eligibility and further analysis (n=110) record identified duplicates and disclaimer (n=8) volume 11, no 1, february 2022, pp. 33-54 39 infinity table 1 show that the identification stage, 477 articles and scientific papers were collected during the initial search stage using the keywords used. these collected documents still contain duplicate documents and ethnomatematic studies throughout indonesia. ethnomathematic identification of sundanese is not enough to be obtained from the title of the article but must be learned from the abstract and the entire document content. after the screening (screening stage) of multiple and incomplete documents, a total of 469 documents remained. then carried out an examination of the documents to choose which ones contained the sundanese ethnomathematic study, which then obtained 110 eligible documents (eligibility stage). then, these 110 documents are presented in a table based on items: title, author, year, research question, cultural setting, method, sample / subject, and findings to be used in the next analysis stage. after that the documents were separated between studies on the relationship of mathematics and culture (54 documents) and studies on the integration of ethnomatematics into the curriculum and teaching (56 documents). these 56 documents are documents analyzed to answer research questions. 3. results and discussion the sundanese ethnomathematics study referred to here is ethnomathematics based on the sundanese cultural background, which encompasses the areas of west java and banten, because sundanese language and culture is taught in both regions. the sundanese culture area, which is based on the sundanese kingdom, includes the jakarta area (sunda kelapa) and the western part of central java. sundanese is still used in central java, bordering west java, although some west java residents use javanese. moreover, in areas like cirebon, there is a combination of javanese and sundanese cultures, forming unique cultural characteristics. in this study, sundanese ethnomathematics covers west java and banten's regional background, both with the general sundanese cultural background and the traditional territory. 3.1. the scope of the sundanese ethnomathematics study there were 72 journal articles and proceedings and 43 student research reports that were used to describe the scope of sundanese ethnomathematics studies. the data used in this section are data that have not gone through the selection process at the eligibility stage. at this stage, it still contained source documents related to sundanese ethnomathematics studies but not related to sundanese ethnomathematics as a teaching approach. referring to the international study group on ethnomathematics, which describes four areas of general interest in ethnomathematics (rosa & shirley, 2016), in general, two areas of interest are contained in existing document sources, namely: (1) field research, where data on mathematics in culture collected and (2) application of ethnomathematics in the classroom. the following is a breakdown of each area of interest in ethnomathematics based on the type of document. table 2. ethnomathematics areas of interest types of document number of documents mathematics and culture curriculum and teaching local journals 18 29 international journals 2 5 scopus-proceedings 7 7 undergraduate theses 24 14 lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 40 types of document number of documents mathematics and culture curriculum and teaching master theses 2 0 dissertations 1 1 total 54 56 based on the source document data in table 2, it is clear that the number of ethnomathematics studies related to the relationship between mathematics and culture and the application of ethnomathematics in curriculum and teaching is quite balanced even though there are large differences in details. of the total source documents, only two authors do not have affiliation with universities in west java and banten, which have a sundanese cultural background. the earliest sundanese ethnomathematics study was conducted in 2013 in the form of a student thesis research. then more studies were conducted in the form of research for student theses and dissertations and journal articles and proceedings. the cultural background or area coverage of west java and banten is relatively evenly distributed. however, there are several backgrounds for sundanese indigenous people, such as the baduy community in banten, kampung naga (naga village) in tasikmalaya, kampung kuta (kuta village) in ciamis, cirebon culture, and the indigenous ciptagelar community in sukabumi. these indigenous peoples' backgrounds are interesting to study to reveal sundanese ethnomathematics heritage preserved because it is indigenous culture. this section will first present studies of sundanese ethnomathematics, which links mathematics and culture. what is the scope of mathematics and culture studies? ethnographic methods carry out almost all research on the relationship between mathematics and culture. ethnomathematics research using ethnographic methods is mostly based on the framework for ethnomathematical research (alangui, 2010). the following are some of the research findings summarized from source documents that are considered presenting studies of the relationship between mathematics and culture, which, according to the authors, provide representative reports and can represent other documents' substances (see table 3). table 3. findings about relationship between mathematics and sundanese culture authors research focuses arisetyawan (2015) this research conducted on the baduy community was focused on exploring various mathematical thinking processes, which are classified into the seven elements of the ethnographic framework as universal anthropological elements of culture including social elements, language elements, scientific systems, technology systems, religious systems, livelihood systems, and art systems. ridwan (2018) research reveals mathematical ideas of geometry and social arithmetic in the baduy weaving culture. rivaldi (2018) this research on salapan village community, karawang revealed the origin of the name salapan (translated into “sembilan” in indonesian language or number “nine” in volume 11, no 1, february 2022, pp. 33-54 41 infinity authors research focuses english language) village and elements of social life influenced by the philosophy of number nine. hermanto et al. (2019) this research conducted on naga village community, tasikmalaya, revealed that the activities of the naga village community as an indigenous community has mathematical nuances include counting activity, measuring activity, build design activity, and provisions in the making/renovating residential houses septianawati et al. (2017) this research on naga village community was conducted to reveal measurement units, including units of length, units of area, and units of volume. umbara et al. (2019) this research conducted on the traditional community of cigugur, kuningan revealed the palintangan device, which is part of the agricultural system knowledge in the sundanese cultural structure. farmers use palintangan to calculate and determine the right days for plantings, such as kolenjer (baduy), tunuk (kampung naga), and tunduk (ciwidey bandung). suprayo et al. (2019) this research conducted on suranenggala kidul village community, cirebon revealed an ethnomathematics activity in agricultural activities, including measuring land and measuring the amount of rice. these two activities are related to the concepts of rank and series, concepts of geometry, and concepts of calculus mustika (2013) this research was focused on exploring the values of traditional games and mathematical ideas contained in kaneker games from the baduy community. febriyanti et al. (2018) research on games for sundanese people such as engklek (similar to hopscotch) and gasing (similar to wooden spinning top) pratiwi and pujiastuti (2020) research on marbles game in sundanese context supriadi and arisetyawan (2020) research on endog-endogan game muchyidin (2016) this research was focused on exploring mathematical ideas in sundanese batik motifs : trusmi cirebon batik mahuda (2020) lebak batik saputra (2017) kuningan batik sudirman et al. (2018) indramayu batik maharani and maulidia (2018) this research was focused on exploring mathematical ideas in traditional house architecture, namely, traditional panjalinan houses lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 42 authors research focuses imswatama and zultiar (2019) resourch on houses in sukabumi prabawati (2016) research reveals mathematical ideas on various motifs of mendong and woven crafts in tasikmalaya puritea (2018) research reveals mathematical ideas on ceramic art findings can be grouped into the framework of the seven universal elements that form culture, namely, social elements, language elements, scientific systems, technology systems, religious systems, livelihood systems, and art systems (arisetyawan, 2015). sundanese ethnomathematics research that focuses on these seven elements can be carried out in an indigenous community that maintains local cultural wisdom from generation to generation and is considered an indigenous community such as the baduy community in banten (arisetyawan, 2015). similar research with arisetyawan (2015) can also be carried out, including in naga village in tasikmalaya, kuta village in ciamis, and ciptagelar village in sukabumi which still running indigenous sundanese culture. sundanese ethnomathematics research results that reveal the relationship between culture and mathematics can be used as teaching and learning resources in schools or education in general to preserve sundanese culture. the next discussion will be directed to how ethnomathematics is applied in curriculum and teaching in schools. 3.2. sundanese ethnomathematisc integration model in the school curriculum mathematics has become a part and practice in every culture. mathematics needs to be integrated into the curriculum in schools. according to the current curriculum model, educators who think critically and creatively can integrate ethnomathematics (brandt & chernoff, 2015). the ethnomathematics approach to the mathematics curriculum aims to make school mathematics more relevant and meaningful for students and to disseminate their knowledge, skills, and attitudes and maintain their cultural identity (balamurugan, 2015). the authors use five different ethnomathematics curriculum models (adam, 2004) to explore the integration model of sundanese ethnomathematics into the school’s mathematics curriculum. based on the results of categorization of 56 documents, it was found that a document tends towards a model even though it has links to other categories. table 4. categorization of documents based on five models of ethnomatematic curriculum integration type of document model 1 model 2 model 3 model 4 model 5 total local journals 9 5 5 10 29 international journals 4 1 5 scopus-proceedings 2 5 7 undergraduate theses 7 2 5 14 master theses dissertations 1 1 total 21 5 14 21 56 description : model 1 : ethnomathematics as a context which is meaningful for the development of thinking skills volume 11, no 1, february 2022, pp. 33-54 43 infinity model 2 : ethnomathematics as a specific cultural content distinct from universal mathematical concepts model 3 : ethnomathematics as a stage in the development of mathematical thinking that the child goes through model 4 : creating a classroom containing a cultural context in the form of values, beliefs and learning theory model 5 : presenting mathematical concepts and practices derived from the culture of students table 4 presents data on the results of document categorization that identifies the curriculum integration model being developed. the distribution of categories shows the dominance of model 5 (37.5%) and model 1 (37.5%), followed by model 4 (25%). based on the analysis results, all documents tended to describe curriculum development at the level of teaching development in the classroom.. none of them specifically discussed the integration model of sundanese ethnomathematics at the school curriculum level, because almost everything is related to integration at the curriculum implementation level at the classroom level. studies related to the sundanese ethnomathematics integration model at the level of the curriculum framework have not provided comprehensive research results.based on the categorization in table 4, the development of sundanese ethnomathematics-based mathematics teaching was mostly done by exploring the culture of students with a sundanese cultural background and elements of sundanese culture in general or in cultural communities, which were then used as teaching materials and resources. several teaching development models can be referenced from several documents whose studies are sufficient and are considered can represent studies in other documents (see table 5). table 5. findings about models of ethnomatematic curriculum integration authors research focuses (supriadi, 2014, 2020) this development of teaching in an ethnomathematics-based curriculum was carried out through a sequence of studies on (a) cultural problems in society, (b) cultural values of the sundanese people, and then presenting them in the form of (c) contextual problems. (supriadi, 2019; supriadi & arisetyawan, 2020) teaching design referred to a contextual approach. the ethnomathematics curriculum integration model developed in this study was relevant to model 1 and model 5, presenting ethnomathematics as a meaningful context relevant to the student's cultural background. although this research was conducted on the students of elementary teacher education, starting from this research, much more research was developed at the elementary school level. (arisetyawan, 2019; febriyanti et al., 2018; supriadi & arisetyawan, 2019; supriadi et al., 2016) this development of teaching in an ethnomathematics-based curriculum was carried out by developing and using mathematics teaching materials, traditional games, and traditional sundanese cultural crafts to build a classroom atmosphere with cultural nuances which is in line with model 4 (supriadi, susilawati, et al., 2019) this development of teaching in an ethnomathematics-based curriculum was carried out to facilitate students' mathematical thinking activities, such as model 3, which can be developed along with the application of other ethnomathematics curriculum integration models lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 44 3.3. sundanese ethnomathematisc-based mathematics teaching approach several possible approaches that can be considered part of the development of ethnomathematics teaching in schools are problem-posing and problem-solving, openended, and realistic mathematics education approach (peni & baba, 2019). in the 56 documents analyzed, there are various approaches, models, or methods used to integrate sundanese ethnomathematics in mathematics teaching in schools. most of the research (40 of the 56 documents) used the experimental method or developmental research methods, namely, didactical design research (ddr), define, design, develop and disseminate (4d), and analysis, design, develop, implement and evaluate (addie). in general, there are two classifications for the use of approaches, models, and methods for ethnomathematics-based teaching: (a) the use of specifically defined approaches, models, and methods to facilitate the integration of sundanese ethnomathematics in mathematics teaching in schools; and (b) the use of non-specifically defined sundanese ethnomathematics approaches, models, and methods or sundanese ethnomathematics-based mathematics teaching. the following is a categorization of the approaches, models, and teaching methods used based on the analysis of 56 documents (see table 6). table 6. categorization of documents based on the sundanese ethnomathematics-based mathematics teaching approaches type of document model 1 model 2 total local journals 11 18 29 international journals 1 4 5 scopus-proceedings 3 4 7 undergraduate theses 3 11 14 master theses dissertations 1 1 total 19 37 56 the results of the categorization showed that there was a tendency to use model 2 (66.07%) than model 1 (33.93%) (see table 6). in model 1, it was shown that there were several teaching approaches, models, or methods that were predominantly used to carry out sundanese ethnomathematics-based teaching which is presented in table 7. table 7. teaching approach, models and method used in sundanese ethnomathematics-based teaching teaching approach, models and method n authors realistic mathematics education (rme) 5 (ardianingsih et al., 2019; irawan et al., 2018; irawan et al., 2019; mahpudin & sunanto, 2019; nugraha & suryadi, 2015) problem based learning 5 (anggara, 2019; fadillah et al., 2019; maulana et al., 2020; perdana & isrokatun, 2019; tuti, 2018) volume 11, no 1, february 2022, pp. 33-54 45 infinity teaching approach, models and method n authors contextual teaching learning 3 (kusuma, 2019b; nugraha et al., 2020; supriadi, 2014) somatic approach, auditory, visualization, and intellectually (savi) 2 (farokhah, 2015; farokhah et al., 2017) process oriented guided inquiry learning (pogil) 1 (fakhruddin & masrukan, 2018) learning cycle approach 1 (sariningsih & kadarisma, 2016) guided invention 1 (permatasari, 2016) hypnoteaching 1 (kusuma, 2019a) total 19 based on table 7, realistic mathematics education (rme), contextual teachingleaning and problem solving, wich were cosidered very relevant for mathematics teaching design. however, the authors will not specifically discuss this model 1 because it is not the focus of this research, and the sundanese ethnomathematics teaching design will adjust to the approaches, models, and methods used. as for model 2, the use and development of ethnomathematics-based teaching did not explicitly show the use of approaches, models, and other methods such as in model 1. in model 2, ethnomathematics teaching design was developed in the form of developing and using teaching material, teaching media, tradisional game, ethnomathematics as teaching approach as presented in figure 3. figure 3. the form of ethnomathematics teaching design the following is a description of sundanese ethnomathematics teaching based on the types of teaching components found from document analysis. 3.3.1. ethnomathematics teaching materials ethnomathematics teaching materials are teaching materials that are systematically organized using sources derived from local culture and wisdom, which can be in the form of printed materials, audio, video, and multimedia (arisetyawan, 2019). for example, teaching materials based on the cirebon people's local culture were a set of mathematics teaching materials packaged using the cirebon community's local cultural content (mahpudin & yuliati, 2019). students were introduced to several objects with cultural and mathematical elements in teaching materials, such as unique local sundanese shapes and objects related to mathematical elements (imswatama & lukman, 2018). the development of teaching materials follows the general stages so that in developing sundanese ethnomathematics11 5 8 14 teaching material teaching media traditional games ethnomathematics as a teaching approach lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 46 based teaching materials, the teacher must explore and select sundanese cultural elements that will be used as a context relevant to the mathematical content being studied by students. however, the teaching materials must contain at least three elements, namely: (a) problems related to the sundanese culture which are related to mathematics; (b) cultural values contained in sundanese culture; and (c) the design of contextual problems that can be solved by mathematics (supriadi, chudari, et al., 2019). 3.3.2. media for ethnomathematics teaching the use of media for teaching mathematics is adjusted to teaching needs that are relevant to teaching design and teaching materials such as using traditional sundanese cultural tools in the classroom even though in the form of artificial models, such as household tools, agricultural tools, clothing materials such as batik, crafts, and art. there are only a few documents that showed the use of media as a focus in research, but the use of media as a teaching component was a part of sundanese ethnomathematics teaching. some documents were considered emphasizing the use of media, one document related to the use of sticks for learning to count (citra, 2017), the rest of them showed the use of computer-based technology to help visualize the context of ethnomathematics-based teaching materials (ferdianto & setiyani, 2018; sudirman et al., 2020). 3.3.3. traditional games apart from teaching materials, the use and development of traditional games for ethnomathematics-based teaching were more dominant than the media, mainly due to the need for an attractive and fun teaching approach in schools. several traditional games had been adopted, such as engklek (similar to hopscotch) (rahmawati et al., 2017), endogendogan (supriadi & arisetyawan, 2019), marbles games (pratiwi & pujiastuti, 2020), and gasing (similar to wooden spinning top) (febriyanti et al., 2018). teaching was designed by including play activities as the main element of learning activities or subsidiary learning activities to build a pleasant atmosphere for students so that they can feel the richness of their culture. 3.3.4. ethnomathematics teaching design in model 1, the development of ethnomathematics teaching design refers to specific approaches, models, and methods used, such as following the principles and designs of teaching rme, pbl, and ctl. while in model 2, the teaching design is designed to adjust to the general design of teaching development and is relevant to teaching materials, media, and types of traditional games used. the following is a model of the stages of ethnomathematics teaching that is quite clearly presented (supriadi, chudari, et al., 2019), relevant to the ethnomathematics curriculum unit framework in figure 1 (adam, 2004). from all the documents analyzed, the rest of them had not developed an original model. on the contrary, they applied a teaching approach, model, and method or developing teaching based on the current national curriculum guidelines by integrating teaching materials, media, and traditional games. it is challenging to conduct research and development of sundanese ethnomathematics teaching designs that are compatible with the indonesian national curriculum model and become one of the curriculum innovations that can be applied in schools. based on the analysis of 56 documents, it was indicated that the implementation of sundanese ethnomathematics based teaching was mostly carried out at the elementary (32 documents) and junior high school (14 documents) levels and rarely at the senior high school level (2 documents) and early childhood level (1 document). some teaching activities were volume 11, no 1, february 2022, pp. 33-54 47 infinity conducted on mathematics students and students of the department of primary school teacher education and mathematics education (7 documents). 3.4. mathematics learning objectives with sundanese ethnomathematics approach d'ambrosio introduces the trivium mathematics curriculum as a concept that combines literacy, matheracy, and technocracy to place mathematics in a sociocultural context that is relevant to students (rosa & orey, 2015). trivium mathematics curriculum recommends a pedagogical activity related to problem-solving, modeling, critical assessment, and understanding mathematical ideas, procedures, and practices from students' sociocultural environment (rosa & orey, 2015). in addition to those activities, which can be used as ethnomathematics learning objectives, the objectives of learning mathematics for all levels of education also include mastery of content standards including numbers, geometry, and measurement as well as data processing and process standards including mathematical understanding, mathematical problem solving, mathematical connections, and mathematical representation mathematical communication skills (joyner & reys, 2000). in another version, it is stated that mathematics learning objectives are called mathematical proficiency, which includes conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (kilpatrick, 2001). learning objectives can also refer to a taxonomy of learning outcomes such as bloom's taxonomy, or various abilities that are relevant to mathematics learning objectives. the following shows some of the competencies and mathematics learning outcomes that are the onjectives of learning mathematics. the data presented may not be by the number of articles because a study may contain more than one competency variable and mathematics learning outcome. therefore, the data is presented cumulatively and not based on document categories. table 8. categorization of documents based on mathematics learning objectives mathematics learning objectives total/ variabel creative thinking 10 general learning outcomes; mathematical understanding; problem-solving 9 mathematical disposition 6 (ethno)mathematical literacy; mathematical modeling 4 geometrical thinking; critical thinking; mathematical communication 3 algebraic thinking; student participation; mathematical connection; respect to cultural values 2 mathematical abstraction; mathematical thinking; metaphorical thinking; cognitive style; reflective thinking; mathematical generalization; hots; self-regulated learning; motivation; mathematical reasoning; resiliency 1 table 8 shows that 25 mathematical competencies are the objectives of ethnomathematics-based teaching, which were found om 56 documents; among the five competencies were the affective domains (mathematical disposition, independent learning, respect for cultural values, motivation, and resilience). the general learning outcomes in question were that the documents did not mention a specific competency but mentioned the mathematics learning outcomes based on general competencies. three salient competencies that appear in the sources as competencies developed through ethnomathematics learning are creative thinking (10), problem-solving (9), and mathematical understanding (9). it shows that ethnomathematics-based mathematics lidinillah, rahman, wahyudin, & aryanto, integrating sundanese ethnomathematics into … 48 teaching has relevance to mathematics learning objectives, especially to develop students' thinking skills. 4. conclusion the studies of sundanese ethnomathematics and its integration into mathematics teaching in schools in indonesia, based on the search results, only started in 2011, while ethnomathematics studies as a field of study began in 1984. this study shows that sundanese ethnomathematics studies are still developing by adopting various thinking frameworks developed in other countries or other communities. ethnomathematics studies in indonesia continue to develop and show the uniqueness that other countries do not have because indonesia is a country with various cultures, ethnicities, and languages, which have the potential for various ethnomathematics studies. ethnomathematics studies and their application in teaching in schools are expected to help students appreciate, preserve, and develop their cultural wealth. the study of ethnomathematics can be broadly divided into two, namely, the study of mathematics and culture and integrating ethnomathematics into the curriculum and teaching in schools. through the study of mathematics and culture, we can explore cultural wealth relevant to mathematics as a cultural heritage reflected in the seven cultural elements including social elements, language elements, scientific systems, technology systems, religious systems, livelihood systems, and art systems. meanwhile, curriculum and teaching development that integrates ethnomathematics can be used as an alternative in order to create more contextual mathematics teaching-learning that is close to student culture so that it can build the ability of various mathematical thinking and mathematical literacy and foster appreciation for the wealthy local wisdom of sundanese culture and diverse indonesian cultures. this systematic literature review research has presented the development of research on sundanese ethnomathematics both on the relationship of mathematics and culture as well as the integration of sundanese ethnomathematics in the curriculum in learning. this research will serve as a reference for future research on this topic. the studies that have been carried out have not shown a complete model of the integration of sundanese ethnomathematics into the school curriculum. even so, alternative approaches and learning models can continue to be developed. although rme, pbl and ctl can be a strong foundation for teaching sundanese ethnomathematics, it is necessary to develop an approach and model that is more relevant to the character of students and the curriculum model that applies in indonesia. references abdullah, a. s. 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(2019). ethnomatematics : how does cigugur traditional community use palintangan on farming. journal of physics: conference series, 1265(1), 012025. https://doi.org/10.1088/1742-6596/1265/1/012025 https://doi.org/10.1088/1742-6596/1521/3/032006 https://doi.org/10.1088/1742-6596/1188/1/012104 https://doi.org/10.26803/ijlter.18.11.9 https://doi.org/10.17051/ilkonline.2020.730747 https://doi.org/10.1088/1742-6596/1567/2/022087 https://doi.org/10.1088/1742-6596/1567/2/022087 https://doi.org/10.53400/mimbar-sd.v3i1.2510 https://doi.org/10.1088/1742-6596/1318/1/012126 http://repository.upi.edu/32594/ https://doi.org/10.1088/1742-6596/1265/1/012025 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p87-102 87 high school students’ error in solving word problem of trigonometry based on newman error hierarchical model thalitha ariesti widhia wardhani*, deshinta p.a.d. argaswari sampoerna university, indonesia article info abstract article history: received jun 29, 2021 revised dec 27, 2021 accepted jan 10, 2022 this study aims to identify students’ errors in solving word problems of trigonometry according to the newman error hierarchical model. this study uses qualitative descriptive data gathered based on fact, and data analysis presented descriptively. the population is students of x ipa 2 at sman 1 cikarang utara, west java, indonesia, and to understand more about the error that happened, the researcher took six students of sman 1 cikarang utara as a sample. the sample is chosen by purposive sampling. instrument tests, interviews, and documentation are the data collected. this study used the newman error hierarchical model to analyze high school students’ errors in solving word problems of trigonometry. the results show that students made errors in comprehension, transformation, process skill, and encoding stage. the errors were caused by their lack of understanding some words and terms in the word problem, lack of ability in transforming the word problem to the mathematical model and strategy, lack of processing the algebra and calculation, lack of motivation, and carelessness. keywords: high school students, newman error, students’ error, trigonometry, word problem this is an open access article under the cc by-sa license. corresponding author: thalitha ariesti widhia wardhani, mathematics education department, faculty of education, sampoerna university jln. raya pasar minggu kav. 16, pancoran, south jakarta, dki jakarta 12780, indonesia email: ariestytalita@gmail.com how to cite: wardhani, t. a. w., & argaswari, d. p. a. d. (2022). high school students’ error in solving word problem of trigonometry based on newman error hierarchical model. infinity, 11(1), 87-102. 1. introduction trigonometry is one of the topics that high school students learn when studying mathematics. it is the study of integrating the relationship between sides and angles (franklin, 2006). furthermore, it covers the study of triangles, the relationships between their sides and angles, the functions of sine and cosine, tangent and cotangent, secant and cosecant (walsh et al., 2017). trigonometry is a critical topic in mathematics in order to prepare students for advanced mathematics, including calculus (hidayat & aripin, 2020; hidayat & riyana, 2021). aside from its importance for advanced mathematics, trigonometry contributes to many applications in non-mathematical fields such as https://doi.org/10.22460/infinity.v11i1.p87-102 https://creativecommons.org/licenses/by-sa/4.0/ wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 88 architecture, civil engineering, cartography, geophysics and other advanced fields (galarza, 2017). therefore, it is important for students to understand trigonometry well. however, students found it abstract and difficult compared to other mathematics subjects (gur, 2009). it is because students have learning complexity in develop the understanding of subject matter content knowledge of trigonometry concept. moreover, students have obstacles about trigonometry, especially when they are given word problems. according to dewanto et al. (2017), word problems are mathematics problems that provide opportunities for students to support them linking the relationship between mathematics and real-life context. it is not only depends on students’ ability to perform the required mathematical operations, but also on the extent to which they are able to accurately understand the text of the word problem (boonen et al., 2016). students need to be able to solve word-problems because it is a crucial learning outcome and is considered an essential skill in mathematics (cuevas, 2000). moreover, word problem tasks can also be used to assess the level of comprehension to the mathematical concepts (khoshaim, 2020). however, students consider solving word problems difficult (fatmanissa & sagara, 2017; haryanti et al., 2019; said & tengah, 2021; sanwidi, 2018). moreover, it is shown that more than 50% of teachers in indonesia complained about students’ difficulties in solving word problem (fatmanissa et al., 2020; fatmanissa & sagara, 2017). a trigonometry word problem is a mathematical word problem related to trigonometry. when solving the word problem of trigonometry, students require to possess both trigonometry understanding and word problem solving skill (arhin & hokor, 2021; hamzah et al., 2021; mensah, 2017). for some cases, students also require doing multi-steps to solve the problem. studies reported that solving a mathematical word problem with multisteps can create more error in obtaining the right answers (dewanto et al., 2017). moreover, the report of the national examination result of high school of 2018 shows that there are only 33.05% of students in indonesia, and 30.03% of students in west java, who answer the right answer when solving the word problem of trigonometry (arlinwibowo et al., 2021; rosidin et al., 2019). therefore, it is important to analyse high school students’ errors in solving the word problem of trigonometry. error analysis is a method that is used to identify the cause of students’ errors when they make consistent mistakes by looking for the pattern of misunderstanding (lai, 2012). error analysis is important because it can be a powerful tool to diagnose learning difficulties and consequently direct remediation in mathematics (borasi, 1987; hasanah & yulianti, 2020). by analysing the errors, teachers can provide instruction targeted to the area of students’ needs. several theories has been found to analyze the error made by students, one of the famous and detail theory is newman error hierarchical model. based on newman (1977), students’ errors involve five levels: 1) reading error, which is the error that students have when they are incomplete in reading the problem; 2) comprehension error, which is the error that students made in understanding and comprehending the problem; 3) transformation error, which is the error that students made when transforming the real problem given in the form of sentence to mathematical form and strategy; 4) process skill error, which is the error that students made in applying the strategy chosen to solve the problem; and 5) encoding error, which is the error that students made when they write the solution incompletely or wrong in giving the final conclusion. this framework is appropriate to be used in analysing students’ errors in solving word problems since newman error analysis provided a framework for considering the reasons that underlay the difficulties students experienced with mathematical word problems and a process that is to be determined where misunderstanding occurred (argaswari, 2016; white, 2010). volume 11, no 1, february 2022, pp. 87-102 89 considering those problems and the urgency that mathematics education community need to improve students’ skills in connecting mathematics and real-life context especially in the topics of trigonometry, this research intended to analyse the high school students’ error of grade x in solving word problems of trigonometry based on the newman error hierarchical model. 1.1. research question the research question to be answered by this research is what are errors made by high school students of grade x in solving word problems of trigonometry based on the newman error hierarchical model? 1.2. research objective the objective of this research is to identify students’ errors in solving word problems of trigonometry according to the newman error hierarchical model. 1.3. significance of research this research has significance for teachers, university, and other researchers. for teachers, this research can be used as a reference to improve and diagnose learning difficulties especially students’ error and consequently direct remediation in the process of learning and teaching of mathematics. moreover, this study can help teachers to design effective practice teaching strategies for the students in learning and solving the word problem of trigonometry. not only that, this research has an important implication for teachers in developing the test and tasks in the classroom so that the students can improve their understanding and ability in learning trigonometry, especially in the type of trigonometry word problems. for other researches, this research will enrich the study and investigation of what error that high school students made when solving the word problem of trigonometry. this research can contribute as a reference for other researchers who conduct research in the same area of study. 2. method in order to address the objective of this study, a qualitative descriptive was used as the research method in this study. according to lambert and lambert (2012), qualitative descriptive methods provide more in-depth examination and understanding of individual learners through their experiences. data gathered based on fact, and data analysis presented descriptively. in this case, researcher gather and analyze the error of individual learners made when solving the word problem of trigonometry in depth through their understanding, then present the data result descriptively. qualitative descriptive gives a comprehensive summarization of specific events experienced by individuals or groups of individuals (lambert & lambert, 2012). therefore, a qualitative descriptive is appropriate to be used in this study since this study aims to analyze the phenomenon of what errors that students made in solving word problems of trigonometry. there are several phases that are used in this study as the research procedure. firstly, the researcher begins with the step of preliminary study. the preliminary study includes identifying the research problem, the research question, the objective or purpose of the study and the literature review. research problem is the problem or issue that guides the need for conducting a study (creswell, 2012). based on the research problem chosen, the researcher wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 90 determines the objective and the research question addressed in the study. the research question of this study is: what are errors made by high school students in solving word problems of trigonometry based on the newman error hierarchical model? after identifying the preliminary study, the researcher develops the instruments to be used for collecting the data. the instruments to be used are instrument test and interview form. the instrument test should be validated by validators, while the interview form is not required to be validated since it was adopted from newman interview procedure that has been translated to bahasa indonesia which the language that students used. after validating the instrument test, researchers collect the data which are test and personal interview. before that, researchers determine the population that will be the target of this study, which is 10th grade students of sman 1 cikarang utara. in collecting the data, there are phases that the researcher conducts which are conducting the test and conducting the interview. before conducting the test, researcher make appointments with the school to process the research letter from the university as the permission to conduct this study. after that, researcher interview the teacher about the students and class context in which topics they are going to learn in mathematics. then, the researcher is allowed to conduct the research in one class. in conducting the test, researcher gives the instrument test to the students that consist of word problems of trigonometry. the test was conducted on march 8th 2021, and conducted through online meeting. after getting the test data, the researcher assesses students’ test and categorizes the student’s score (low, medium, high) in order to determine the sample to be interviewed. then, the researcher conducts a personal interview to the sample of six students which are about 20% of the total students in a class, which are 2 students with lower score, 2 students with medium score, and 2 students with higher score. the phase of collecting data interviews was conducted on a different day which is not far from the day of the test in order to avoid the unexpected event response of students such as forgetting about the questions, having low motivation to do the interview, etc. before analyzing the data, researchers need to check whether the data have completed or not. if not complete, the researcher should do the iteration to the first step that is needed. if the data have completed, then the researcher continues to analyze the data. after getting the complete data collection, the researcher analyzes the data based on newman error hierarchical model. this has be done by calculating the points on which error on the stage of newman’s whether it is on the reading, comprehension, transformation, process skill, or encoding error. finally, the researcher draws the conclusion and suggestion and presents it to the research paper. the chart flow of research procedure is given in figure 1. volume 11, no 1, february 2022, pp. 87-102 91 figure 1. research procedure wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 92 2.1. instrument and data collection technique there are two kinds of data that will be used. there are questionnaires in the form of a worksheet about the word problem of trigonometry, and personal interviews that will help researchers to investigate types of students’ error deeply. the instrument test contains four questions that are constructed of triangle trigonometry subtopic. the personal interview aims to identify subject’s emotions, feelings, and opinions regarding the particular problem that showed in their worksheets. the main advantage of personal interview is the involvement of direct contact between interviewers and interviewees so that the interviewers are able to get accurate screening, capturing verbal and non-verbal responses. some certain questions were prepared to guide the interview towards the objective of the research based on newman (1977). the instrument test is validated, and the face validity will be applied. face validity is the extent to which a test is subjectively viewed as covering the concept it purports to measure (taherdoost, 2016). the worksheet / instrument test has been validated by three validators. the researcher takes validity from the experts who are stakeholder in mathematics and also mathematics teachers in school. 2.2. data analysis the analysis of qualitative data used miles and huberman method (miles et al., 2018). this method consists of three stages: 1) data reduction, 2) data presentation, and 3) conclusion. the process of data reduction is based on the data result which includes the aspects that are not related with this research. stage of data presentation, researcher will present data in the form of words, sentences, and tables. the researchers used a scoring scale adopted from rohmah and sutiarso (2018) to see the scores of students working on questions. this scoring scale is done by referring to the results of the five steps in solving the trigonometry problem. 3. results and discussion 3.1. results the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. the results should include the rationale or design of the experiments as well as the results of the experiments. results can be presented in figures, tables, and text. in order to see the overview students error of the population, the researcher collected the instrument data from 31 students grade x at sman 1 cikarang utara. the data instrument results of students’ error in solving trigonometry word problems are analysed using newman error hierarchical model. the summary of the data results is presented as the table 1. table 1. summary of students’ error indicators problem total (%) 1 2 3 4 reading errors 0 0 0 0 0 0 comprehension errors 6 22 12 22 62 17.77 volume 11, no 1, february 2022, pp. 87-102 93 indicators problem total (%) 1 2 3 4 transformation error 8 25 15 31 79 22.63 process skill errors 18 25 30 31 104 29.8 encoding errors 18 25 30 31 104 29.8 total 50 97 87 115 349 100 table 1 shows a summary of the data results of students’ errors in solving word problems of trigonometry. based on the newman error hierarchical model, there are five types of errors which are reading error, comprehension error, transformation error, process skill error, and encoding error. the percentage column shows the percentage of errors frequency for each type of error made, from the total number errors made by students. based on the data, there are found 0% in the reading error, 17.77% in the comprehension error, 22.63% in the transformation error, 29.8% in the process skill error, and 29.8% in the encoding error out of all students’ error. after that, six students from each level of low, medium and high are chosen and interviewed deeply to identify the students’ errors. the six students are the sample to be interviewed, and it has been discovered that there are some errors types students made in solving the word problem of trigonometry. since the newman error hierarchical model classified errors based on the level of hierarchy, so the initial error made might cause the subsequent errors in the next level of errors. the coding result of error types that the subjects have in each problem can be seen on the table 2. table 2. matrix code of initial students’ error subject sine cosine tangent elevation depression 1 2 3 4 a7 transformation error transformation error comprehension error comprehension error a21 transformation error transformation error comprehension error comprehension error b4 transformation error comprehension error comprehension error b3 comprehension error process skill error comprehension error c30 process skill error comprehension error c23 process skill error comprehension error wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 94 3.1.1. comprehension error all the samples experienced the comprehension error. the error happened when students worked on problems 2, 3, and 4. students who have errors in comprehension are mostly caused by their lack of understanding and comprehending some words on the problem. student a7 has an error in comprehending the problem. she writes the information given about what is known incompletely. she did not write and understand the angle of elevation which is the essential information component in the word problem. not only that, she also did not comprehend the word “from the same ground height” which is the set-up component of the word problem. the set-up component gives the condition of the problem situation. it gives the setting of the problem condition that the angle between the building and ground is 90 degrees (see figure 2). therefore, it caused her not to use the right triangle in solving this problem as shown in the picture. she also did not write the question asked. because of that, she was also wrong in transforming the mathematical model. she drew the angle wrongly. figure 2. students’ comprehension error on problem 3 moreover, she did not understand the concept of trigonometry ratio in the right triangle. she said in the interview that the ratio of tangent is opposite over the hypotenuse. she also cannot decide the part of opposite and hypotenuse. based on her understanding, she assumes that the hypotenuse is adjacent. this error causes the error on the process skill and gives the wrong answer. student a21 has an error in comprehending the problem. in the written task, she writes the information given of what is known incompletely. she only wrote that the angles formed are 30 degrees and 60 degrees without any detailed explanation about what type of angles are given. she also did not understand the meaning of the angle of depression. it is shown in the written task that she drew the angle formed between the vessel’s view to tower and the horizontal line below the vessel (see figure 3). in the interview, it is shown that her understanding about the angle of depression was wrong. other sample students also experience the same typical error details. figure 3. students’ work on problem 4 volume 11, no 1, february 2022, pp. 87-102 95 3.1.2. transformation error it is found that the common transformation error is transforming the word problem into the visual mathematical representation. students with lower scores have errors in determining the position of length of the kite’s string, the position of the objects given, and the appropriate angle position in the mathematical model. not only that, some of them were also wrong in determining the strategy used in solving the problem. moreover, they were still confused about the position of sohcahtoa, in which part the adjacent, opposite, and hypotenuse lead on the right triangle. it refers to the concept of triangle trigonometry. for students with medium scores, they made errors in positioning the length into the visual mathematics model. even though they know the information component given and interview, they have difficulties in transforming it. for the students with higher scores, they have errors in transformation but not as the initial error. student a21 has an error in transforming the problem into a mathematical model. in the written task and the interview, she knows that the given information is the length of the kite's string is 120 m, the angle formed is 60 degrees, and the asked question is to find the distance between the ground and the kite. however, she cannot transform the kite’s string into the model well. she assumes that the length is always transformed as the horizontal line. she said in the interview that the concept is similar to the concept of a rectangle in which the length of the rectangle is located as the horizontal line, while the width of the rectangle is located as the vertical line. it is not true that the length is always the horizontal line. therefore, she has errors in transforming the mathematical model. as a consequence, she was wrong in deciding the strategy plan to solve the problem. she uses tangent instead of using the ratio of sine (see figure 4). she also was wrong in the process skill and the encoding in solving the problem. therefore, she has errors in transformation, comprehension, process skill, and encoding. figure 4. students’ work on problem 1 3.1.3. proces skill error students who have errors in process skill commonly are caused by their lack of calculation and doing the algebra. some students were wrong in finding the variable of asked questions in the algebra concept. not only that, some of them are also wrong in deciding the value of the trigonometry ratio. student b3 has errors in the process skill. after succeeding in reading, comprehension, and transformation; she failed to obtain the right ratio value of cos 30 degrees. in the written task, she writes 𝑐𝑜𝑠 30° = √2 2 . the right ratio for cos 30° should be √3 2 (see figure 5). because of that, the researcher interviewed the student to investigate this error. based on the written instrument task and the interview, the student was wrong in obtaining the ratio number of cos 30°. it is because she forgot the value of trigonometry ratio even though the value of trigonometry ratio table was provided in the instrument. as a result, wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 96 she was also wrong in giving the final solution. therefore, student b3 has errors in the process skill and encoding. figure 5. students’ work on problem 3 3.1.4. encoding error as can be seen from table 2, students made error in encoding not as the initial error, but as the consequence of the previous errors. it happens on subjects who has error in the previous stage of errors. it is found that mostly students did not write and make the conclusion for the problem solution. it is because they confused about the final solution since problem number 3 and 4 require multi-steps to be solved. as the consequence, students cannot decide the final solution, then cause the low motivation to finish the solution. 3.2. discussion the results show that the students made some errors in solving the word problem of trigonometry. the data result follows the newman error hierarchical model to analyse the errors. it shows that if students make errors in the first stage, the following stages will result in further errors. based on the results, students of sman 1 cikarang utara have errors in comprehending the problem, transforming the problem into mathematical models and strategy, processing the operations, and encoding the solution when solving the word problem of trigonometry. students have no reading errors in solving the problem. this finding is in line with the previous study of usman and hussaini (2017) and fatmanissa and sagara (2017) that high school students have no error in reading the problem. both studies show there were no reading errors found since high school students have good reading skills. the students were able to read the information and mathematical symbols completely. although the students were able to read the problem, they have errors in comprehending the mathematics word problem. it means that although they can read the problem completely, but they cannot understand the meaning of some words in the problem. all the six subject samples made initial errors in comprehending the problem. most of them made errors in comprehending the angle of depression. it is because students are not familiar with those words in daily real life. they did not understand, for the students with low and medium scores, and have misunderstanding, for students with high scores, about the definition of depression angle so that it caused the error. another error made by students with lower scores is they cannot interpret the key word in the information component of simple terms. according to sumule et al. (2018), students have this error because they cannot understand and interpret key words in the problem. while for students with medium score, students do not read the questions carefully so that there is unreadable information that leads them to carelessness. it can be concluded that students made error in comprehending the set up and information component, which is not explained in the previous research (dewanto et al., 2017) which is only explained that students have comprehension error because the volume 11, no 1, february 2022, pp. 87-102 97 problem is too long without any explanation in which problem component that students made the error. another error found in this research is the transformation error, where students were not able to transform the mathematical word problem into mathematical representation. it is found in student a7 where she was wrong in transforming and imagining the visual representation. it occurred because there is a difficulty of the students in recognizing and imagining the context in which a word problem is set, or their approach is altered by the context in which the word problem is given (gooding, 2009). not only that, there are students who have errors in transforming the mathematical word problem into mathematical strategy. they were wrong in determining the appropriate strategy in order to solve the problem. it is in line with nanmumpuni and retnawati (2021), that students were confused in choosing the right concepts of trigonometry to solve the problems. they were confused in deciding the trigonometry ratio that might be the strategy to solve the problem. it is because students have a lack of understanding the triangle trigonometry concept. they were confused in deciding which one is the opposite, adjacent, and hypotenuse. it is in line with the previous study that there are still many students who incorrect to determine ratio trigonometry (erlisa & prabawanto, 2019). process skill error is the error that students make when they are not able to process the procedural and the computation in carrying out the problem after deciding the mathematical strategy. students who have errors in process skill commonly are caused by their lack of calculation and doing the algebra. some students were wrong in finding the variable of asked questions in the algebra concept. it occurred because students have a lack of ability in doing algebra, while algebra takes an important role to be connected in trigonometry (demir & heck, 2013; hidayat & aripin, 2020). for the calculation error, students were wrong in determining the value of trigonometry ratio and calculating the operation. it is caused by the carelessness and lack of focus resulting in less precision (hidayat & riyana, 2021; kelly & mousley, 2001). in the encoding stage, there are some students who did not write and make the conclusion for the problem solution. for example, student a7 made encoding errors in a problem that required multi-steps in attempting the solution. she was confused about the final solution so that she did not find the distance between two objects. this is related to (dewanto et al., 2017) that problems requiring multi-steps will tend students to make errors. the fact that it can be solved by adding the result of the previous work that she has solved if she writes and comprehends what is asked. 4. conclusion the objective of this research is to identify students’ error in solving word problems of trigonometry according to newman error hierarchical model. the population of this research are 31 students of grade x ipa 2 sman 1 cikarang utara, and the sample are six students of grade x ipa 2. based on the instrument test results and interviews, it can be concluded that high school students made errors in solving word problem of trigonometry which are: 1) the percentage of errors in comprehension stage is 17.77% where students made error in comprehending and understanding the problem; 2) the percentage of errors in transformation stage is 22.63% where students made error in transforming the real problem given in the form of sentence to mathematical form and strategy; 3) the percentage of errors in process skill stage is 29.8% where students made errors in applying the strategy chosen to solve the problem especially in doing the algebra and calculation; 4) the percentage of wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 98 errors in encoding stage is 29.8% where students made error when they write the solution incompletely or wrong in giving the final conclusion. students who have errors in comprehension are mostly caused by their lack of understanding and comprehending some words and terms on the word problem. mostly they have errors in comprehending the set up and the given information component which led them to make errors in comprehending the situation and essential information of the problem given. then, students who have errors in transformation are caused by lack of ability in transforming the word problem to the mathematical model and strategy. moreover, they are wrong in determining the strategy used in solving the problem. it is because they were still confused about the position of sohcahtoa, in which part the adjacent, opposite, and hypotenuse lead on the right triangle. it refers to the concept of triangle trigonometry. while, students who have errors in process skill commonly are caused by their lack of processing the algebra and calculation. some students were wrong in finding the variable of asked questions in the algebra concept. not only that, some of them are also wrong in deciding the value of the trigonometry ratio. meanwhile, students who have errors in encoding are mostly caused by their lack of motivation in deciding and finishing the final solution. some students are also made error caused by carelessness. acknowledgements we would like to express our gratitude and thanks to all lecturers in the mathematics education department, faculty of education, sampoerna university for the help and support during the project. we also thank the students and teachers at grade x sman 01 cikarang utara, especially ibu nurita as the mathematics teacher who helped and allowed us in conducting the research. lastly, we would like to thank ibu ani, mathematics teacher sman 01 cikarang utara, and ibu namirah fatmanissa, mathematics lecturer sampoerna university, who helped us in validating the research instruments. references argaswari, d. p. a. d. 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(2010). numeracy, literacy and newman's error analysis. journal of science and mathematics education in southeast asia, 33(2), 129-148. https://doi.org/10.1088/1742-6596/947/1/012053 https://doi.org/10.1088/1742-6596/947/1/012053 https://doi.org/10.2139/ssrn.3205040 https://doi.org/10.9790/5728-1302040104 https://doi.org/10.1080/03323315.2017.1327361 wardhani & argaswari, high school students’ error in solving word problem of trigonometry … 102 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p285-296 285 project-activity-cooperative learningexercise model in improving students' creative thinking ability in mathematics m. afrilianto1*, tina rosyana1, linda1, tommy tanu wijaya2 1institut keguruan dan ilmu pendidikan siliwangi, indonesia 2guangxi normal university, china article info abstract article history: received feb 13, 2022 revised sep 19, 2022 accepted sep 20, 2022 mathematical creative thinking ability is essential to be mastered by students to create good learning results and achievement. to install creative thinking in students' minds, they are highly requested to apply an innovative learning model. among innovative learning models, project-activity-cooperative learning-exercise (pace) is the one to be considered suitable for improving the students' mathematical creative thinking. this study is an experimental research using a pretest-posttest control group design. it was conducted in three classes of ikip siliwangi students by applying different learning models. the first class was given the pace model treatment with geogebra, the second class was given the pace model treatment, and the third class was given direct learning. the instrument used tests. based on the data analysis, we can conclude that there are differences in improving creative thinking abilities among the students who get the pace model learning with geogebra (pace-g), the pace model, and direct learning (dl). the improvement of creative students who get pace and pace-g models is better than those who get dl. the progress of students' mathematical creative thinking abilities obtained from pace and pace-g models is a high category. in contrast, improving students' creative thinking abilities who acquire dl is categorized as a medium. keywords: mathematical creative thinking, pace this is an open access article under the cc by-sa license. corresponding author: m. afrilianto, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi jl. terusan jenderal sudirman no. 3, cimahi city, west java 40526, indonesia. email: muhammadafrilianto1@gmail.com how to cite: afrilianto, m., rosyana, t., linda, l., & wijaya, t. t. (2022). project-activity-cooperative learning-exercise model in improving students' creative thinking ability in mathematics. infinity, 11(2), 285-296. 1. introduction science, technology, engineering, and mathematics are important terms in mathematics education. as one of the implications of the statement, analyzing the four terms above simultaneously can help achieve meaningful knowledge needed in dealing with the development of science in this digital era. https://doi.org/10.22460/infinity.v11i2.p285-296 https://creativecommons.org/licenses/by-sa/4.0/ afrilianto, rosyana, linda, & wijaya, project-activity-cooperative learning-exercise model … 286 according to afrilianto et al. (2019), the teaching materials for mathematics at the college level are typically more difficult than those for mathematics at the school level. it is crucial for the students to be able to complete their math assignments at the college level, especially for their subsequent math course subject (luttenberger et al., 2018; sun et al., 2018; xu & dadgar, 2018). mastering arithmetic can benefit children in a variety of ways to shape their character or personality (rosyana et al., 2018). as a result, the student required learning activities that are appropriate. with pace learning activities, it is intended that the model will strongly motivate individuals to grow their knowledge. since this knowledge can only be communicated to active receivers, it is possible for individuals to construct knowledge that they have already learned. it is preferable for students to acquire the content first, present it in front of the class, and engage further with other peers (konopka et al., 2015; litster et al., 2020; lopez-caudana et al., 2020; maass et al., 2019). this way, the learning environment will be more active and dynamic under the direction of the teacher. as a result, when the teacher distributes the materials, they lack the necessary expertise to engage in class discussion with their classmates. students are obliged to have mathematical creative thinking skills. garrison et al. (2001) said that when creative and reflective thinking skills are developed, people tend to seek the truth, open minded, be tolerant of new ideas, and be able to analyze problems well, big, systematically, and individually critical thinking. creative thinking skill develops in someone, it produces a lot of ideas, making connections, has a lot of perspective on things, makes and does imagination, and cares about results (garrison et al., 2001). creative thinking is required to produce something relatively new. evans (1991) said that creative thinking looks at how we perceive things from a different perspective. briggs (2007) suggests that creative thinking can be identified by its aspects of novelty, productivity, and impact or benefit. novelty refers to the problems solving strategy used is relatively unique. productivity refers to the construction of the ideas or approaches that are generated as much as possible, while the impact or benefit refers to the benefit of the ideas that have been generated. this mathematical thinking skill is very relevant, considering that real-world problems are generally not simple and convergent, but are often complex and divergent, even unpredictable. the creative thinking skill is important in analyzing, synthesizing, and evaluating all arguments needed to make rational and responsible decisions (aizikovitshudi & amit, 2011; ersoy & başer, 2014; krisdiana et al., 2019; ülger, 2016). students should be directed to achieve this high level competence through varying, contextual, and open learning activities. based on the factors analysis, guilford (carbonell-carrera et al., 2019; nurdiana et al., 2020) found that there are five characteristics of creative thinking: (a) fluency, the ability to produce multiple ideas; (b) flexibility, is the ability to propound some solution or approach to the problems; (c) originality, is the ability to making decision of the ideas in originality, not cliche; (d) elaboration, is the ability to elaborate things in detail; and (e) redefinition, is the ability to review the problems from different perspective to what many already know. as for indicators of mathematical creative thinking skill used in research by tandiseru (2015), are fluency, simplicity, originality, and elaboration. the indicator of the mathematical creative thinking skills used are eloquence, flexibility, and novelty (hidayat et al., 2018). based on the description, the indicators used to measure mathematical creative thinking in this study are fluency, simplicity, originality, and elaboration. according to afrilianto et al. (2019), independent students will be able to locate the necessary learning resources. the student will look for a variety of learning barriers, such as volume 11, no 2, september 2022, pp. 285-296 287 poor learning environments, unclear content, and challenging subject matter, but these can be overcome so that student learning outcomes improve. lee (1999) created the pace model for statistical learning. there are four learning stages in this model: project (project), activity (activity), cooperative learning (cooperative learning), and exercise (exercise). additionally, lee (1999) found that pace model learners were more engaged in group projects and class discussions. the pace paradigm is built on the following tenets. lee (1999) said that prioritizing active learning when solving problems, (2) practice and feedback are crucial components in grasping new concepts, and (3) independent knowledge production under the lecturer's guidance. exercising a model project, activity, and cooperative learning (pace). pace model learning consists of four essential parts, specifically: (1) project. it is crucial to learning using the pace model (lee et al., 2000). according to laviatan (2008), the project is an example of creative learning that relies on problem-solving activities; (2) activity. the pace model's activities are designed to introduce students to new knowledge or ideas (lee et al., 2000). cooperative learning. through cooperative learning, there is a complementary exchange of information between students; and (3) exercise. through practicing, students can strengthen the concepts that have been constructed at the activity and cooperative learning stages. pace model will be good in collaboration using geogebra application. geogebra is software that is freely available for teaching and learning mathematics with features suitable for topics such as geometry and algebra (azizul & din, 2018). geogebra software is an interactive media that allows students to explore various mathematical concepts (kusumah et al., 2020). the use of geogebra in learning can help teachers improve student understanding of mathematical concepts and procedures (zulnaidi & zamri, 2017). kusumah et al. (2020) the use of geogebra can improve students' mathematical communication skills, it is recommended for mathematics teachers to use geogebra in geometry learning, especially in probability concept material. 2. method this study employs experimental research that uses pretest and posttest control groups design. it is conducted in three classes of ikip siliwangi students by applying different learning methods. the first class (experiment 1) is given the pace model treatment with geogebra software, the second class (experiment 2) is given the pace model treatment, and the third class (control) is given direct learning (direct instruction). quantitative data is collected through test giving. observations are done twice, the first one is before the learning process, which is called pretest and the second one is after the learning process, which is called posttest. in this study, the independent variables are learning (pace-g model, pace model, and dl), while the dependent variable is the mathematical creative thinking ability. 3. result and discussion 3.1. result table 1 shows the results of normality test for pretest data are presented. afrilianto, rosyana, linda, & wijaya, project-activity-cooperative learning-exercise model … 288 table 1. pretest data of normality test mathematical creative thinking ability grades kolmogorov-smirnov statistic df sig. pace-g 0.171 46 0.002 pace 0.145 39 0.039 dl 0.214 38 0.000 the results of the normality test showed that the three classes were not normally distributed (see table 1), this resulted in further testing using the kruskal-wallis test. the kruskal-wallis test was conducted to see if there was a difference in overall mathematical creative thinking ability between classes using the pace and geogebra models (experiment 1), pace model (experiment 2), and dl (control). the summary of the kruskal-wallis test on increasing mct ability is presented in table 2. table 2. pretest data of kruskal-wallis test of mct ability based on learning mean sig. h0 pace-g pace dl 2.36 2.35 2.10 0.715 accepted table 2 show that the significance values obtained are more than 0.05, then h0 is accepted. therefore, as the result, based on the pretest, there are no differences in mathematical abilities of creative thinking of students between the three classes. the results of testing the pretest data showed that there was no difference in students' initial mathematical creative thinking abilities, then analyzed the posttest data for mathematical creative thinking abilities with the results of the normality test presented in table 3. table 3. posttest data normality test of mathematical creative thinking ability class kolmogorov-smirnov statistic n sig. h0 experiment-1 0.123 46 0.080 accepted experiment-2 0.159 39 0.015 rejected control 0.215 38 0.000 rejected the results of the normality test of two classes that received pace (experiment 2) and direct learning (dl) learning obtained a significance value of 0.015 and 0.000 (see table 3). thus, the data is not normally distributed for classes taught using the pace model and direct learning (dl), while for other classes (experiment 1), the results obtained a significance value of more than 0.05. thus, the data is normally distributed for the class that learns using the pace model with geogebra (pace-g). because there are data that are not normally distributed, it is continued with the kruskal-wallis test to see whether there are volume 11, no 2, september 2022, pp. 285-296 289 differences in the mathematical ability of creative thinking between classes using the pace and geogebra (pace-g) learning model, the pace model, and direct learning (dl) as a whole. the summary of the mct ability difference test based on the posttest is presented in table 4. table 4. posttest data of kruskal-wallis test of mct ability based on learning mean sig. h0 pace-g pace dl 12.10 10.84 7.21 0.000 rejected the results of the posttest data show that there are differences in students' initial mathematical creative thinking abilities (see table 4), then the data on the gain of mathematical creative thinking abilities with the results of the normality test is presented in table 5. table 5. gain data normality test of mathematical creative thinking ability learning kolmogorov-smirnov statistic n sig. h0 pace-g 0.073 46 0.200 accepted pace 0.099 39 0.200 accepted dl 0.151 38 0.028 rejected based on the results of the normality test (see table 5), it was found that there was one class that was not normally distributed (dl), this resulted in further testing using the kruskal-wallis test. the kruskal-wallis test was conducted to see whether there was a difference in the overall increase in mathematical creative thinking skills between the classes using the pace model with geogebra (experiment 1), pace model (experiment 2), and dl (control). the summary of the kruskal-wallis test on increasing mct ability is presented in table 6. table 6. gain data kruskal-wallis test of mct ability based on learning mean sig. h0 pace-g pace dl 0.718 0.623 0.367 0.000 rejected table 6 show a significance value of less than 0.05. to put it another way, groups of students (population) that learnt using the pace-g, pace, and dl models showed varying increases in their mathematics creative thinking abilities in this study. then, obtained student responses in learning pace learning. supporting findings related to student opinions about the implementation of pace model learning were obtained from questionnaires and interviews. based on the results of the interviews, it was revealed that the pace and pace-g models generally made a positive contribution in improving students' mathematical problem posing and creative thinking skills. students admitted that afrilianto, rosyana, linda, & wijaya, project-activity-cooperative learning-exercise model … 290 learning the pace and pace-g models actually helped in improving their understanding of the cone slice material. they are very enthusiastic in participating in every stage of learning the model which is supported by the existence of the lkm. the learning role of the pace model, which has a positive contribution to understanding the cone slice material, is also strengthened by the results of an open questionnaire. the results of the study of open questionnaires (free comments) showed that all students had feelings of pleasure towards the lectures they attended. students feel that pace model learning provides opportunities for them to complete projects through learning activities with cooperative learning and individual and group exercises, thus making it easier for them to understand the material. likewise, the impression of students in learning the pace model with geogebra (pace-g) turned out to be student interest and it was seen both while studying and after lectures. these results can be seen from the student's comments, one of which is revealed in figure 1. figure 1. student comments on learning besides that, obtained of student responses from closed questionnaires that were filled directly by students were also obtained by choosing answers very agree (ss), agree (s), disagree (ts), and very disagree (sts), can be presented in figure 2. figure 2. recapitulation of student opinions related to pace and pace-g figure 2 shows that the percentage of students who give a very high agree response is 60%, meaning that students are interested in learning with the pace and pace-g models. in addition, there are also the working on mathematics problem from students who have 30% 60% 9% 1% sts ss ts s volume 11, no 2, september 2022, pp. 285-296 291 learned pace, pace-g, and dl. for indicators of mathematical creative thinking ability, namely fluency. the overall average achievement of students' mathematical creative thinking skills who received pace-g and pace learning models was higher than students who received direct learning (dl). in other words, students who received the overall paceg and pace model learning on the "fluency" indicator experienced lower difficulties in solving mathematical creative thinking skills than students who received direct learning (dl). to strengthen the descriptive results, it is necessary to analyze student answers next. in order to obtain further analysis related to the difficulties experienced by students in solving mathematical creative thinking skills on the "fluency" indicator, the analysis of student answers will be carried out based on the level of student learning independence. the question of mathematical creative thinking skills that reveals "fluency" is in number 1, namely: the equation of the parabola is x2 + 8x – 4y – 16 = 0 determine the coordinates of the extreme poin, focal point, directrix equation, and the lotus rectum. for students with a high level of learning independence (tkb) who receive paceg and pace learning models, generally do not experience significant difficulties, only the accuracy factor makes the answer wrong. for example, the following answers are presented by m-1 students (see figures 3 and 4), as representatives of students with high early learning independence who received pace-g and pace learning models. figure 3. m-1 student answers on test about fluency indicator figure 4. m-1 student answers based on geogebra afrilianto, rosyana, linda, & wijaya, project-activity-cooperative learning-exercise model … 292 for students with moderate tkb who receive pace-g and pace learning models, generally they do not experience significant difficulties, but sometimes they are not checked in detail. for example, in the following, the answers of students m-17 are presented as representatives of students with moderate tkb who are learning the pace-g and pace models. figure 5. m-17 student answers related to fluency indicators figure 5 show that m-17 students did not experience too many difficulties, but they were not described in detail. after checking, it turned out that the answer was correct, but the score obtained was not optimal because it was not described or checked again. this finding was strengthened by the results of interviews with representatives of students with moderate tkb who received pace-g and pace learning models, that they admitted that they did not experience too many difficulties, only that they wrote the answers directly on the answer sheet without elaborating or re-checking in detail. as a result, the score obtained by the student is not optimal. meanwhile, for students with tkb level who received dl, some were still confused in answering questions or were not careful in answering questions, so the answers were wrong. for example, in the following, the answers of students m-112 are presented as representatives of students with low tkb who received dl (see figure 6). figure 6. m-112 student answers related to fluency indicators volume 11, no 2, september 2022, pp. 285-296 293 the students with low tkb who get the pace-g model learning, and pace on the "fluency" indicator, generally experience lower difficulties than students who get dl in solving mathematical creative thinking skills problems. this finding is supported by the achievement scores and the improvement of students' mathematical creative thinking skills with low tkb on the "fluency" indicators of learning (pace-g, pace, and dl models) which concludes that students with low tkb who get learning the pace-g and pace models have a higher average of achievement and improvement than students who received dl. based on the overall analysis, it can be seen that most of the students who received the pace and pace-g models did not experience difficulties in working on mathematical creative thinking skills on the "fluency" indicator, both students with high, medium, and low tkb. it's just that it still requires better accuracy. for students who received dl varied, namely students with high and medium tkb, in general they did not experience difficulties, although they had to be more careful. meanwhile, for students with low tkb who received dl, some students were still confused in answering questions or were not careful in answering questions, so the answers were wrong. this is of course the fact that the index/difficulty level of mathematical creative thinking skills for the “fluency” indicator is 0.287 and is categorized as difficult. 3.2. discussion based on the analysis of research results about the achievement and improvement of students 'mathematical creative thinking abilities influenced by learning factors with the project-activity-cooperative learning-exercise (pace) and direct learning (dl) models, it was found that there are some differences. the achievement and improvement of mathematical creative thinking abilities of students who learn by learning the pace and pace-g models are better than those learning with dl. in addition, there are differences in the achievement and improvement of students' mathematical creative thinking abilities based on the level of learning independence. this invention is based on the average score of achievement and improvement of students' mathematical creative thinking abilities in terms of learning factors. the results of this study indicate that students who learn by learning the pace model are helped in developing mathematical creative thinking skills through stages: (a) projects, (b) activities, (c) cooperative learning (cooperative learning), and (d) exercise (exercise). for groups that receive pace-g model learning, students are also helped by the use of geogebra software in completing student worksheets / lkm at learning meetings. the effect of pace model learning on the achievement and improvement of mathematical creative thinking abilities due to the characteristics of pace model learning is also focused on fostering and developing students' mathematical creative thinking actively through project assignments, activities in cooperative learning, and exercises. stages in learning the pace model can develop mathematical creative thinking ability. creative thinking abilities need to be developed especially in facing the information age. someone with creative thinking abilities will grow healthy and face challenges (behnamnia et al., 2020; yaniawati et al., 2020). to develop creative thinking abilities, lecturers must create class conditions that stimulate students' sensitivity through assignments by raising several questions, such as: "how if," "what is wrong," "what will you do," and to settle the problems with variety of ways (krulik & rudnick, 1999). the implementation of project-activity-cooperative learning-exercise (pace) and pace with geogebra (pace-g) learning models have a positive influence on mathematical creative thinking abilities, so that they are worthy of being used as learning models at afrilianto, rosyana, linda, & wijaya, project-activity-cooperative learning-exercise model … 294 campus. these learning models can be used to improve mathematical creative thinking abilities. judging from the study, these findings are similar to the findings of previous studies. the finding is that the improvement of various mathematical abilities of students who received pace model learning was overall better than students who received conventional learning (lee et al., 2000; pearce & cline, 2006). the research findings of lee (1999) suggest that the pace model learning is able to train students to be able to construct new concepts by themselves by applying previously owned mathematical concepts (assimilation process) or even modifying other mathematical methods or concepts through the process. exploration in constructing new (accommodation process). hartman (1997) explains the relationship between the concepts of assimilation and accommodation with cooperative learning. assimilation is the entry of new information into an existing schema through a process of continuous exploration. meanwhile, accommodation is a change to the previous schema or the creation of a new schema so that we are ready to adapt it to the new information. the learning factor of the pace model is more instrumental in developing students' mathematical creative thinking abilities. this shows that learning the pace model makes different contributions to the level of ability called the zone of proximal development (zpd). vygotsky and cole (1978) defines the zone of proximal development as the distance between the actual level of development determined by the individual's ability to solve problems independently and the level of potential development determined by the individual's ability to solve problems with the help of others who are more mature or by collaborating with a partner who is more capable. 4. conclusion the study's findings indicate that students who receive direct learning (dl) on the probability concept, the pace model (pace), or both exhibit disparities in the growth of their mathematics creative thinking skills. students who use the project-activitycooperative learning-example (pace) and pace-g models for learning attain higher levels of mathematics creative thinking than students who use direct learning (dl). the success rate of students' mathematical creative thinking abilities as measured by the projectactivity-cooperative learning-example (pace) and pace-g models is high, whereas the success rate of students' direct learning (dl) abilities falls into the medium category. references afrilianto, m., sabandar, j., & wahyudin. 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(2017). the effectiveness of the geogebra software: the intermediary role of procedural knowledge on students’ conceptual knowledge and their achievement in mathematics. eurasia journal of mathematics, science and technology education, 13(6), 2155-2180. https://doi.org/10.12973/eurasia.2017.01219a https://doi.org/10.3390/math8122163 https://doi.org/10.2147/prbm.s141421 https://doi.org/10.1007/s11858-019-01048-6 https://doi.org/10.1007/s11858-019-01048-6 https://doi.org/10.1088/1742-6596/1567/2/022049 https://doi.org/10.22460/infinity.v7i1.p1-6 https://doi.org/10.1016/j.iheduc.2017.09.003 https://doi.org/10.16986/huje.2016018493 https://doi.org/10.1177/0091552117743789 https://doi.org/10.3991/ijet.v15i06.11915 https://doi.org/10.12973/eurasia.2017.01219a sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p271-284 271 online learning interaction of mathematics teacher in junior high school: a survey in the covid-19 pandemic ahmad*1, zulkifley mohamed2, eka setyaningsih1, chumaedi sugihandardji1 1universitas muhammadiyah purwokerto, indonesia 2universiti pendidikan sultan idris, malaysia article info abstract article history: received may 26, 2021 revised aug 1, 2021 accepted aug 6, 2021 this study aims to determine the online learning interactions carried out by junior high school teachers in the classroom during the covid-19 pandemic. the quantitative using survey was used as a research methodology. 141 mathematic teachers was selected as the subject of this research. a questionnaire of classroom interaction practice in an online class was used as a data collection technique. the result found that mathematic teachers’ interaction activity in online courses has a different level. the interaction process that mathematic teachers use is in preparing the students to join an online course and leading the discussion with the mean of 4.2 and 4.3. in contrast, the lowest interaction happens in interaction in giving feedback and interaction in closing activity with an average of 2.5. the research also found that 78.70% of mathematic teachers always provide direction to the students in starting the online class. 40.30% of them never ask students to correct incorrect assignments during online learning. keywords: classroom interaction, covid-19, mathematics teachers, online learning copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: ahmad, departement of mathematics education, universitas muhammadiyah purwokerto jl. kh. ahmad dahlan, purwokerto, central java 45133, indonesia email: ahmad@ump.ac.id how to cite: ahmad, a., mohamed, z., setyaningsih, e., & sugihandardji, c. (2021). online learning interaction of mathematics teacher in junior high school: a survey in the covid-19 pandemic. infinity, 10(2), 271-284. 1. introduction the use of online learning is currently being used in various fields, including in education, especially in the times of covid-19 (bao, 2020). online learning is growing because technology makes the learning process can be done more efficiently in terms of time, distance, and cost (abidah et al., 2020). despite the advantages of online class, it has also a limitation in the term of interaction (moore et al., 2011). the online learning interactions that occur in the classroom through communication media that utilize internet technology (alabdulaziz, 2021; borba et al., 2016; haleva et al., 2021; mulenga & marbán, 2020; rosa & lerman, 2011; sullivan et al., 2020). this can lead to a lack of interaction between teachers and students and students with others. classroom interaction is actually the most important element in all aspects in running the education (markee, 2015). without good interaction there will be no social life, social https://doi.org/10.22460/infinity.v10i2.p271-284 ahmad, mohamed, setyaningsih, & sugihandardji, online learning interaction … 272 groups nor the social system (schwarz et al., 2009). this kind of interaction will only occur when these individuals or human groups to cooperate, talk, competition or even holding disputes and so on. classroom interaction is a dynamic process that which concerns the relationship between individuals and individuals, between individuals and groups, or between groups and groups. teaching and learning interactions contain a meaning of the interaction activities of the teaching staff who carry out the task teaching on the one hand, with learning groups (students, students / learning subjects) who are carrying out learning activities on the other side (mercer & howe, 2012). in government regulation number 19 of 2005 concerning standards national education mentioned that the learning process in the class is held in an interactive, inspirational, fun, challenging, motivating students to participate actively, and provide sufficient space for initiative, creativity, and independence according to talents, interests, and physical and psychological development of students (ulum, 2020). to achieve a good classroom interaction in online class, the role of teachers is very important (webb, 2009). teachers in online class can do some pattern of interaction namely, asynchronous pattern and synchronous pattern (naranjo et al., 2017; rex & schiller, 2009). the difference in interaction patterns is characterized by the time in accessing learning activities. the asyncronous pattern is usually carried out by the teacher by providing all teaching materials and learning display materials so that students can access them at different times (bao, 2020). students also can respond to teaching materials and broadcast materials at different times. while syncronous is a pattern where the process of delivering subject matter is carried out simultaneously (bennett et al., 2008). educators present learning materials that are also directly accessible to students simultaneously. so, there is a collaborative process between students and teachers when learning activities happen (mawad, 2020). teachers also should be able to put the interaction in every stage of learning, namely: opening stage, connection stage, action stage and reflection and evaluation stage. in opening stage, the teacher provides instructions about the identity of the subjects and the learning objectives that must be achieved (hendriana, prahmana, & hidayat, 2018; irfan et al., 2020; nasution et al., 2021; van manen, 2016; watkins & scott, 2012). after this stage is completed, then the teacher provides the second stage, namely connection, which is to give positive questions about students' initial knowledge of the material to be discussed and connect the previous material with the material to be studied (payler, 2007). at this stage the teacher also builds psychological relationships with students. the action stage becomes the stage where students access the resources and teaching materials prepared by the teacher in the form of videos, interactive slides and e-books (van manen, 2016; watkins & scott, 2012). teachers need to provide more time for students to understand and read the material as well as provide time for consultation through applications such as zoom aplication, chatting, and so on. this process is a syncronous process between students and teachers. the next activity is reflection and evaluation; teachers can use a quiz or other online assessment (swan et al., 2006). from the data presented above the key to the interaction in online class is very important to investigate for the betterness of the intruction. thus, this study aimed at finding out the classroom interaction of mathematic class in online learning. 2. method descriptive quantitative using survey was used in this research. the subject of this research was 141 mathematic teachers from banyumas regency. the number was based on volume 10, no 2, september 2021, pp. 271-284 273 slovin’s formula to calculate the sample size necessary to achieve a certain confidence interval when sampling a population this research was done on september 2020. questionaire was used in this reaseach to investigatethe interaction process in online class during covid-19. the questionnaires contain closed questions type to respond by teachers with 20 questions. the organization of the questionnaire was adjusted to the instrument indicators based on the theoretical studies carried out. the indicators of the questionaire include : preparing the students to join online class, guiding students during the activity in online class , leading the discussion, giving feedback, and interaction in closing activity. the questionnaire in this research was made by likert scale, each variable provide 4 alternative answers such as very often (ss), often (s), seldom (j), and never (tp). the respondents fill one of the option from the alternatives that is suitable with them. after filling out the data of the questionaire, the data then calculated and analyze using descriptive stattstics to find out the average score and the percentage of the result. 3. results and discussion 3.1. results statistic descriptive analysis used for analyzing data by describing collective data from each variable studied after the research was carried out so that it is easier to understand. the average of the teachers’ interaction process in online class during covid-19 times (see table 1). table 1. teachers’ classroom interaction in online class no interaction indicators average level of interaction 1 preparing the students to join online class 4.2 high 2 guiding students during the activity in online class 3.5 medium 3 leading the discussion 4.3 high 4 giving feedback 2.5 low 5 interaction in closing activity 2.7 low table 1 show that teachers’ classroom interaction in online class are varied in the level of interaction. preparing the students to join online class and leading the discussion have the high level of interaction with the mean of 4.2 and 4.3. guiding students during the activity in online class has a medium level of interaction with the average 3.5. the low interaction was found on giving feedback and interaction in closing activity with 2.5. in detail the classroom interaction in every indicators can be seen in following explanation. 3.1.1. preparing the students to join online class the result of mathematic teachers’ interaction in preparing the students to join online class can be seen on the table 2. ahmad, mohamed, setyaningsih, & sugihandardji, online learning interaction … 274 table 2. preparing the students to join online class number of items statements always often sometimes never 1 i give a direction to students in starting online class 78.70% 11.40% 9.40% 0.50% 2 i make sure all students are ready to start the online class 64.80% 9.70% 3.90% 21.60% 3 i contacted students who were unable to participate in online class 12.30% 50.40% 35.60% 1.70% 4 i provide solutions to students who have difficulty starting online class 59.00% 5.50% 32.20% 3.00% table 2 describes the classroom interaction conducted by mathematics teacher in preparing the students to join online class. the result showed that is always given by the tecahers is giving the direction to the students in starting the online class with 78.70%. 50.40% of tecahers often contacted students who were unable to participate in online class. moreover, 32.20% of teachers sometimes provide solutions to students who have difficulty starting online class. finally, 21.60% of tecahers never make sure all students are ready to start the online class. this shows that online learning is currently not used by students and teachers, so it is still necessary to adjust the use of media to be better in the online teaching and learning process (hebebci et al., 2020; herliandry et al., 2020; johns & mills, 2021; könig et al., 2020; rodríguez-muñiz et al., 2021). 3.1.2. guiding students during the activity in online class the result of mathematic teachers’ interaction in guiding students during the activity in online class can be seen on the table 3. table 3. guiding students during the activity in online class number of items statements always often sometimes never 1 i discuss the learning objectives students during online learning 28.70% 31.40% 29.40% 10.50% 2 i communicate learning materials to students 24.80% 39.70% 13.90% 21.60% 3 i guide every activity in online class 12.30% 50.40% 35.60% 1.70% 4 i check the activeness of student in online class 49.00% 25.50% 12.20% 13.00% volume 10, no 2, september 2021, pp. 271-284 275 table 3 describes the interaction in guiding students during the activity in online class. the result is varied. the majority of teachers always check the activeness of student in online class with 49%. 50.40% often guide every activity in online class, and 35.60% of them sometimes do it. finally, 21.60% of teahers never communicate learning materials to students. this happens because teachers are not used to interacting in online learning, making it difficult for teachers to guide students individually (hasan & khan, 2020; hebebci et al., 2020; johns & mills, 2021; könig et al., 2020; mehall, 2020; mumford & dikilitaş, 2020; rodríguez-muñiz et al., 2021; van den berg, 2020). 3.1.3. leading the discussion the result of mathematic teachers’ interaction in leading the discussion can be seen on the table 4. table 4. leading the discussion number of items statements always often sometimes never 1 i opened a questions and answer forum with students in online class 68.70% 11.40% 19.40% 0.50% 2 i asked students to ask their friend if they had trouble during online class 53.80% 9.70% 14.90% 21.60% 3 i chat directly with students who have difficulty during a learning 52.30% 40.40% 5.60% 1.70% 4 i answer all student questions in online classes 43.20% 5.50% 32.20% 19.10% table 4 discuss about the mathematics teacher classroom interaction in leading the discussion. it can be seen that the majority of mathematic teachers always used classroom interaction in every activity of leading the discussion with the students with 68.70%. 40.40% of them often chat directly with students who have difficulty during a learning. in addition, 32.20% of teachers sometimes answer all student questions in online class. finally, 21.60% of teachers never asked students to ask their friend if they had trouble during online class. this shows that the discussion in class is still dominated by the teacher. students still do not feel brave to ask questions or respond to questions from teachers or friends. thus, the ability to ask questions in mathematical learning still needs to be a concern to be developed better (bosch et al., 2018; franke et al., 2009; hendriana, 2017; hendriana, hidayat, & ristiana, 2018; hendriana, rohaeti, & hidayat, 2017; lim et al., 2020; mccarthy et al., 2016; steyn & adendorff, 2020; way, 2008). ahmad, mohamed, setyaningsih, & sugihandardji, online learning interaction … 276 3.1.4. giving feedback the result of mathematic teachers’ interaction in giving feedback can be seen on the table 5. table 5. giving feedback number of items statements always often sometimes never 1 i clearly communicate the assessment/ assignment given to students 2.20% 16.90% 39.40% 41.50% 2 i provide direct feedback on the results of online student assignments 1.80% 8.70% 63.90% 25.60% 3 i write a comment to each student regarding the assessment being done 10.30% 10.40% 55.60% 23.70% 4 i ask students to correct incorrect assignments during online learning 13% 16.50% 32.20% 40.30% table 5 discuss the mathematic teachers’ interaction in giving feedback to their students. the result showed that mathematic teachers who always and sometimest write a comment to each student regarding the assessment being done is 10.30% and 10.40%. in addition, 63.90% of them provide direct feedback on the results of online student assignments. the last, 40.30% of mathematic techers never ask students to correct incorrect assignments during online learning. this shows that in online learning, teachers are still adapting and need to improve their technological capabilities in managing online learning (hebebci et al., 2020; herliandry et al., 2020; johns & mills, 2021; könig et al., 2020; rodríguez-muñiz et al., 2021). so, the teacher can conduct a comprehensive assessment of all students (irfan et al., 2020; jackson et al., 2013; lee, 2014; ryve, 2011). 3.1.5. interaction in closing activity the result of mathematic teachers’ interaction in interaction in closing activity can be seen on the table 6. table 6. interaction in closing activity number of items statements always often sometimes never 1 i reflect on learning together with students 14.20% 21.90% 39.40% 24.50% 2 i give students the opportunity to work together to make a summary of the lesson 3.80% 9.70% 64.90% 21.60% 3 i ask students to discuss with each other in summarizing 12.30% 20.40% 35.60% 31.70% volume 10, no 2, september 2021, pp. 271-284 277 number of items statements always often sometimes never learning outcomes during online learning 4 i motivate students to always be motivated to follow online learning 33.00% 35,50% 31.20% 0.30% table 6 shows that the interaction in closing activity. 33.00% and 35.50% of mathematic teachers motivate students to always and sometimes motivate students to always be motivated to follow online learning. moreover, 64.90% of them sometimes give students the opportunity to work together to make a summary of the lesson. finally, 31.70% of mathematic teachers never ask students to discuss with each other in summarizing learning outcomes during online learning. this shows that the habits of teachers and students in online learning need to be improved, so that the online learning process will then become more meaningful (hebebci et al., 2020; herliandry et al., 2020; irfan et al., 2020; johns & mills, 2021; könig et al., 2020; rodríguez-muñiz et al., 2021). 3.2. discussion the teaching and learning process carried out in the classroom so far is often oneway, where students only listen to what the teacher says (tularam & machisella, 2018). therefore, the interaction in the class, especially in online class is low. interaction is also an important point in teaching and learning activities because not only students get the benefit, but also the teachers also get feedback (feedback) whether the material presented can be received by students well (lockyer & dawson, 2011). there are some components of interaction that should be prepare by teachers, namely: interaction in preparing the students to join online class, guiding students during the activity, leading the discussion, giving feedback and interaction in closing activity (payler, 2007). among those factors, this study revealed that interaction in preparing the students to join online class and leading the discussion was highly used by mathematic teachers, while interaction in guiding students during the activity is medium. the lowest interaction happened in interaction in giving feedback and closing activity. classroom online interaction skills to open lessons are activities carried out by teachers to mentally prepare and generate student attention (mawad, 2020). this is so that students focus on the things to be learned. activities to open lessons must not only be carried out by the teacher at the beginning of class hours but also at the beginning of each part of the activity from the core of the lesson given during that lesson (smith & higgins, 2006). to prepare students mentally for the lesson that will be studied, the teacher can make provide the students references and making connections between the subject matter that has been mastered by students with the new material to be studied. students who are mentally ready to learn are those who already know the objectives of the lesson, and the steps for learning activities to be studied. therefore, teachers should be warm and enthusiastic in building a good interaction with the students in the beginning of the lesson (fein & logan, 2003; metros, 2008). guiding discussion activities in learning is one of the teaching skills that must be mastered by the teacher, because through discussion students are encouraged to learn actively, learn to express opinions, interact, respect each other, and practice being positive (cohen, 1994; dallimore et al., 2004; hendriana, prahmana, & hidayat, 2018; irfan et al., ahmad, mohamed, setyaningsih, & sugihandardji, online learning interaction … 278 2020; nasution et al., 2021). through the discussion of the teacher's role, the impression that it is too dominating the conversation will automatically disappear. with discussion, students and teachers are both active, even though discussion it can facilitate an active learning process (cobb et al., 2001; jesionkowska, wild, & deval, 2020; koh & kan, 2021; tan et al., 2020; wang, 2020). giving feedback must be possessed by teachers (burnett & mandel, 2010; kulik & kulik, 1988). competent teachers can do an effective interaction in giving feedback for their students in order to create an effective learning environment and achieve a good student learning outcomes (ellis, 2009; montgomery & baker, 2007). student participation in learning should be given feedback by the teacher so that students are motivated to repeat these activities with better quality. thus, a teacher must be able to maintain student motivation in order to achieve optimal results when carrying out a learning process. the activity of closing lessons is an activity carried out by the teacher to end the core activities of the lesson (payler, 2007). efforts to close the lesson are intended to provide a comprehensive picture of what students have learned, to find out the level of student achievement and the level of success of the teacher in the teaching and learning process. efforts that can be made by the teacher include summarizing or asking students to summarize and evaluate the subject matter that has just been given (downer et al., 2010). like opening lessons, closing lessons must be carried out by the teacher not only at the end of class hours but also at the end of each part of the activity from the core of the lessons given during that lesson. like opening lessons, closing lessons also does not include sequences of routine activities such as giving assignments at home, but activities that have direct activities with the delivery of lesson material. 4. conclusion the conclusion in this study shows that the interactive activities of mathematics teachers in online classes have different levels. the interaction process used by mathematics teachers is in preparing students to take online courses in leading discussions. in contrast, low interaction occurs in interactions in providing feedback and interactions in closing activities. this happens because in indonesia, during the current covid-19 pandemic, all sectors do their work online, so the education sector is also affected by online learning. the impact is that teachers who are not familiar with online learning conditions are forced to keep up with speedy technological developments in the teaching and learning process. acknowledgements the authors would like to express my special thanks to universitas muhammadiyah purwokerto for supporting the facilities and financial to accomplish this paper. references abidah, a., hidaayatullaah, h. n., simamora, r. m., fehabutar, d., & mutakinati, l. 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(2009). the teacher's role in promoting collaborative dialogue in the classroom. british journal of educational psychology, 79(1), 1-28. https://doi.org/10.1348/000709908x380772 https://doi.org/10.36835/syaikhuna.v11i1.3845 https://doi.org/10.17718/tojde.803411 https://doi.org/10.4324/9781315417134 https://doi.org/10.1016/j.compedu.2020.103935 https://doi.org/10.1093/acprof:osobl/9780199740529.001.0001 https://doi.org/10.1348/000709908x380772 ahmad, mohamed, setyaningsih, & sugihandardji, online learning interaction … 284 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 1, february 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i1.236 77 the analysis of diagnostic assessment result in pisa mathematical literacy based on students self-efficacy in rme learning khaerunisak 1 , kartono 2 , isti hidayah 3 . ahmad yusril fahmi 4 1 universitas selamat sri, kendal east java, indonesia 2,3 mathematics education semarang state university, semarang east java, indonesia 4 teacher mts walisongo, pekalongan east java, indonesia 1 igetmydreams@yahoo.co.id, 2 pakarunnes@yahoo.com, 3 isti.hidayah@yahoo.com, 4 fahmiahmad@madrasah.id received: january 2, 2017 ; accepted: january 29, 2017 abstract this research aimed to test rme learning with an effective scientific approach to improving mathematical literacy and self-efficacy, obtaining an overview of the mathematical literacy diagnostic assessment results that has high, medium and low self-efficacy as well as student difficulties in learning rme with a scientific approach. this research using mix method concurrent embedded with the subject of research is students class viii. the research begins with a mathematical literacy diagnostic assessment and self-efficacy inventory then performed rme learning in experimental class and conventional learning in control class. quantitative analysis was conducted to test the effectiveness of learning and deepened with the interview as a qualitative analysis. learning rme with a scientific approach effective is marked by the achievement of classical completeness, the proportion of students' mathematical literacy, self-efficacy and the difference in pre-post students’ mathematical literacy on rme learning better than conventional learning. the results of students’ mathematical literacy diagnostic assessment fit the criteria of self-efficacy students except for medium mathematical literacy that having high self-efficacy. student difficulties in rme learning with the scientific approach are based on the results of mathematical literacy diagnostic assessment, namely language skills problem, the capacity to understand, create strategies, and create the algorithm. keywords: mathematics literacy, self-efficacy, rme, diagnostic abstrak penelitian ini bertujuan untuk menguji pembelajaran rme dengan pendekatan saintifik efektif meningkatkan literasi matematika dan self-efficacy, memperoleh gambaran hasil penilaian diagnostik literasi matematika yang memiliki self-efficacy tinggi, sedang dan rendah serta kesulitan siswa dalam pembelajaran rme dengan pendekatan saintifik. metode penelitian ini menggunakan mix method cuncurrent embbeded dengan subyek penelitian siswa kelas viii. penelitian diawali dengan penilaian diagnostik literasi matematika dan inventori self-efficacy selanjutnya dilakukan pembelajaran rme pada kelas eksperimen dan pembelajaran konvensional pada kelas kontrol. analisis kuantitatif dilakukan untuk menguji keefektifan pembelajaran. dan diperdalam dengan wawancara sebagai analisis kualitatifnya. pembelajaran rme dengan pendekatan saintifik efektif ditandai dengan tercapainya ketuntasan klasikal, proporsi literasi matematika siswa, self-efficacy siswa dan selisih literasi matematika awal-akhir pada pembelajaran rme lebih baik daripada pembelajaran konvensional. hasil penilaian diagnostik literasi matematika siswa sesuai kriteria self-efficacy siswa kecuali untuk literasi matematika sedang yang memiliki self-efficacy tinggi. kesulitan siswa dalam pembelajaran rme dengan pendekatan saintifik berdasarkan hasil penilaian diagnostik literasi matematika, yakni kesulitan kemampuan bahasa, kemampuan memahami, membuat strategi, dan membuat algoritma. kata kunci: literasi matematika, self-efficacy, rme, diagnostik khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 78 how to cite: khaerunisak, kartono, hidayah, i. & fahmi, a. y. (2017). the analysis of diagnostic assesment result in pisa mathematical literacy based on students self-efficacy in rme learning. infinity, 6 (1), 77-94. introduction education has an important role in educating the human resources to be able to compete globally in the development of science and technology. this is in accordance with national education goals outlined in act number 20 year 2003 on national education system. through education, students are equipped with knowledge and skills needed in school and its application in real life. students have certain aspects that can be measured and the results can provide useful information for improving the quality of education through the study. for example, the study programme for international student assessment (pisa), the international literacy study is a form of skills and knowledge evaluation that are designed for students aged 15 years, were carried out every 3 years under the auspices of the organization for economic co-operation and development (oecd). pisa aims to assess the 15-year-old students in oecd countries and other countries in the achievement of reading proficiency, mathematics literacy, and science to make a contribution towards its member of his country (wilkens, 2011). the results of pisa study in 2003, indonesia was ranked 39 out of 40 countries and the following year was also not encouraging. pisa results in 2009 showed that the mathematics literacy score for indonesia students ranked 61 out of 65 participating countries and the results of the latest pisa in 2012, indonesia was ranked 64 out of 65 survey participants countries. although the results of pisa 2015 have elevated points with indonesia was ranked 64 out of 72 countries. this indicates a mathematical literacy smp/mts students in indonesia is still low. according to dzulfikar, asikin & hendikawati (2012), mathematical subjects for many students still considered as a difficult lesson, scary, and less useful in everyday life, such as for many students a math lesson seemed difficult and unattractive. this case makes many students became less motivated to learn math and have an impact on student difficulties in solving mathematical problems caused by the inability of students to understand or remember the basic concepts of mathematics ever learned before. particularly with the condition of students in indonesia are not familiar with modeling form question, which requires the ability to translate everyday problems in the form of formal mathematics to completing it. thus, students’ mathematical literacy skills need to be cultivated so that the result of student learning outcomes increased in mathematics learning. cases that are often encountered by mathematics teachers, especially in smp n 2 wonopringgo, many students have not reached the minimum completeness criteria in math tests despite being held remedial against students who do not achieve completeness. based on the interview with one mathematics teacher at smp n 2 wonopringgo, the percentage of students who scored pure mathematics in daily tests above 67 is not more than 25%. thus, the proportion of students who achieve completeness learning is still low. this occurs when students are faced with a math problem associated with real problems, students have difficulty in interpreting the real problems into mathematical models or can be said mathematical literacy smp students is low. volume 6, no. 1, february 2017 pp 77-94 79 student difficulties in interpreting the real problem into a mathematical model needed to be diagnosed the source of the problem and held a follow-up to resolve the issue. one of them by having a diagnostic assessment. the diagnostic assessment in the form of diagnostic tests are given to know the strengths and weaknesses in learning (hughes in suwarto, 2013) so that learning can be improved and the learning objectives are achieved. sion & jingan (suwarto, 2013) states the diagnostic test as a test that provides information to teachers on students' prior knowledge and misconceptions before starting the activity. with the diagnostic assessment can know things that need to be repaired and improved and things that need to be maintained in the implementation of learning. assessment obliges teachers to gather information as complete as possible for the purpose of decision making of teaching so that teaching decisions can be precisely targeted, one of them is diagnostic assessment (hidayat, sugiarto & pramesti, 2013). according to shute, graf & hansen (2006) there are three aspects in the diagnostic evaluation that includes diagnostic process of determining the nature of a child's ability in a learning activity, the diagnostic process should be able to classify the students' cognitive abilities, diagnostics is part of a larger learning process with the main aim to identify problems and help overcome learning problems. based on the results of diagnostic assessments, needs to be followed in determining the strategies and appropriate learning methods about mathematical literacy. judging from the characteristics of mathematical literacy that is often associated with contextual issues appropriate if applied with a realistic mathematic education (rme) learning. in addition to the application of rme, need to hold scaffolding for some students who do not meet the kkm. in this process, the teacher explains the material that has not been mastered by students without looking at the concepts, principles, and procedures that are not yet fully understood by students. the application of the scientific approach that adopted scientists measures to build knowledge through scientific methods, 2013 curriculum can help students' skills in reasoning subject matter based on the evidence of the observable, empirical and measurable object. 2013 curriculum were applied in smp n 2 wonopringgo because included in the pilot schools (pilot project) from kemendikbud pekalongan. rme learning can be applied at smp 2 wonopringgo due to suitable with curriculum 2013 regarding rme characteristics that in line with the character of a scientific approach (implementation of 2013 curriculum). the pessimistic tendency of students in learning mathematics because students' views of mathematics that are still considered difficult and scary subjects make the daunting obstacles when students solve a math problem. students who thus have a timid soul, less bold in making decisions, and less daring responsible for the actions that have been carried out. therefore, it takes a strong self-efficacy on students so that they can succeed in learning mathematics. self-efficacy beliefs influence the choice of duty, endurance and persistence efforts and achievement. according to hacket & betz (nicolaidou & philippou, 2003) stated that the influence of selfefficacy on performance in mathematics as strong as the effect of general mental ability. self-efficacy has an impact on motivation, so it is also related to the success of students. a student who has high self-efficacy, if given the learning they will be enthusiastic/strive to demonstrate its ability to achieve success or otherwise (wigfield & eccles, 2001). many researchers report that self-efficacy (se) students correlated with the construction of the motivation, performance and student achievement. one of them is research done by zimmerman, bandura & martinez-pons (1992), that self-efficacy influences academic khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 80 achievement directly by increasing the value of students interest. pintrich and de groot (1990) found that students who believe that they can perform academic tasks using cognitive and metacognitive strategies more and keep doing better than students who do not believe. self-efficacy is making a difference in the way people act, as a follow-up of feelings and thoughts. people who believe that they can do something that has the potential to transform environmental events are more likely to act and more likely to succeed than those with low self-efficacy. behavior is influenced by the extent to which one believes can perform the actions required by certain situation. learning certainly requires the right strategy for learning optimally implemented. although learning has been applied, the need for improvement of learning to optimize student learning outcomes, the next step to improve learning by taking into account the results of a diagnostic assessment of realistic mathematic education (rme) learning approaches. rme developed by hans freudenthal have two views, i.e. mathematics must be connected to reality and mathematics as human activity (gravemeijer in tandililing, 2010). based on that idea, mathematics should be close to the students and should be relevant to everyday life situations. the situation that is relevant to everyday life will help the learning process that is meaningful to students so that students were able to find their own concepts and ideas of mathematics, must be mapped. in addition, he also emphasizes that mathematics as a human activity, so students should be given the opportunity to learn to perform activities of all the topics in mathematics. as a consequence, teachers must be able to develop interactive teaching and give students opportunities to contribute to their learning process. in connection with the background that has been described, presented some of the research questions as follows: (1) is the rme learning with scientific approach effective against to mathematical literacy and self-efficacy? (2) how do the results of the students' mathematical literacy diagnostic assessment in rme learning with scientific approach that has high, medium and low self-efficacy students? (3) how do the results of the students' mathematical literacy diagnostic assessment in conventional learning with a scientific approach that has high, medium and low self-efficacy students? (4) how is the student's difficulties on rme learning with a scientific approach based on the results of the mathematical literacy diagnostic assessment? method this study is a combination of qualitative and quantitative research. combinations model used in this study is the type of concurrent embedded strategy. in this study, quantitative research as the primary method while quantitative research as a secondary method. the population in this study were students of smp negeri 2 wonopringgo the second semester of the academic year 2014/2015. from classes viii in smp negeri 2 wonopringgo 3 classes randomly selected as samples in accordance with the study design, the first experimental class, which in the classroom experiment applying the rme learning model. secondly, the control class where the learning applying the learning model used by their teacher and the third is a trial class is a class that is used for the analysis of test trials. the scaffolding application is given when students are in the zpd. it is aimed so that is actual ability can be increased to a potential ability. determination of the students who are in zpd area through the calculation of an average student and a standard deviation of class, with intervals of the mean-sd<zpd<mean+sd. students located above the zpd does not need given the scaffolding because it has a high capability, while students are under the zpd volume 6, no. 1, february 2017 pp 77-94 81 can not be granted scaffolding because of his ability is too low so that students continue to experience difficulties if given assistance. there were 17 students who get the scaffolding from the results of the mathematical literacy diagnostic assessment at the end of each learning of meeting i, whereas in the meeting ii and iii there are 5 students who get the scaffolding. mathematical literacy diagnostic assessment 1 results got an average value of 5 and a standard deviation of 2, so the zpd area is located in 3<zpd<7. mathematical literacy diagnostic assessment 2 results get the average value of 3.58 and standard deviation of 6.35, so the zpd is located on 3<zpd<10. mathematical literacy diagnostic assessment 3 results got average value of 6.46 and standard deviation of 3.37, so the zpd is located on 3.1<zpd<9.8. the results of the mathematical literacy diagnostic assessment 4 scored an average rating of 7 and a standard deviation of 2. implementation of mathematical literacy diagnostic assessment 4 as habituation, so that the results of mathematical literacy diagnostic assessment 4 is not analyzed because the next meeting is already entering a final mathematical literacy diagnostic assessment. mathematical literacy diagnostic assessment final results get an average score of 73 and a standard deviation of 13, so the zpd is located at 6<zpd<86. data collection techniques in this research consisted of observation, testing, and interviews. data analysis was performed at a stage before the field up to the analysis stage during in the field. analysis before on the field is done by device and research instruments validation. analysis during in the field was compiled in a systematic quantitative and qualitative data that obtained from observation results, diagnostic tests, and interviews. quantitative data analysis that obtained from diagnostic tests data to determine the effectiveness of rme learning with scientific approach consists of completeness test with z-test, class completeness proportion test, different mean se samples class and different test of deviation mean pre-post ability of class sample. while the qualitative data analysis done by reducing the data, presenting the data, and draw conclusions from data that has been collected and verified these conclusions. recapitulation of data validation results are presented in table 1. table 1. recapitulation data validation results no type of data value category 1 learning media a. syllabus b. rpp c. teaching materials d. lkpd 93,5 90,7 87 89,1 highly valid highly valid highly valid highly valid 2 3 diagnostic mathematical literacy test question self-efficacy inventories 87,5 80 highly valid valid the calculation results of test validity question number 1 to number 13 a are all valid items except the items number 8. from the calculation of reliability test obtained price r_11 = 0.7064, and = 0.404. from the results of the reliability values valid bila . so the questions that was created form reliable questions. intake of such questions with consideration of the validity, reliability, level of difficulties, and distinguishing questions that meet the criteria. from the results of consideration, in this study used as many as 12 question i.e. question numbers 1, 2, 3, 4, 5, 6, 7, 9, 10.11, 12, and 13, who meets the criteria and includes components on mathematical literacy. khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 82 results and discussion results characteristics of mathematical literacy diagnostic question can be viewed from three aspects, namely the component processes, mathematical skills, and component context. table 2 is a table of data about the characteristics of diagnostic mathematical literacy. table 2. characteristic data of diagnostics mathematical literacy question characteristic count percentage students' average value process 1. formulate 3 79% 7.86 2. employ 4 67% 6.72 3. interpret 5 78% 7.47 mathematical capability 1. communication 1 95% 9.68 2. mathematising 1 80% 7.14 3. representation 1 85% 8.93 4. reasoning & argument 3 66% 6.79 5. devising strategies for solving problems 1 75% 6.11 6. using symbolic, formal and technical language and operations 1 65% 6.75 7. using mathematical tools 1 80% 6.93 context 1. private 6 77.14 7.71 2. work 3 65.95 6.60 3. social 1 76.07 7.61 4. science 3 72.86 7.29 percentage of mathematics literacy assessment framework in pisa 2012 is shown in figure 1 below. volume 6, no. 1, february 2017 pp 77-94 83 figure 1. the average percentage of mathematical literacy contain mathematical capability explanation: a = communication b = mathematising c = representation d = reasoning and argument e = devising strategies for solving problems f = using symbolic, formal and technical language and operation g = using mathematics tools according to the component context, the focus of pisa is meant as a situation which is reflected in a problem can be seen in figure 2 below. figure 2. the average percentage of mathematical literacy contain component context the question of mathematical literacy diagnostic test used in the study analyzes the learning process and students' mathematical literacy skills on geometry material class viii smp 0 10 20 30 40 50 60 70 80 90 100 a b c d e f g presentase 60 62 64 66 68 70 72 74 76 78 pribadi pekerjaan sosial ilmu pengetahuan p e rs e n ta se r a ta -r a ta khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 84 negeri 2 wonopringgo with sub-subject material cubes and blocks considering three large components identified in pisa, i.e. based on the components of the content or material learned in school. in this study, the researchers consider parts of space and form only because this section relates to the geometry subject. the questions of mathematical literacy ability test include three components which were tested in the pisa study as well as paying attention to the seven important things that support the occurrence of the process ability and also such questions created by observing the fourth component context the focus in pisa. the following will be outlined identifications numbers mathematical literacy diagnostic test item. question number 1 included in the category space and shape content components, process component parts of interpret, and included in a social context as well as the types of questions that could measure the highest value communication ability that achieved by eighth grade students when working on question 1 is 10 corresponding to maximum value of questions number 1, there were 25 eighth grade students who get the maximum value, but there is one student who received a minimum grade of 5. average correct answer reached 9.7 or equivalent to 97% of the average of correct answers obtained by the students in the class. question number 2 is a space and shape content components, process component that parts of formulate also include the types of questions that can measure the representation ability, and are included in the personal context. the highest value of question 2 is 10 which is the maximum value, while the minimum value is 3. the average of correct answers reached 8.9, equivalent to 89% of the average correct answer obtained by the students. question number 3 is a space and shape content components, process component that parts of intepret also include the types of questions that can measure the ability of using symbols, formal and technical language and operations. question number 3 is included in a social context. question number 3 highest score is 10 and also the maximum score while the minimum value is 2. the average of correct answers reached 7.6, equivalent to 76% of the average correct answers obtained by the students. question number 4 is a space and shape content component, process component that parts of formulate and types of question that can measure the ability of communication and include in a personal context. the highest value of question 4 is 10, and the maximum score number 4 is 10, while the minimum value is 1. the average correct answer of 7.1, equivalent to 71% of the average correct answer obtained by the students. question number 5 is a space and shape content component, process component that parts of employ and types of questions that can measure the ability of reasoning and argument also included in a personal context. question number 5 the highest score is 10 and the minimum value of question 5 that is 1. average of correct answers was 6.8, or an average of correct answers obtained by the students is 68%. question number 6 is a space and shape content component, process component that parts of employ, and include in the personal context as well as a type of mathematical literacy problems involving the ability to use mathematics tools. the highest value achieved was 10, whereas the maximum score that can be achieved by students when working on question 6 is 10, while the minimum value obtained graders viii to question 6 is 3. the average answer correctly answered by the student reaches 7,0 or equal to 70%. volume 6, no. 1, february 2017 pp 77-94 85 question number 7 is a space and shape content component, process component that parts of employ, and include mathematical literacy question that can involve the ability of devising strategies for solving problems, and included in the work context. the highest value a question number 7 is 10, while the minimum value is 1. the average of correct answers reaching 6.9, equivalent to 69% of the average correct answers obtained by the students. question number 8 is a space and shape content component, process component that parts of employ and include the types of mathematical literacy question that can involve the ability in devising strategies for solving problems, and is included in the personal context. the highest value question number 8 is 10, while the minimum value is 1. the average of correct answers up to 6.2, equivalent to 62% of the average correct answers obtained by the students. question number 9 is a space and shape content component, process component that parts of formulate and include the types of mathematical literacy question that can involve the ability of reasoning and arguments, and including in the science context. the highest value question number 9 is 10, while the minimum value is 3. the average of correct answers reaches 7.5, equivalent to 75% of the average correct answers obtained by the students. question number 10 is a space and shape content component, process component that parts of interpret and include the types of mathematical literacy question that can involve the ability of reasoning and arguments, and included in the work context. the highest value question number 10 is 10, while the minimum value is 0. the average of correct answers was 6.1, equivalent to 61% of the average correct answers obtained by the students. question number 11 is a space and shape content component, process component that parts of interpret and include the types of mathematical literacy question that can involves the ability of devising strategies for solving problems, and included in the work context. the highest value about the number 11 is 10, while the minimum value is 0. the average of correct answers was 6.8, equivalent to 68% of the average correct answers obtained by the students. problem number 12 is a space and shape content component, process component that parts of interpret and include the types of mathematical literacy question that can involve the ability of devising strategies for solving problems, and including in the science context. the highest value question number 12 is 10, while the minimum value is 2. the average of correct answers, equivalent to 7.2% of the average correct answers obtained by the students. percentage of each level in pisa is shown in figure 3 below. figure 3. mathematical literacy level students based on pisa 0 2 4 6 8 10 12 level 0 level 1 level 2 level 3 level 4 level 5 level 6 b a n y a k n y a s is w a khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 86 to get a more in-depth overview of related research results, students' mathematical literacy were grouped into three categories: low, medium, high. frankel and wallen (2009) stated that almost all score in the normal distribution is on average range minus 3 times the standard deviation and the average plus three times the standard deviation. from the results of mathematical literacy diagnostic tests done there are 2 students had a high category of mathematical literacy, 21 students had a moderate category, and 5 students had a low category of mathematical literacy. this results in mathematics literacy score is presented in table 3 below. table 3. category of mathematical literacy no students code category 1 e-10, e-13 high 2 e-01, e-02, e-03, e-04, e-05, e-06, e-07, e-08, e-09, e-11, e-12, e-14, e-15, e16, e-19, e-22, e-23, e-24, e-25, e-27, e-28 medium 3 e-17, e-18, e-20, e-21, e-26 low self-efficacy is also categorized into high, medium, and low self-efficacy with the provisions of the students with the average score as follows. table 4. categorized score students’ self-efficacy no score value category 1 40 ≤ x ≤ 93,3 low 2 93,3< x ≤ 146,3 medium 3 146,3 < x ≤ 200 high figure 4 grouping selfj-efficacy on the results of initial and final assessment volume 6, no. 1, february 2017 pp 77-94 87 based on figure 4 grouping of self-efficacy at the preliminary and final assessment results it appears that the students' self-efficacy assessment results have increased. increased selfefficacy assessment results found in the experimental class that implements rme learning model. furthermore, the final self-efficacy assessment results will be examined more deeply about mathematical literacy through interviews and the results of final mathematical literacy diagnostic test, then the subjects will be selected as follows. a. 3 subjects in the low self-efficacy category that meet the first, second and third quartiles as the subjects of the experimentation, i.e. e-17 and e-21 because there are only 2 subject categories low self-efficacy and subject k-14. k-16 and k-22 as control subjects. b. 3 subjects in the middle self-efficacy category wthat meet the first, second and third quartile as subjects of the experimentation, i.e. e-5, e-04 and e-20 and the subject k-23, k-13, k-9 as control subjects. c. 3 subjects in the high self-efficacy category that meet the first, second and third quartile as subjects of the experimentation, i.e. e-25, e-08 and e-13 and the subject k-27, k-24, k-8 as control subjects. based on interviews with students, resulting in that students who have high levels of selfefficacy if they supported by a responsive environment, then there is a behavior change in the form of student successfully carry out tasks according to its ability or good results in mathematical literacy diagnostic test. however, students who have high levels of self-efficacy if it is not backed by a responsive environment, then these students strive to change the environment to becomes responsive or even forcing a change to get a great result in the next literacy test through diligently inquired and diligently practicing and carry out tasks assigned by the teacher. while students who have low levels of self-efficacy if supported by environmentally responsive, these students become resigned and was not able to work on math literacy question and not even have spirit and reluctant to do the work assigned by the teacher and refused to write the results of discussions/duties despite encouragement, motivation and the spirit that has been given by the teacher. also with the students who have low levels of self-efficacy if it is not backed by a responsive environment, the students have low self-esteem to see their friend can work on the literacy question that considered difficult. for students who have high levels of self-efficacy were either supported or not supported by the responsive environment, the behavior of the students is more likely to be higher or lower depending on the condition of the individual. this is consistent with the theory of bandura (feist & feist, 2006) that the high-low self-efficacy correlated with the environment that is responsive and unresponsive, as follows: (a) if the high self-efficacy and environmentally responsive, the most results can be expected is a success, (b) when the self-efficacy is low and environmentally responsive, people can become depressed when they observe others successfully accomplish tasks which according to them is difficult, (c) when the high selfefficacy meet with the unresponsive environmental situation, humans usually will strive to change the environment, for example in protest, social activism, (d) when low self-efficacy in combination with an environment that is not responsive, humans will conduct apathy, give up easily, feeling powerless. discussion 1. quantitative analysis the calculation results of learning completeness the experimental class using right parties proportion test obtained z-value = 1.75. at α = 5% was obtained z_ (0.5-α) = z0,45 = 1.64. because z-value > z_ (0.5-α), then the mathematics literacy of experiment class students reaches a minimum completeness criteria for more than 75%. based on the calculation, the khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 88 value for z_value = 5.36. at α = 5% was obtained z_critical = 1.64. because z_value> z_critical, then the proportion of mathematics literacy diagnostic test results experimental class students is better than the proportion of mathematics literacy diagnostic test of control class. based on the calculation, the t¬value = 2.025 and t-critical = 1.67. because 2.025>1.67 so t-value> t-critical. this information shows that self-efficacy of experiment class students better than the control class. t-value = 1.913 and t-critical = 1.67. because t-critical > t-critical then the difference between preliminary and final mathematical literacy value of experimental class students better than difference between preliminary and final mathematical literacy value of control class. rme learning model with scientific approach effective against students' mathematical literacy. this is because (1) the percentage of students in rme learning has reached completeness, i.e. more than 75%; (2) the proportion of diagnostic mathematical literacy tests results of experimental class is better than the diagnostic mathematical literacy tests results of control class; (3) students' self-efficacy of experimental class is better than self-efficacy of control class; and (4) the average difference between preliminary and final students' mathematical literacy of experiment class better than the control class. figure 5. photo of application of using realistic mathematics education this study supports previous research including by dewanto (2008) conveys that the higher the students' self-efficacy, the higher the multiple representations of its mathematical ability, meaning that self-assurance was positively correlated with mathematical ability. stacey (2011) said that the additional reporting category of pisa 2012 will enhance the usability of the results for the development of public policy and provide further insight into mathematics learning in schools that are expected to affect the mathematical literacy ability. this is also consistent with wardono and mariani (2014) that the realistic learning device that innovative with character education and pisa assessment that has been developed can be categorized as valid, practical, and effective way to enhance the mathematics literacy problem solving of smp students, as well as quality of learning categorized good and students' character rise better. according to tarigan (2006), the general approach of rme is oriented approach towards the students' reasoning that are realistic and aimed at the development of practical mindset, logical, critical and honest with a math-oriented reasoning in solving a problem. with pmr model learning, students gave more positive response and can develop creative solution of a problem (krismiati, 2013) and students understand mathematical concepts through the completion of a problem (haji & abdullah, 2015). volume 6, no. 1, february 2017 pp 77-94 89 2. qualitative analysis mathematical literacy diagnostic test results and interviews were used to analyze mathematical literacy students based on self-efficacy is divided into 7 indicators, namely communication, mathematising, representation, reasoning and argument, devising strategies for solving problems, using formal and symbolic, technical language and operation, and using mathematics tools. here is a snippet of the students' answers on rme learning indicating that the student e-17 has the poor mathematising ability and low self-efficacy. figure 6. sample of matematising e-17 from figure 6 it is shown that students e-17 less able to change the real world problems into mathematical form or just the opposite, namely students e-17 are less able to interpret a result or mathematical models into the original problem. figure 10 is a snippet of the students' answers on rme learning indicating that the student e-04 has the good mathematising ability and medium self-efficacy. from figure 10 shows that the students e04 capable of changing the real world problems into mathematical form or just the opposite, namely student e-04 is able to interpret the results or mathematical model to the original problem. figure 7. sample of mathematising e-04 khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 90 figure 7 is a snippet of students' answers on rme learning indicating that the student e-13 has the good mathematising ability and high self-efficacy. from the figure is seen that students e-13 capable of changing the real world problems into mathematical form or just the opposite, namely student e-13 is able to interpret the results or mathematical model into the original problem. figure 8. sample of mathematising e-13 in rme learning, group of students who have low self-efficacy largely can not reach seven indicators of mathematical literacy and contains low mathematical literacy category anyway. for most of the students who have medium self-efficacy was reached seven indicators of mathematical literacy, it's just only in his achievements on each indicator is insufficient e.g. in the achievement indicators number 6, students who have medium self-efficacy has reached indicator number six but could not understand between relationship of problem context with problem representation. medium self-efficacy was having medium mathematical literacy category and can only achieve level 3, while most of the students who have high self-efficacy has reached maximum seven mathematical literacy indicators. high self-efficacy has medium and high mathematical literacy and has reached level 5. in the study by collins (mukhid, 2009) about self-efficacy revealed that children that mathematical capable, has stronger self-efficacy beliefs. the same was stated by somakim (2011) says that there is significance in increasing students' mathematical self-efficacy ability between learning using realistic mathematics approach and usual mathematical approach. additionally, dzulfikar (2013) shows that the importance of mathematically self-efficacy to be owned by each student is also mandated in the purpose of mathematics courses given to students is that they have respect for the usefulness of mathematics in life, i.e. have curiosity, attentive, and interest in learning mathematics, as well as a tenacious attitude and confidence in problem solving. peters (2013), show that students who have high self-efficacy also have high mathematics achievement. mathematical literacy in conventional learning, where the conventional application of the model in question is the discovery learning. discovery learning (dl) can be applied to achieve mathematical literacy, just not so recommended to be applied in improving mathematical literacy because it's should be applied for higher education. the results of this volume 6, no. 1, february 2017 pp 77-94 91 study support previous research conducted by alfieri, brooks, aldrich & tenenbaum (2011) in his research, he mentions that the dl model has the potential formation of misconception, because when students were left to find the information or facts and new knowledge itself, it is feared the lesson will undergo a series of mistakes, misconceptions, making them frustrated and confused about what is being studied. discovey learning also has the potential formation of cognitive overload (make it difficult for students who need more structured learning). it is accordingly presented by kirschner, sweller & clark (2006) says "cognitive load theory suggests that the free exploration of a highly complex environment may generate a heavy working memory load that is detrimental to learning". in the control class, group of students who have low self-efficacy largely can not reach all mathematical literacy indicators. most of the students in this category still find it difficult to accept and act upon the given question. low self-efficacy had lower math literacy category anyway and can only reach level 1 but some are not reaching the level of mathematical literacy. for medium self-efficacy, most of the students have achieved four of the seven mathematical literacy indicator are met with good although even in the achievement of the indicators of mathematical literacy is still less than the maximum. the results of his work looks less clear and not easy to understand. medium elf-efficacy had medium mathematical literacy category and can reach level 3 but some are not reached the mathematical literacy level. for high self-efficacy, some defecate students who have high self-efficacy has achieved some mathematical literacy indicators properly. in this category students can provide answers and acceptable solution although less than perfect. high self-efficacy has medium or and low mathematical literacy category can only reach level 2. diagnostic tests in this research also used to analyse the students difficulties. four student difficulties in rme learning with scientific approach based on the results of the diagnostic assessment of mathematical literacy in each category of self-efficacy. first, the difficulty the ability to translate problems into mathematical language occurred in the group of students who have low self-efficacy. the reason is that students are less careful in reading the questions and students are less careful in expressing information that exists on the question, not uncommon among them did not write down any information that known from question given by teachers. second, the ability to understand the difficulties occurred in the group of students who have low self-efficacy. most of them are less able to write formulas or concepts used in determining the answer to the given question. the reason is that students do not understand the explanation given. third, the difficulty in the ability to make the strategy happen on a group of students who have low and medium self-efficacy. the reason is that students do not understand the concepts and principles that have been studied and students are less thorough in completing the answer. fourth, difficulties in the ability to perform troubleshooting steps occurred in the group of students who have low and medium selfefficacy. the reason is that students are less scrupulous and often students do not write a conclusion on the results of the answer, he just wrote the final results of the calculation operation. this is in line with previous studies, namely hidayat, sugiarto & pramesti (2013) states that one of the common mistakes student made in solving the question is a misconception, meanwhile satoto, sutarto & pujiastuti (2012), which also states that the mistakes of understanding the problem is a common mistakes made by students. (3) ability to create strategies/identify the stages of problem solving (strategy knowledge). the reason is that students do not understand the concepts and principles that have been studied and students are less thorough in completing the answer. (4) ability to carry out stages problem solving khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 92 (algoritmic knowledge). the reason is that students are less scrupulous and often students do not write a conclusion on the results of the answer, he just wrote the final result of the calculation operation. conclusion rme learning with scientific approach proven effective to mathematical literacy and selfefficacy. the results of the mathematical literacy diagnostic assessment fit the criteria of students' self-efficacy except for middle mathematical literacy that having high self-efficacy. students' self-efficacy need to be inculcated through the creation of environmental conditions that are responsive and personal approach to students. mathematical literacy is one of the most important skills in learning mathematics. students' mathematical literacy are not directly grown well and needed proper exercise to train this ability to develop properly. one way to practice math literacy and create a responsive environment is to adopt rme learning with scientific approach. teachers need to do an analysis of student difficulties in resolving the question to improve the next learning and materials in finding solutions for student difficulties. acknowledgments the author would like to thank smp 2 wonopringgo which had allowed the execution of this research at the school. references alfieri, l., brooks, p. j., aldrich, n. j., & tenenbaum, h. r. 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(2014). the realistic learning model with character education and pisa assessment to improve mathematics literacy. international journal of education and research, 2(7), 361-372. wigfield, a & eccles, j. (2001). development of achievement motivation. san diego: academic press. wilkens, hendrianne, j. (2011). textbook approval systems and the program for international assessment (pisa) results: a preliminary analysis. iartem e-journal, 4(2), 63-74 khaerunisak, kartono, hidayah & fahmi, the analysis of diagnostic assesment … 94 zimmerman, b. j., bandura, a., & martinez-pons, m. (1992). self-motivation for academic attainment: the role of self-efficacy beliefs and personal goal setting. american educational research journal, 29(3), 663-676. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p137-148 137 analysis of mathematic representation ability of junior high school students in the implementation of guided inquiry learning yumiati 1 , mery noviyanti 2 1,2 open universitas, jl. cabe raya ciputat, tangerang 15418, indonesia 1 yumi@ecampus.ut.ac.id, 2 merrynov@gmail.com received: june 13, 2017 ; accepted: june 19, 2017 abstract the purpose of this research is to analyze the difference in the improvement of students’ mathematic representation ability by guided inquiry learning and conventional learning. the subject of the research was eight grade students of ut dharma karya junior high school. there were 19 students chosen as controlling class and 20 students as experiment class. this research used quasi-experimental methods with non-equivalent control group design in one school. before and after the learning, both of the classes were given mathematic representation. the result of the research showed that the mathematic representation ability of experimental group students with guided inquiry experienced an increase of 0.41; it was included in medium category. meanwhile, students’ representation ability of conventional learning group was increased 0,26 and it was included in low category. the results of this study indicated that the mathematical representation ability of the guided inquiry learning group students was significantly better than the students of the conventional learning group. keywords: guided inquiry learning, mathematic representation ability. abstrak tujuan penelitian ini adalah untuk menganalisis perbedaan peningkatan kemampuan representasi matematis siswa dengan pembelajaran inkuiri terbimbing dan siswa dengan pembelajaran konvensional. subjek penelitian adalah siswa smp dharma karya ut kelas 8. terpilih 19 orang sebagai kelas control dan 20 orang sebagai kelas eksperimen. penelitian ini menggunakan metode eksperimen semu dengan nonequivalent control group design di satu sekolah. sebelum dan sesudah pembelajaran, kedua kelas diberikan tes representasi matematis. hasil penelitian menunjukan bahwa kemampuan representasi matematis siswa kelompok eksperimen dengan pembelajaran inkuiri terbimbing mengalami peningkatan sebesar 0.41, termasuk kategori sedang. sementara itu, kemampuan representasi siswa kelompok pembelajaran kontrol (konvensional) mengalami peningkatan sebesar 0,26 dan termasuk kategori rendah. hasil penelitian tersebut menunjukkan bahwa kemampuan representasi matematis siswa kelompok pembelajaran inkuiri terbimbing lebih baik secara signifikan dari siswa kelompok pembelajaran konvensional. kata kunci: kemampuan representasi matematis, pembelajaran inkuiri terbimbing. how to cite: yumiati & noviyanti, m. (2017). analysis of mathematic representation ability of junior high school students in the implementation of guided inquiry learning. infinity, 6 (2), 137-148. doi:10.22460/infinity.v6i2.p137-148 mailto:1%20yumi@ecampus.ut.ac.id yumiati & noviyanti, analysis of mathematic representation ability … 138 introduction math equipped students to deal with world challenge, which was more growing, and as one tool used was reasoning. this matter was suitable with the statement of ayalon & even (2010), namely math served students a group of unique powerful tool to comprehend and to change the world. these tools scoped logically reasoning, problem solving ability and abstract thinking ability. in relating with math representation ability of students, it was found that math representation ability of students still had problem, particularly in translating from graphic to verbal and from table to verbal, as expressed by anastasiadou (2008). ozyildirim, ipek & akkus (2009) stated that the easiest translation, apparently, it was translation from diagram’s representation to algebra, meanwhile the most difficult translation was from table representation to algebra. the problem found relating with student representation ability for example as follow. when students were questioned: “s and t were two numbers and s was eight more than t. wrote down the equation indicated a relation between s and t”. most student answered “s + 8 = t”. it showed that students still had a weakness to change from verbal to algebra representation. in order to upgrade student representation ability, it was necessary to done math learning which gave student opportunity for reasoning and solving problem independently. according to suryadi (2005), math learning was more emphasizing on reasoning and problem-solving aspects, which enable to produce high performance students on math test, conducted by timss, like in japan and korean. learning qualified above characters was inquiry learning. gulo (trianto, 2009), stated that inquiry strategy meant that a sequence of study activities which involved entire students ability maximally to seek and to inquiry systematically, critically, logically and analytically, so that they could formulate their own invention by fully confidence. hereby, student math representation ability would be trained. suitable inquiry learning to junior high school student still needed a study with dominantly teacher guidance was guided inquiry learning. based on the above outline, then problem occurred within this research as follow “was student who gained the guided inquiry learning, obtained the higher upgrade of math representation ability compared than student obtained conventional learning?” math representation ability (mra) goldin & shteingold (2001) divided the representation into two systems, external and internal representation system. external representation is a kind of signs or symbols, characters, or object to symbolize, depict, encode, or represent something other than itself. external representations can be: 1) notation and formal, such as the number system, algebraic notation, equations, function notation, derivative, and integral calculus; 2) visual or spatial, such as the number line, cartesian graph, polar coordinate system, box plots of data, geometrical diagrams, and computer-generated images of fractals; and 3) the words and sentences, written or spoken. figure “5” is an external representation sample that can represent a set consisting of five objects, or may also represent the location or the result of the measurements. cartesian graphs can describe the data set, or it can represent a function or solution set of algebraic equations. thus, one thing can represent many things. according to goldin (2002), internal representation system consists of several types, which are: 1) verbal-syntactic system, describes the ability of both mathematical and nonvolume 6, no. 2, september 2017 pp 137-148 139 mathematical natural language and the usage of grammar and syntax; 2) imagistic system, including visual and spatial forms, or "mental images"; 3) formal notation system, the internal configuration associated with conventional mathematical symbol system and the rules for manipulating it. for example, students mentally manipulate numbers, perform arithmetic operations, or visualize the symbolic steps in solving algebraic equations; 4) the system of planning, monitoring and executive controlling or heuristics process and strategy to solve mathematical problems. for example, children develop and manage mental "trial and error" or "working backwards" when solving problems; and 5) the affective system, changes in students' emotions, attitudes, beliefs, and values about mathematics or about themselves in relation to mathematics. goldin & shteingold (2001) states that someone's internal representation can not be observed directly. however, it can be seen through the students' interaction with their external representation. guided inquiry learning inquiry learning is a process in which the students are engaged in their learning, formulating questions, investigating widely and then creating understanding, meaning and new knowledge. through those activities, students will create or construct understanding, meaning and new knowledge. this is in compliance with constructivism theory that all the knowledge we gain is acquired by ourself. sund, trowbridge, and lieslie (gani, 2011) divided inquiry learning into three types, according to the magnitude of the intervention or guidance from teachers to students, which are: a) guided inquiry: students get guidance from their teacher to understand the concept, then students independently complete the relevant tasks by having discussion or individually; b) free inquiry: students are free to determine the problem to be observed, to find and to resolve the problem independently by designing the procedures or steps required with limited or no guidance from their teacher; c) modified free inquiry: collaboration or modification of guided inquiry and free inquiry method. guidance provided by the teacher to the student is less than that of guided inquiry model and is unstructured. based on the definition and description of the three types of inquiry methods mentioned above, guided inquiry type is allegedly more appropriate to be applied to junior high school students. the steps of guided inquiry learning used refer to sanjaya’s premise as mentioned by afgani & sutawijaya (2011), which are: 1) orientation; 2) formulating problem; 3) formulating hypothesis; 4) collecting data; 5) testing hypothesis; and 6) drawing conclusion. method this research conducted by applying quasi experiment method with nonequivalent control group design where the experimental class and the control class were not chosen at random (sugiyono, 2011). the design is: o1 x o2 experimental class o1 o2 control class remarks: o1 = pretest o2 = posttest x = guided inquiry learning yumiati & noviyanti, analysis of mathematic representation ability … 140 the subjects of the research are students of dharma karya ut middle school at 8 th grade. selecting for middle school student as subject conducted according to the following considerations. middle school students are having 11 – 16 years old. according to piaget, children on these ages already had formal or abstract mind level. this matters are suit with the representation refer to abstraction forming. in addition to that, the implementation of guided inquiry learning is most appropriate to be implemented for middle school student, considering that middle school student still need dominantly guidance within session learning. selecting for 8 th grade by considering as follow: 1) students at this class have more homogeny within their basic competencies; 2) students at 8 th grade have not been undergoing national examination (un) so that it wouldn’t disrupt their preparation; 3) students at 8 th grade have been more adapted with new school environment (from elementary up to middle school) compared with student at 7 th grade. dharma karya ut middle school has four 8 th grades. 19 students selected as control grade and 20 students selected as experiment grade. research instrument covers: a) ability test of mathematic representation; b) observation sheet; and c) interview guideline. early step conducted in making instruments are making instrument summary and designing research instrument. all of the instruments was validated by validator. to analyze mathematical representation of students, the learning tools was developed. the tools developed to facilitate teachers and students in implementing the guided inquiry learning. these tools also used to guide teachers in implementing the learning that can improve the ability of mathematical representation of students. developed learning tools consist of learning implementation plan (rpp) and the student worksheet (lks). rpp developed to guide teachers for learning implementation according to the steps of guided inquiry learning model, while lks developed to guide students in conducting learning activities. prior to use, the learning tools e must first be validated learning. once validated, revised learning tools based on the input of the validators. validation conducted by math education expert from university of bengkulu and indonesian education university. data in this research comprises of two kinds, are namely qualitative and quantitative data. qualitative data gained from validation result of expert to research instrument, observation result to teacher and student activities, and interview result of teachers. qualitative data analyzed descriptively to backup the completeness of quantitative data and to answer the research questions. quantitative data gained through trial result analyzing to view reliability, validity, difficult level and differentiation power for test instrument, along with analysis regarding student respond upon student mathematic representation test. results and discussion results the purpose of this research is to comprehensively analyze differences in increasing of math representation ability (mra) of students who got a guided inquiry learning and who received conventional learning. data relates with the pretest, posttest, and n-gain mra students presented in the following figure. volume 6, no. 2, september 2017 pp 137-148 141 figure 1. diagram of mra student data based on figure 1, it indicated that the average pretest students mra relatively similar between the experiment class students and control class student. however, after learning mra posttest scores, obtained that experiment class students is higher than the control class. n-gain mra experiment class student is 0.41 including medium category, while n-gain mra control class student is 0.26, which included as low category. based on 1, achievement and improvement of mra experiment class students is higher than the control class students. trial result of wider model indicates that extensive guided inquiry learning is more effective in achieving and increasing mra student than conventional learning. this is indicated by any significant difference between the achievement and improvement mra between student group of guided inquiry learning with student groups of conventional learning. achievement and increasing mra student group of guided inquiry learning group is higher than student group of conventional learning. the magnitude of the increasing in mra student group of guided inquiry learning group is 0.41 included as medium category. based on statistical calculation, the difference increased mra ability of students in both groups of study obtained the following results. table 1. results of normality n-gain data of mra data group n avarage dev. stand. sig. (2-way) h0 n-gain of control group 19 0.26 0.119 0.612 accepted n-gain of experimental group 20 0.41 0.167 0.042 rejection based on table 1, one of the mra data is not normally distributed. therefore, to know the difference between the two groups, namely guided inquiry learning and conventional learning mann-whitney test was used. different test increasing (n-gain) math representation’s student (mra) in both groups using the mann-whitney ho generate revenue. this means, by increasing the mra of students taught using the guided inquiry learning better than students taught using conventional learning. this is shown in table 2 below. yumiati & noviyanti, analysis of mathematic representation ability … 142 tabel 2. results of mann-whitney test of mra data group avarage u mann whitney z sig. (2-way) h0 n-gain of control group 0.26 77.000 -3.218 0.001 rejection n-gain of experimental group 0.41 in other words, guided inquiry learning effect on increasing the mra of students. meanwhile, the increasing mra student group of conventional learning is 0.26 included as low category. although the guided inquiry learning is more effective in achieving mra student, but these achievements is not maximum. posttest scores obtained by student group of guided inquiry learning group is 14.65, which still under median of maximum score (17.5), while the posttest scores mra student group of conventional learning is 10.05, which still under the score of mra student group of guided inquiry learning. the research result is consistent with alhadad (2010) which concludes that improvement the ability of multiple representations of mathematical learning of students who receives open-ended approach is better than the students who receives the usual learning, reviewed from whole students. mathematical representation ability as used in this study using indicators as follow. a) to use the symbolic notation, visual or spatial, and words or phrases in solving mathematical problems; and b) to change from one representation form to another one. to provide a clearer description of mra students, the following given examples of the students' answers and analysis of faults based on mra indicators. test relates with first mra indicator is namely: the following given some student’s answer relate with the test. figure 2. correct answer sample of mra student indicator 1 this student has understood the symbolic notation used in the test, and able to use these notations in solving the problem. meanwhile, the following students still have trouble to understand symbolic notations. comparison of length, width, and height of a rectangular prism is 5: 1: 2. if the prism volume is 1,250 cm3, then defines surface area of a prism! volume 6, no. 2, september 2017 pp 137-148 143 figure 3. fault answer sample of mra student indicator 1 test relates with second mra indicator is namely: here are sample of student answers that able to change from image representation to verbal representation. figure 4. correct answer sample of mra student indicator 2 although students have not completed solving the problem, but they have shown their abilities in mathematical representation. meanwhile, the following students are still weak resentation capability figure 5. fault answer sample of mra student indicator 2 it known cylinder with radius is r cm and height is t cm as shown below. cone has a radius and a height equal to the cylinder. determine the ratio between the cone volume and cylinder volume. yumiati & noviyanti, analysis of mathematic representation ability … 144 discussion guided inquiry learning model all guided inquiry learning ineffectiveness can be explained through the stages of learning as follows. first stage: orientation at this stage, the main activity of learning is the teacher motivates student through the explanation of the topic, objectives, learning outcomes expected to be achieved by students, explanation of the main points of guided inquiry learning activities that must be performed by the students to achieve the goal. at this stage the teacher also performs apperception, which is reminiscent of the material relating to the material will be learned. for example, when it will discuss the topic of the surface area of the prism, the related materials is flat building region area. through the question and answer, the teacher reminds the area of triangle, rectangle, parallelogram, and so on. this activity relates to the deductive reasoning of students. thus, activities orientation can make mra students better. second stage : problem formulating problem formulating is a step to bring students to an issue that contains puzzles. for example, when students given the following problem. figure 6. sample of problem formulating when formulating the problem, students must know what information is contained in matters provided by teacher and what you want to achieve from the settlement of the issue. the information contained in problem 1 is: a. tent is a triangular prism shape. b. tent dimension consists of: rib base and height of a triangle which is the prism base, as well as high prism c. what want to be achieved from problem 1 is the surface area of the prism this activity is deductively reasoning, because at here the students put the premises and attempt to draw a conclusion. in formulating the problem, it required to form images, graphics, or verbal statement to clarify the premises. for example, in problem 1, students must understand the overall shape of a tent image, and image form the sides of the prism. students also need to understand the numbers shown in the image, which indicate what size. these activities require the ability of the student representation. thus, the activities to formulate the problem can practice mra students. have you ever been camping? what shape of your tent? if your tents like the tent picture on the side, can you calculate the area of the fabric needed to make a tent including tent pads? volume 6, no. 2, september 2017 pp 137-148 145 third stage: hypothesis formulating the hypothesis is temporary answer of a problem being studied. in the reasoning activity, this activity called also composing a conjecture. when composing a conjecture, students observe and analyze whether found a pattern or whether the problem can be generalized. for example, in the completion of problem 1, students asked to formulate a conjecture about the surface area of the prism through props forms prism made of cardboard. prisms of the cardboard cut by students based on the corresponding ribs, so that will be formed prism nets. students observe and analyze the general pattern of the surface area of the prism from prism nets formed. this activity is reasoning inductively. the formulation of this hypothesis can also be done by guessing intuitively, thus this activity is an activity intuitive reasoning. in activities discover patterns or generalize sometimes students do it with the help of images, tables, graphs, or verbal form, so that the representation ability plays a role in this activity. in other words, through this hypothesis formulating, mra can develop or increase. sample of student activities in making prism nets. (a) (b) figure 7. sample of prism nets made by students fourth stage: data collecting data collecting is activity to capture information needed to test the proposed hypothesis. an activity of collecting data filled with intuitively reasoning activities. students intuitively collect and arrange data needed to test the conjecture. for example, when completing issue 1, students have created a conjecture through observation and analysis of prism nets. to test the truth of the conjecture, students search the required data, for example, shape and size of ribs from the sides of the prism. these data can also be a visual form (tables, images, and graphics). in inquiry learning, data collecting is a mental process that is very important in the intellectual development. the data collecting process not only requires a strong motivation to learn, but also requires persistence and the ability to use the potential of thinking. examples of student activities figure 3. for data collect that when students are looking for a hypotenuse of prism base triangle. hypotenuse is one of the prism ribs. yumiati & noviyanti, analysis of mathematic representation ability … 146 figure 8. example of data collecting student activities fifth stage: hypothesis testing hypothesis testing is to determine the answer that considered acceptable in accordance with the data or information obtained based on the data collecting. hypothesis testing also means developing the ability to think rationally. the meaning is the truth of the answers given not only by argument, but must be supported by the data found and accountable. this hypothesis testing must use rules, nature, and existing definition, so that produces a new rule or a trait. this activity filled with activities of deductive reasoning. in testing hypothesis, it often used representation of images, graphs, tables, and verbal forms. sixth stage: conclusion formulating conclusion formulating is the process of describing the findings obtained based on the results of hypothesis testing. rules and nature that have been concluded is then extended to apply to the more complex problems. this latest activity is a deductively reasoning activity. whe n the conclusions obtained reinforced by applying to the more complex issues, the role of representation is also very important because in the resolution of complex problems need to be presented in the form of pictures, graphs, tables, and verbal forms. example of conclusion given to students relates with surface area of the prism as shown below. figure 9. student conclusion example remark: lp = prism surface area la = pad area ka = pad surrounding tp = prism height examples of more complex issues related to the surface area of the prism are namely: figure 10. complex issues related to the surface area of the prism entire area of prism side formulation by dividing into several prisms, calculates area surface of these materials a. b. volume 6, no. 2, september 2017 pp 137-148 147 thus, the habits conducted by students at each stage in the overall guided inquiry learning can improve students' mra. these habits if done continuously will contribute to a good impact for the development of students' thinking skills that needed to jump into the community later. in conventional learning, teacher explains all the material and students just listen. then the teacher gives examples of questions, and students then do the exercises. meanwhile, the material not given directly at the guided inquiry learning,. students are actively thinking to discover facts, concepts, and procedures. this is in line with the theorem construction of bruner (hudojo, 1988) which states that the best way of thinking for students to begin learning the concepts and principles in mathematics is to construct their own concepts and principles learned. thus, students taught through guided inquiry learning, have more deepened understanding of newly learned material than conventional learning. strong understanding can improve students' mra. in addition, students taught through guided inquiry learning, have longer memory storing regarding newly learned material than conventional learning. this is in line with dahar (1988) which states that some of the advantages of learning is the knowledge discovered last long or easier to remember, has an effect of better transfer. the results of this study will reinforce and complement the findings of related research on inquiry learning, namely research conducted by gani (2007), and wardani (2009). gani (2007) concludes that the inquiry-learning model of alberta can improve comprehension and problem-solving abilities of high school students. research conducted by wardani (2009) found that the inquiry-learning model of silver can develop creativity and problem solving skills at mathematical high school students. conclusion mathematical representation is important ability in mathematics. one of the reasons why it is important is because representation refers to abstrack formation and demonstration of mathematical knowledge, as well as illustration of mathematical problem solving situations. the use of different modes of representation and relationship between them illustrates the starting point in mathematics education in which the students use a symbolic system to expand and to understand others, and many other reasons to convince that representation is an ability that must be mastered by students in learning mathematics. guided inquiry learning model can be used to improve the ability of mathematical representations. inquiry learning is a series of learning activities which maximally involve the student's ability to search and investigate the problem systematically, critically, logically, analytically, so that they can formulate their own findings confidently. allegedly, by applying guided inquiry learning and mathematical representation, the student's math reperesentation skill can be enhanced. it may be caused by the activities in inquiry learning that is filled with reasoning and manipulation from one representation to another representation. references afgani, a. s. j., & sutawijaya, a. (2011). pembelajaran matematika. jakarta: universitas terbuka. alhadad, s. f. (2010). meningkatkan kemampuan representasi multipel matematis, pemecahan masalah matematis, dan self esteem siswa smp melalui pembelajaran yumiati & noviyanti, analysis of mathematic representation ability … 148 dengan pendekatan open ended (doctoral dissertation, universitas pendidikan indonesia). anastasiadou, s. d. (2008). the role of representations in solving statistical problems and the translation ability of fifth and sixth grade students. international journal of learning, 14(10), 125-132. ayalon, m., & even, r. (2010). mathematics educators’ views on the role of mathematics learning in developing deductive reasoning. international journal of science and mathematics education, 8(6), 1131-1154. dahar, r. w. (1988). teori-teori belajar. departmen pendidikan dan kebudayaan, direktorat jenderal pendidikan tinggi, proyek pengembangan lembaga pendidikan tenaga kependidikan. gani, r. a. (2007). pengaruh pembelajaran metode inkuiri model alberta terhadap kemampuan pemahaman dan pemecahan masalah matematika siswa sekolah menengah atas. (doctoral dissertation, universitas pendidikan indonesia). gani, w. (2011). pembelajaran inkuiri. available at: http://widodoalgani. blogspot. com/2011/09/pembelajaran-inkuiri. html [25 april 2012].. goldin, g., & shteingold, n. (2001). systems of representations and the development of mathematical concepts. the roles of representation in school mathematics, 2001, 1-23. goldin, g. a. (2002). representation in mathematical learning and problem solving. handbook of international research in mathematics education, 197-218. hudojo, h. (1988). mengajar belajar matematika. jakarta: depdikbud. ozyildirim, f., ipek, s., & akkus, o. (2009). seventh grade student's translational skills among mathematical representations. international journal of learning, 16(3), 197206. sugiyono, d. (2011). metode penelitian kuantitatif, kualitatif dan kombinasi (mixed methodes). bandung: alfabeta. suryadi, d. (2005). penggunaan pendekatan pembelajaran tidak langsung serta pendekatan gabungan langsung dan tidak langsung dalam rangka meningkatkan kemampuan berpikir matematik tingkat tinggi siswa sltp (doctoral dissertation, universitas pendidikan indonesia). trianto, m. p. (2009). mendesain model pembelajaran inovatif-progresif. jakarta: kencana. wardani, s. (2009). pembelajaran inkuiri model silver untuk mengembangkan kreativitas dan kemampuan pemecahan masalah matematik siswa sekolah menengah atas. (doctoral dissertation, universitas pendidikan indonesia). sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p217-234 217 students’ geometric thinking on triangles: much improvement is needed joanne ramirez casanova, claudeth cathleen canlas cantoria, minie rose caramoan lapinid* de la salle university manila, philippines article info abstract article history: received apr 7, 2021 revised jun 12, 2021 accepted jun 15, 2021 a look into students’ misconceptions help explain the very low geometric thinking and may assist teachers in correcting errors to aid students in reaching a higher van hiele geometric thinking level. in this study, students’ geometric thinking was described using the van hiele levels and misconceptions on triangles. participants (n=30) were grade 9 students in the philippines. more than half of the participants were in the van hiele’s visualization level. most students had imprecise use of terminologies. a few had misconceptions on class inclusion, especially when considering isosceles right triangles and obtuse triangles. very few students correctly recognized the famous pythagorean theorem. implications for more effective geometry teaching are considered. keywords: geometric thinking, misconception, triangles, van hiele levels copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: minie rose caramoan lapinid, faculty of the science education department, de la salle university manila 2401 taft ave, malate, manila, 1004 metro manila, philippines email: minie.lapinid@dlsu.edu.ph how to cite: casanova, j. r., cantoria, c. c. c., & lapinid, m. r. c. (2021). students’ geometric thinking on triangles: much improvement is needed. infinity, 10(2), 217-234. 1. introduction geometry is the branch of mathematics that studies shapes and measurement. it has been a subject of practical application in surveying, navigation, architecture, and engineering, among others. however, scholars assert that there is more than the practical application of geometry that makes it a “must-know”. it is also a training ground to sharpen reasoning and problem solving skills (johnston-wilder & mason, 2005) students’ difficulty in mastering geometric concepts, particularly in formal proving, has been a dilemma shared across the globe. studies (fitriyani et al., 2018; fuys et al., 1988; gutiérrez et al., 1991; senk, 1989) have reported that a majority of students who finished a formal geometry class did not completely reach van hiele’s level 4 on formal deduction. the study by senk (1989) revealed a large percentage of students finishing high school in the united states only acquired a van hiele level 1 or 2. a study in the philippines, conducted by contreras (2009), disclosed that many students fall under level 2 while some are in transition between levels 3 and 4. https://doi.org/10.22460/infinity.v10i2.p217-234 casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 218 atebe (2008) classified students’ misconceptions in geometry based on his review of literature of previously conducted studies. some illustrations were provided to explain each type of misconception in geometry. nonetheless, an in-depth closer look into students’ misconceptions specifically focused on triangles can help shed light regarding why they have very poor geometric thinking and how these can be addressed to aid students in reaching higher van hiele geometric thinking level. knowledge of these misconceptions can help teachers become proactive to consider preventive measures or to be reactive and remedy these errors. research shows that teachers who take into account students’ prior knowledge in planning their lessons can better promote conceptual understanding (banerjee & subramaniam, 2012). 1.1. the van hiele theory of geometric thinking the van hiele theory is an empirically tested theory of learning in geometry asserting that students pass through five hierarchical levels of geometric thought upon proper instruction. the van hiele theory also recommends a particular order of instruction that can enable students to progress to a consecutively higher level (see table 1). it is grounded on the premise that a student undergoes five developmental sequences of geometric thinking (mason, 1998; sarama et al., 2011). table 1. the van hiele levels of geometric thinking (van de walle et al., 2019) level students can 1 – visualization recognize shapes merely by their appearances. 2 – analysis recognize shapes and figures by their parts but unable to explain the relationship among these. 3 – informal deduction comprehend relationship between and among geometric properties; give own formal definitions or concepts; and give informal reasoning using “if-then” statements. 4 – formal deduction grasp geometry fully as a system; can do a formal proof. 5 – rigor understand the relationship between various geometry systems; compare, analyze, and prove in different geometry systems even in the absence of concrete objects. the van hiele model has proven to be a valid framework to assess and describe students’ progress of geometric understanding and for designing instructional activities that cater to that level (jones, 2003; van de walle et al., 2019). if a teacher knows at which van hiele level the student is, the educator understands where the student is operating and should be heading to next (lim, 2011). volume 10, no 2, september 2021, pp. 217-234 219 1.2. misconceptions crawford (2001) defined misconceptions as “conceptual or reasoning difficulties that hinder students’ mastery of any discipline” and drews et al. (2005) described it as the result of “a misapplication of a rule, an overor under-generalization, or an alternative conception of the situation.” misconception can occur as a natural stage of conceptual development (swan, 2001), but must be corrected to overcome difficulties in understanding concepts (van der sandt & nieuwoudt, 2003). on the other hand, not all errors are consequences of misconceptions as some of these may arise from carelessness, misinterpretations of symbols or text (swan, 2001), or from making wrong assumptions (confrey, 1990). misconceptions can be formatively assessed so teachers can design and deliver remedial instruction to correct them in time for the summative assessment (atebe & schafer, 2010). effective teachers take these misconceptions as powerful learning opportunities (luneta, 2015). they understand that their critical role is to anticipate these misconceptions in their lesson planning and to have an array of approaches at their disposal to address headon, common misunderstandings before these misconceptions stay on, worsen, and undermine confidence (bamberger et al., 2010). atebe (2008) generated a summary of the different misconceptions held by students in triangles and quadrilaterals (see table 2). table 2. students’ misconceptions in geometry as classified by atebe (2008) misconception description imprecise terminology lack of proper vocabulary identification/classification of basic shapes failure to correctly identify the name of a shape (mayberry, 1983) class inclusion inability to recognize the inclusion of shapes within a larger category; this impedes geometric progress on reasoning about relationships. parallelism and perpendicularity failure to correctly identify angle relationhsips formed when parallel lines are cut by a transversal or any properties brought by perpendicular lines angle sum of a triangle failure to use this relevant theorem in finding the measure of the third angle given the measures of the other two angles properties of shapes inability to describe explicitly the properties of triangles which includes relating the sides, angles, and the type of triangle the choice for the topic under study is triangles. french (2004) argues that triangles are the key building blocks of geometric configurations and are known for its feature of being most stable and rigid. students’ poor understanding of the concepts of triangles, as a basic polygonal shape, consequently leads to poor performance of subsequent polygonal shapes such as the quadrilateral. casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 220 1.3. statement of the problem this study aimed to describe the geometric thinking and misconceptions of students as they think about triangles. particularly, it addresses the following qustions: (1) what are the students’ van hiele levels of geometric understanding of the triangles? (2) what are students’ misconceptions of triangles? 2. method participants (n=30) consisted of grade 9 students (average age=15 years old) from a regular public high school in the philippines. an intact class of students was provided by the school principal after permission to conduct the study was granted. a written informed consent was secured from their parents to allow voluntary participation in the study. from the initial 35 participants who gave their parents’ consent, 30 came for the actual test. all students were assigned numbers and thus remained anonymous. all students had already gone through formal instruction in geometry prior to this study. in the philippines, students are taught two-dimensional basic shapes as early as grade 1, measurements in grade 3, lines, line segments, angles and quadrilaterals in grade 4, sides and angles of polygons in grade 7, the axiomatic structure of geometry, triangle congruence, parallel and perpendicular lines in grade 8, and parallelograms and triangle similarities in grade 9. each participant went through the test one at a time for 20-30 minutes to allow observation and further probing of student responses for their justifications of their answers. they were given the liberty to use a language with which they were comfortable. all students reside in the metropolitan area of the philippines, belong to lower income families and attend free public education, can speak filipino as their mother tongue, and speak english as their second language. the philippines during this study was using the mother tongue-based multilingual education as its banner program with filipino as the medium of instruction from kindergarten to grade 3, and then the use of both filipino and english as the language of instruction after grade 3 (metila et al., 2016). since the study sought to describe students’ geometric understanding of triangles in each van hiele level, and their misconceptions in triangles were identified using students’ verbatim response and proofs, the study employed a descriptive research design. there was no intervention introduced and the study was conducted in the respondent’s natural environment. the scope did not go beyond the formal deduction level of van hiele since the highest level which is rigor, by theory, requires non-euclidean geometry systems and these are not included in the high school curriculum. the students’ van hiele level on geometric understanding was measured by a set of open-ended questions that probed students’ conceptual understanding and reasoning skills that typify the level. questions in level 1 asked students to identify which of the given figures are triangles and which ones are right triangles. level 2 questions asked students to apply the triangle angle sum theorem, to state the pythagorean theorem and identify the hypotenuse. level 4 items asked students to prove two triangles congruent and similar. the van hiele test was adapted from senk (1989), contreras (2009), mayberry (1983) and de villiers (2010) on triangles. it was validated by two content experts. their comments were taken to revise and improve the items. a pilot-test was conducted with eight students who were pre-service teachers. this allowed the researchers to further refine the questions. volume 10, no 2, september 2021, pp. 217-234 221 table 3. rubric in scoring students’ proofs adopted from brandell (1994) number of points experiments 5 the proof is correct as written. 4 the proof is correct for the most part, but it is missing a minor point; a “statement” may have an incorrect “reason”. 3 the proof is generally correct, but it is missing a few minor points or a major point. 2 the proof goes in a direction that is totally incorrect. 1 the proof restates the “given” information but contains very little else. the transcript of the interviews and solutions in the tests were analyzed by the first two researchers. in the event that scores disagreed, the third researcher broke the tie. responses to each question in levels 1 to 3 were scored using a rubric. the rubrics were prepared by the researchers and approved by two mathematics teachers who had more than 5 years of teaching experience. a different rubric adopted from brandell (1994) was used to score students’ proofs in the fourth part of the van hiele test (see table 3). the success criterion for each van hiele level was based on mayberry’s (1983), except in levels 1 and 2. a student was considered to have attained the level if the score was at least the required percentage score in that van hiele level. success criteria in percent score in level 1, 2, 3, and 4 were 75, 70, 65 and 60, respectively, out of 100. since the lower level items were easier than those in the higher levels, a greater percentage of the total points in the lower levels must be earned compared to the higher levels. the highest level attained by a student was considered the van hiele level that he was able to achieve. students’ reasons were systematically and objectively characterized and compared against the correct reasoning. misconceptions were classified based on atebe’s (2008) list in table 2 taking into consideration strict observance of the descriptions in each category. 3. results and discussion 3.1. results 3.1.1. students’ van hiele levels table 4 summarizes students’ van hiele test results per level in comparison to the success criteria score. students’ mean score indicates that as a cohort, they failed to reach the success criteria score in all four levels. the predominantly attained level was visualization (level 1) as this had the greatest number of students who passed the required score. it can be seen that there are fewer students who passed a van hiele level as we go from level 1 through level 4. casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 222 table 4. students’ scores in the van hiele test level 1 level 2 level 3 level 4 success criteria score (out of 100) 75 70 65 60 students’ mean score (out of 100) 76.00 57.55 48.33 35 number of students who passed (n=30) 20 12 9 7 the results revealed that, while at their high school, students were expected to have attained level 4 – formal deduction. yet, only seven out of the 30 students reached this level. most students (18) were only able to reach level 1 (visualization). the majority of students reaching only level 1 is indicative that most of them could only recognize geometric shapes based on their appearance and not on their properties. level 1 – visualization twenty-six students were able to correctly identify which shapes are triangles (see figure 1). but four of them failed to give the correct justification. for example, s29 said shapes c and d are triangles because shape c is an equilateral triangle and d is a representation of an isosceles triangle. here, using “kind of triangle” as justification was uncalled for, thus, only three out of five points were given to this kind of answer. figure 1. question 1.1 in the van hiele test in question 1.2, the correct answers are the shapes a, b, d, h and i (see figure 2). most of the students identified shapes h and d because of the 90° and the right-angle symbol, respectively. twenty-three students answered b because they noticed that the sum of the two acute angles add up to 90°, leaving the third angle measuring 90° (see figure 2). volume 10, no 2, september 2021, pp. 217-234 223 figure 2. question 1.2 in the van hiele test figure 3. students’ responses to question 1.2 seventeen students recognized that shape i is a right triangle (see figure 3), but only seven of them were able to explain the reason. an example of correct reasoning is shown in the following transcript. s13 : kasi congruent ‘tong side na ‘to at ito. ang measure nito ay 45 and 45 din ito kaya ang natitirang measure niya ay 90 degrees. (because this side is congruent to this, [pointing to the sides with tick marks], its measure is 45 [referring to the angle with 45⁰ label] and this is also 45 [pointing towards the other angle], so what’s left with the third angle is 90 degrees. half of the respondents did not notice triangle a as another correct answer. only three students were able to provide a correct argument. see an example below. s19 : kinuha ko yung x + 2x + 3x is equal to 180 kasi yung sum ng interior angles of triangle is 180. so 6x po. divide both sides by 6, x is 30. then 30 times 3, 90 sya so right triangle sya. ayon po sa definition ng right triangle. (i got x + 2x + 3x equal to 180 because this is the sum of interior angles of a triangle. so 6x. then divide both 0 8 15 23 30 a b c d e f g h i n u m b e r o f s tu d e n ts w h o c h o se t h e s h a p e shape labels casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 224 sides by 6, x is 30. then 30 times 3 is 90. so, it is a right triangle based on the definition of a right triangle.) level 2 analysis question 2.1 asked for the measure of one of the base angles of an isosceles triangle given the measure of its vertex angle, which is 140⁰. only 50% of the students gave the correct answer using the triangle angle sum theorem. question 2.2 asked for the relationship of the sides of a right triangle; whether or not a right triangle has a longest side, and if yes, which one. unexpectedly, only three out of the 30 respondents recognized and stated the pythagorean theorem. thirteen respondents knew a right triangle has a longest side and correctly named it as the hypotenuse. the rest recognized the existence of a longest side, but failed to give its name. level 3 – informal deduction students in this level were expected to recognize the relationship of shapes and their formal definitions. question 3.1 asked whether a right triangle can be isosceles. a majority of the students answered “yes” although some reasons were found to be inconsistent with their answer and a few of them did not give any reason. for instance, s25 said that this case is possible if one side is not congruent to any of the other sides. other students answered “no” with the following reasons: s17 : because in an isosceles triangle, you can’t form a square (referring to the perpendicular symbol) while in a right triangle you can form a square inside. s18 : because an isosceles triangle has all equal parts or sides. figure 4. question 3.2 in the van hiele test in question 3.2 (see figure 4), only one student (s29) said “sometimes” to the completion of the statement and explained "…because one of the other sides may be greater than one side of the other triangle.” the rest of the respondents correctly answered “always,” although some of their justifications were questionable as some admitted they cannot recall this postulate. other students gave “parallel postulate,” “isosceles triangle theorem,” and “sss postulate.” in question 3.3 (see figure 5), only eight of the participants recognized that the relationship is similar, albeit all except one gave reasons that were questionable. two of them simply stated the premise: “because the bases are parallel” and did not elaborate how this makes similar triangles. s01 was unable to identify relevant necessary and sufficient conditions to justify why the triangles are similar. instead, he tried to describe the corresponding congruent angles and corresponding proportional sides of similar triangles. volume 10, no 2, september 2021, pp. 217-234 225 figure 5. question 3.3 in the van hiele test as can be seen in s01’s statement, properly stating proportional corresponding sides is a common difficulty among students. s01 : because [of] angle acf and angle ade, their [δade and δacf] sides are proportional to each other. assuming the side cf is proportional to segment de and side af is proportional to segment ae. s04 and s11 based their judgment on their observation and intuitive understanding of similar triangles: s04 : parehas ng shape. ang pinagkaiba lang yung laki. (the shapes are the same. the only difference is their size). s11 : because they have the same angles, they only differ in size. level 4 formal deduction the first proving item was on triangle congruence. in figure 6, s14 was able to give correct statements except in statement #3. the pairs of congruent corresponding angles do not follow from any of the preceding ones. the student incorrectly assumed that the two triangles are isosceles based on how the figures were drawn. instead of pairing corresponding angles of the two triangles, s14 paired two angles of one triangle. the reason for concluding “vertical angles are congruent” was incorrectly referred to as the “vertical triangle theorem.” moreover, the student stated “triangle addition postulate” instead of using sas to justify the triangle congruence. casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 226 figure 6. an example of proof with flaws by s14 figure 7. an incomplete proof by s06 the second proving item was on triangle similarity. eight respondents gave incorrect proofs and two students did not answer this. figure 7 shows an example of an incomplete proof. s06 correctly deduced the two right angles are congruent from the given right triangles. however, he could have first established that the angles bdc and bda are right angles because the given triangles are right triangles. instead, he incorrectly gave “definition of a right angle” (a right angle measures 90 degrees) as the reason. there was a missing statement before one can conclude that the angles are congruent – for example, angles bdc volume 10, no 2, september 2021, pp. 217-234 227 and bda are right angles. moreover, the student did not know how to proceed (see figure 7). figure 8. an example of proof with incorrectly stated reasons by s10 in the proof by s10 (see figure 8), the right angle labeled 2 is not relevant in arriving at the desired conclusion as this is not an angle of either of the triangles under consideration. the student’s use of pairs of similar segments is indicative of irrelevance as there is no such concept and this cannot be assumed from the given premises. 3.1.2. students’ geometric misconceptions imprecise terminology students either had difficulty recalling the correct term or they simply mixed up the concepts. for example, a student defining an obtuse angle as having less than or equal to 90 degree measure, suggests that he was aware of the measure being in a range of values but got the different terms for the kinds of angles mixed up. students were not accurate enough to identify which reason was the most appropriate to justify their claim. for example, (1) the general definition of congruence was used when the student meant that the two triangles are congruent; (2) the right angle theorem was used to refer the pythagorean theorem; (3) the isosceles triangle theorem was used instead of the asa triangle congruence; students chose a name that was close or quite related to what they were trying to prove. students had difficulty distinguishing between similar and congruent triangles. this may be due to students’ difficulty in distinguishing between a conditional statement and its converse. students’ failure to identify the correct reason in their proofs may also be due in part to their failure to dissect or unpack the definitions, postulates, and theorems into premise and conclusion parts. casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 228 students had a tendency to take chances and guess if they could possibly mention the correct justification to support their claim. some students used “postulate 7” and “theorem 19” as reasons in their proofs. the tendency of students to use the numbered postulates and theorems had been found to be accepted by their geometry teachers during seatwork and exercises where the study was conducted. the effect of such a strategy to name a postulate or a theorem may be detrimental. instead of focusing on reasoning, students tend to be preoccupied with memorizing the postulate or theorem number. it also raises doubts whether students actually understand the principle stated in the postulate or theorem. this seems to encourage guessing. some students consistently interchanged terms that were closely related. the term “equal” instead of “congruent” when describing the relationship between two triangles was a common error. students were also confused when to say definition, postulate, and theorem. mathematical notations are symbolic representations of mathematical objects and processes. it is a language and a shorthand to communicate and express mathematical concepts and ideas. it has precise semantic meanings that are crucial in mathematical discourse. however, the study revealed students used improper notations. similar segments were denoted as 𝐵𝐷̅̅ ̅̅ 𝐴𝐵̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ 𝐵𝐶̅̅ ̅̅ even though there is no such concept of similar segments. the student must have meant sides are proportional which should be properly denoted as 𝐵𝐷 𝐴𝐷 = 𝐴𝐷 𝐵𝐶 . identification/classification of basic shapes most respondents could easily identify triangles. right triangles were easily identified when there is an indicated right angle (either with the perpendicular symbol or with the 90⁰ measure). however, students failed to recognize shapes that involve additional concept and procedure such as those with the triangle angle sum, algebraic solutions and isosceles triangle theorem in shapes a and i of question 1.2. students gave naïve conceptions and informal understanding as they tended to judge the shapes by their mere appearance: with an l-shaped angle and the sides look like the hands of a clock at three o’clock. students assumed that based on the drawing, the angle in the triangle looked like they were perpendicular even if neither the perpendicular symbol nor the 90⁰measure was written. this can be due in part to students’ difficulty in determining what can or cannot be assumed, what was given and what was not. parallelism and perpendicularity parallelism is one of the prerequisite concepts for understanding the principle of similarity. in question 3.3, a given a pair of parallel lines is cut by a transversal forming congruent corresponding angles. another pair of congruent angles can be deduced by reflexivity. these two pairs of congruent angles make the triangles similar by aa triangle similarity. none of the respondents was able to recognize that each of the two sides of the larger triangle can serve as transversals of the parallel sides. instead, some students showed informal understanding of similarity as indicated in their responses: “triangle ade is larger than triangle acf…because the measure of the sides is larger” and “triangle acf expanded, become ade”. students’ inability to apply the concept they previously learned from parallel lines to triangle similarity indicates that their knowledge seemed to be compartmentalized. to them, these concepts were not linked or connected to one another. it also connotes poor understanding as this knowledge from previous lessons were not retained and used when the situation called for it. volume 10, no 2, september 2021, pp. 217-234 229 class inclusion misconceptions under this category are those that involve misunderstandings on the family of triangles. these are mostly caused by students’ tendency to operate on properties exclusive to a specific type of triangle. most of them can recognize a right triangle, but some of them cannot point out that a right triangle can also be isosceles. a minority of them said a right triangle cannot be isosceles and gave reasons that are not correct. properties of a triangle recognition of the properties of a shape is necessary to relate shapes to each other. misconceptions in this category are further classified into the following. kinds of triangles students misunderstood the definition of isosceles triangle and its parts as seen in some responses such as “all angles of an isosceles triangle are equal” and “a base angle and its vertex angle are congruent.” in another instance, students had difficulty handling the definition of acute and obtuse angles and the triangle angle sum theorem concurrently as seen in the following responses: “in an obtuse triangle, the sum of the two acute angles is equal to the obtuse angle.” and “in an obtuse triangle, the sum of the two acute angles is greater than 90.” this difficulty seems to show that students were seldom engaged or not engaged at all in higher order thinking discourse and problem solving. angle sum of triangle most of the students can only handle one or two but not all of the given conditions. for instance, in question 2.3, a number of students overlooked the given conditions that an angle in the triangle is obtuse and the rule on sum of interior angles. they were fixated on the given two acute angles, disregarded the type of triangle being obtuse and said it is possible that the sum of these acute angles in the triangle is greater than 90⁰. pythagorean theorem the pythagorean theorem has been an important and popular concept in geometry. however, very few (5 out of 30) students were able to state this when they were asked to relate the three sides of a right triangle. those who were able, answered the question correctly by naming the relationship in a procedural manner: “given a, b, and c as sides, a squared plus b squared is equal to c squared.” it was not clear whether they referred to the side whose length is labeled c as the hypotenuse and the sides with lengths a and b, as the legs. this is worthy of note because this is fundamental to other succeeding concepts such as the hypotenuse-leg triangle congruence theorem. relationship between two triangles students described congruent and similar triangles in their own words based on their mere appearance as opposed to analytically describing these in terms of their corresponding angles and sides. students had informal understanding that similar triangles have the same shape but different sizes as most of them said “congruent triangles are not similar triangles.” students failed to recognize the difference between triangle congruence and similarity. casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 230 plausible causes are their lack of knowledge of conditional statement and its converse, and failure to identify the necessary and sufficient conditions for each relationship. aside from the misconceptions that contributed to students’ difficulty in proving, students lacked the necessary cognitive strategies to proceed from the premises. some tacit premises in the figure that were relevant to use in proving were not recognized by some students: congruent angles due to reflexivity, vertical angles that are congruent, transversal line cutting parallel lines that form congruent angles, among others. instead, irrelevant information which was not given and could not be assumed (congruence of angles in statement #3 in figure 6 and figure 8, and proportionality of sides in statements 4 and 5 in figure 8) was used. 3.2. discussion consistent with the findings in other international studies (atebe, 2008; luneta, 2015), less than 24% of the students reached the formal deduction stage. the percentage of grade 9 students who reached the level 4 is a slightly better than that of grade 6 students (20.7%) in taiwan (ma et al., 2015) and that of another local study in the south of the philippines by solaiman, magno and aman (2017) with no single grade 9 respondent who reached the informal deduction (level 3). this implies that most were clearly behind their expected van hiele levels after taking euclidian geometry in grade 9. more concerning is that more than half of the respondents were still operating in level 1. aside from the general notion of students’ poor geometric thinking, the study described detailed errors committed mostly in levels 2 and 3. results in this study points toward how geometry is being taught in lessons that deal with analysis (level 2), abstraction (level 3) and eventually formal deduction (level 4). the study explored, revealed, and described students’ different misconceptions and difficulties in learning triangles. the use of imprecise terminologies could be caused by poor understanding of definitions, unmindful use of terms or preference of using informal language. while the use of informal language can help students gain intuitive understanding, eventually the regular use of the proper mathematical language and notation in textbooks and classroom discourse should be encouraged. the use of notation was introduced to lessen the use of texts and words. yet, students take for granted the proper use of notation, this very thing that makes mathematics less cumbersome and teachers underrate students’ difficulty of acquisition of notations in students’ learning (edwards, 2000). language and notation play an important role in the development of conceptual understanding since instructional materials, resources, mathematics textbooks, and the like, use these in concept development. knowledge of the correct technical terms and notations is necessary for learners to be able to communicate their ideas clearly and for them to be receptive of class discussions (atebe & schafer, 2010). most of the participants exhibited a lack of knowledge on geometric properties. this was due to either students lack communication skills in expressing their ideas or students knew the names of the various definitions, theorems, and postulates but did not know what they meant. students also had difficulty handling a lot of concepts and relating them to one another. students’ knowledge was mostly compartmentalized and they failed to see which concept is relevant to use to defend their answers. most of the students in this group had superficial understanding and seem to regard geometry as a collection of unrelated concepts, rules, and properties. this may also explain why for them, mathematics in general is a difficult subject because concepts are interrelated. since each concept is built on another, failure to master a previous concept adds to greater difficulty in understanding the volume 10, no 2, september 2021, pp. 217-234 231 succeeding concepts. to say the least, a surface acquisition of a concept does not guarantee its recall and application in an unfamiliar problem. results of the study support radatz’s (1979) claim that learners’ misconceptions are due to semantic differences between natural language and mathematical language, limited spatial abilities, failure to master the prerequisites, incorrect associations, lack of cognitive control and strategies, and application of rules or irrelevant ideas. 4. conclusion on the basis of the results obtained, grade 9 students have not attained the desired learning competencies expected of their level as far as the triangle concept is concerned. they are not ready to learn the concepts intended for the grade 9 curriculum – quadrilaterals, its classifications and properties since they have not reached the van hiele’s formal deduction level on triangles. an in-depth analysis of their answers also reveals various misconceptions held by students on triangles. students do not use the correct terminologies, mixed up concepts, write incorrect notations, grappled with simultaneous properties in a single figure, failed to connect previous concepts on parallel and perpendicular lines in triangles, and can’t use tacit premises in their reasoning. consequently, we first recommend remedial work in geometry for 9th grade students, starting with level 2 of the van hiele model may be incorporated to the syllabus. only once this is mastered should educators proceed to level 3. it should not be assumed that just because students receive instruction in geometry at a young age that they come to 9th grade with a firm grasp of the ideas. the manner by which the topics in geometry are being taught in the earlier grades needs to be re-examined. some of these misconceptions may be deeply seated from early grade instruction. also, lim (2011) asserts that a major cause of misconception is in the communication line between the sender (teacher) and the receiver (student) when they operate at different van hiele levels. for example, a teacher gives examples of two triangles having the same shape but different sizes and says that these are similar triangles. this may inadvertently contribute misconception in students’ overgeneralization that for two triangles to be similar, they have to be of different sizes for as long as they have the same shape. teachers’ given examples and those not given can cause students’ misconception (bamberger et al., 2010). teachers could form learning communities to share their list of students’ misconceptions and discuss effective ways to counter or correct misconceptions. future studies on the van hiele levels can include investigations in geometry topics other than the triangles and their properties, development of comprehensive test that can assess a wider scope of a particular geometric concept, and interventions to improve students’ reasoning skills. both pre-service and in-service teachers may also be assessed for their van hiele levels and misconceptions. if teachers hold misconceptions, they are more likely to be unable to recognize errors students make and their instruction may inadvertently perpetuate these misconceptions (graeber et al., 1989). the van hiele theory models the hierarchical property of geometric understanding. as noted by van hiele (atebe, 2008) the movement between two levels is not natural but undergoes a formal teaching-learning process and depends on the factors within the direct control of the teacher and the curriculum (senk, 1989). learners go through visualization before formal definitions, postulates, and theorems are introduced. as suggested by barrett and clements (2003) and groth (2005), teachers can adapt the phases of learning: information, guided orientation, explication, free orientation, and integration, in conscientious preparation of their instructional plan. rather than presenting the lesson in a casanova, cantoria, & lapinid, students’ geometric thinking on triangles: much … 232 transmissive manner, teachers may employ strategies to help learners be more prepared to do complicated and challenging tasks. references atebe, h. u. 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(2001). dealing with misconceptions in mathematics. in p. gates (ed.). issues in mathematics teaching (pp. 147–165). london: routledge. https://doi.org/10.4324/9780203469934 van de walle, j. a., karp, k. s., & bay-williams, j. m. (2019). elementary and middle school mathematics: teaching developmentally (10th ed.). london: pearson education uk. van der sandt, s., & nieuwoudt, h. d. (2003). grade 7 teachers' and prospective teachers' content knowledge of geometry. south african journal of education, 23(3), 199205. https://doi.org/10.25255/jss.2017.6.3.603.609 https://doi.org/10.4324/9780203469934 infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p195-206 195 language literacy and mathematics competence effect toward word problems solving namirah fatmanissa 1 , rahmat sagara 2 1 universitas pendidikan indonesia jl. dr. setiabudhi no.229, bandung. indonesia 2 institut teknologi dan bisnis kalbis jl. pulomas selatan. dki jakarta. indonesia 1 namirahf@student.upi.edu received: may 12, 2017; accepted: july 02, 2017 abstract this study aims to know the effect of language literacy and basic mathematics competence toward students’ ability to solve word problems. the research was done by giving three sets of questions; language literacy (ll) set, basic mathematics competence (bm) set, and word problems (wp) set; to the research sample. research sample was 315 tenth grade students from five schools in jakarta. score of students in each set was analyzed as research data. score of ll set was treated as data of independent variable 1, score of bm set was treated as data of independent variable 2, and score of wp set as data of dependent variable. preliminary data analyses, such as normality, validity, and reliability test, were done. then, data was analyzed using wilcoxon test and calculation of r-square. the result shows that each of independent variable affects dependent variable with bm variable has more effect on wp variable. it is expected that in solving word problems, the way students use their basic mathematical competencies is supported together with their ways of using language literacy. keywords: basic mathematics competence, language literacy, word problem. abstrak penelitian ini bertujuan untuk mengetahui pengaruh kecakapan bahasa dan kemampuan dasar matematika terhadap kemampuan siswa dalam menyelesaikan soal cerita. penelitian dilakukan dengan memberikan tiga paket soal, yaitu paket soal kecakapan bahasa (kb), kemampuan dasar matematika (kdm), dan kemampuan soal cerita (ksc)kepada sampel penelitian. sampel penelitian ini adalah siswa kelas x dari satu sekolah di setiap area di jakarta, yaitu jakarta utara, jakarta selatan, jakarta barat, jakarta timur, dan jakarta pusat. dari lima sekolah tersebut diperoleh sampel penelitian berukuran 315 siswa. data yang dianalisis adalah nilaiyang diperoleh siswa pada tiap paket soal. data nilai untuk paket soal kb sebagai data variabel independen pertama, data nilai untuk paket soal kdm sebagai data variabel independen kedua, dan data nilai untuk paket ksc sebagai data variabel dependen. hasil uji wilcoxon menunjukkan bahwa variabel independen berpengaruh terhadap variabel dependen, dan dari nilai r-kuadrat didapatkan bahwa variabel kdm memiliki pengaruh yang lebih besar terhadap variabel ksc. dalam menyelesaikan soal cerita, diharapkan siswa dapat didukung dalam menggunakan kemampuan dasar matematikanya sekaligus dengan kecakapan bahasanya. kata kunci: kecakapan bahasa, kemampuan dasar matematika, soal cerita. how to cite: fatmanissa, n., & sagara, r. (2017). language literacy and mathematics competence effect toward word problems solving. infinity, 6 (2), 195-206. doi:10.22460/infinity.v6i2.p195-206 fatmanissa & sagara, language literacy and mathematics competence effect … 196 introduction mathematics is an essential knowledge and its application is used in almost all branches of science. in indonesia education, mathematics is one of knowledge that should be mastered by all students. the purpose of this mastery is to make students use mathematics as a tool to understand daily life (conway & sloane, 2005). education world nowadays has realized students’ reluctance on understanding mathematics. the reason behind this case is that mathematics is considered as collection of formulas and rigid, abstract procedures that is hard to be understood (schwanebeck, 2008). students argue that mathematics is not important for their life and it is enough to be learned by students who are indeed capable of it. however, in fact, mathematics is integrated deeply in every aspect of life (gouthro & griffore, 2004). due to that reason, mathematics learning in the class is aimed not only to make students understand its concept, but also to apply it in life (gouthro & griffore, 2004). learning process in the class should relate mathematics materials to its use in solving daily problems. in other words, the lesson objectives are not the mastery of mathematics numerical operations, but also the mastery of solving problems related to real life. in this case, word problem become one of the tools to assess students ability in solving such problem. word problem is generally defined as collection of words and structures which creates a problem. specifically, word problem has several characteristics that make it more complicated compared to other forms of problem. the first characteristic is that word problem is contextual, which means it contains elements exist in daily life. word problem offers daily life problem which should be solved by students by using mathematical approach. the second characteristic is that word problem needs not only one single step to solve it. one of the ways to make students able to apply mathematics in daily life is by optimizing their ability in solving word problems because through word problems, students can grasp the concrete feeling of mathematics (lave, 2016). however, in indonesia, students’ ability to solve word problems are considered to be low (rindyana & chandra, 2012; huda & kencana, 2013; sutarni, 2011). pppptk for mathematics, center for the development and empowerment of teachers and educational staff in indonesia, stated that over 50% of indonesian teachers complaint about the difficulties of students in solving word problems. rindyana & chandra (2012) found that this problem was due to the lack of understanding of the meaning of words contained in the problems. they found that more than 84% of students being studied did not understand the meaning of words and/or did not know the purpose of the question. sutarni (2011) addressed the same focus by finding that students being studied were not accustomed to read carefully and thus resulting on the lack of ability to solve word problems. huda & kencana (2013) addressed different aspect by finding that the low ability was due to the lack of mathematical concept understanding. to optimize the ability to solve word problems, the knowledge of factors influencing students’ ability to solve word problems is needed.if a student is given a word problem and cannot answer it correctly, here come a question whether this student cannot answer the problem due to his/her inability to understand the context of word problem and thus cannot construct steps to solve it, or due to his inability to do the steps to get the answer although he/she understands the context. this question leads to the discussion of factors influencing students’ ability to solve word problems. volume 6, no. 2, september 2017 pp 195-206 197 language literacy factor language literacy is one’s ability to read, write, speak, and listen to reach a particular purpose. related to word problem, the abilities being used are reading and writing ability. reading ability is the ability to understand meaning of collection of written letters and words. while writing ability is the ability to express what is known and what is needed to be delivered in a written form. language literacy, the ability to read and write, is used when students read the word problem, understand it, and then write its mathematical model. gardner (2004) highlighted students’ language literacy as verbal-linguistic intelligence, one of eight intelligences in his multiple intelligences theory. gardner said that one’s language literacy could be seen from four aspects. they are convincing someone that information is true, reminding someone of an information, explaining something, and reflecting idea to another form of language. in solving word problem, aspect of explaining something and reflecting an ideato another form become important. aspect of explaining something consists of how someone expresses and understands some information either orally or written. in relation with word problem, this aspect is shown when student tries to understand the word problem. when student reads the written word problem, the process of changing written material to information stored in the brain happened. whereas aspect of reflecting idea consists of how someone changes information become another form with the same meaning. this aspect is contained in the process of making mathematical model from written sentences of word problem. a correct reflection process will result on mathematical model which has the same meaning with the sentences in the problem. without an accurate model, there may be mistakes in the next stages of solving problem. because of that, students’ lack of language literacy may cause misinterpretation of the meaning and purpose of the given word problem. context understanding that is necessary in solving word problem was analyzed by clement (2008) who stated that students with difficulty in solving word problems tend to question the meaning of the problem or discuss its interpretation. clement also said that students who failed to correctly answer word problem were mostly the ones who did not pay attention to its context, but directly did the mathematical operation that they considered to be appropriate. these statements show that language literacy, which is indicated by information and context understanding, is the factor that is needed to be examined in order to escalate students’ ability in solving word problems. lee (2006) stated that in understanding written information, there are two things should be put into concern, vocabularies and syntax or sentence structure. in mathematical text, vocabularies are very varied. kersaint, thompson, & petkova (2014) had similar opinion with lee that vocabularies in mathematics could be classified into vocabulary that has similar meaning with daily words and vocabulary that has different and specific meaning with its daily usage. vocabulary such as limit, supplementary, and positive have different meanings between their use in mathematical text and in daily usage. while vocabulary like smaller than, addition, and greater than have the same meanings both in daily life and mathematical text. the ability to understand the difference and similarity of vocabulary meaning has role in one’s language literacy. fatmanissa & sagara, language literacy and mathematics competence effect … 198 the understanding of syntax or sentence structure is also an important factor. people often use active voice in daily conversation, while in fact mathematical text mostly uses passive voice rather than active voice (kersaint, thompson, & petkova, 2014). in active voice sentence, subject of the sentence is stated clearly, while in passive voice sentence, sentence subject is not stated explicitly. mathematical text that has passive voice structure demands someone to adjust his/her understanding of active to passive voice, which make understanding mathematical text more difficult. basic mathematics competence factor gardner (1999) defined logic-mathematics ability as an ability of someone to analyze problem logically, do mathematical operation, and do scientific inquiries. these abilities are needed in solving word problems. this is inline with piaget’s cognitive development theory. piaget explained that students mathematics competence develops gradually through four development stages i.e. sensorimotor, pre-operational, concrete operation, and formal operation. in the second and last stages lies the relation between basic mathematics competence and ability to solve word problems. on pre-operational stage, student is only able to solve problem using one step (ojose, 2008). concerning the complex steps of solving word problems, in this stage student is only able to do procedural calculating operation. on concrete operation stage, student can convert real life mathematics problem into symbols and mathematics equations (moursund, 2007). on solving word problems, this stage shows “plan” process when students try to design strategy to solve problems. whereas on the last stage, formal operation stage, student can do process of abstraction (moursund, 2007). students are able to understand abstract concepts without involving concrete example of the concept. from this discussion, it is known that basic mathematics competence can be monitored from pre-operational stage. this is because student does not do process of converting information from concrete (real life context) to abstract (mathematics symbols and equations), but only organizing information provided using mathematical operation. beside cognitive development, basic mathematics competence is also related to the topics or branches of mathematics that are tested in word problems. basic mathematics competence is identified from branches which are the foundation of other branches. one of these is arithmetic. arithmetic has been agreed as a fundamental branch and foundation of mathematics (marjanović, 1999). beside arithmetic, logic is also a fundamental branch of mathematics. this is because logic is used in almost all reasoning of other mathematics branches concepts or known as logical reasoning. together with arithmetic and logic, algebra becomes the basic of mathematics learning content. algebra contains fundamental principles in solving word problems (wilson, 2009). another branch to be considered, as niss (1998) said to have important role in mathematics learning as shown on the number of research discussing reasoning of its concept and role in daily life, is geometry. other factors one of other factors that is considered to affect the process of solving word problems is level of difficulty. level of difficulty can be seen from two aspects. first is the use of syntax or term and word preference in the problem (xin, 2007).the more complex and unfamiliar the volume 6, no. 2, september 2017 pp 195-206 199 word for the students, the more difficult for students to interpret the objective of the problem and the greater possibility for students to obtain wrong information. second is the existence of element that has been learnt by students before. the more difficult the element to be recognized, the more difficult students catch the information correctly. beside level of difficulty, students’ perception toward word problem is considered affecting the ability to solve word problems (schwanebeck, 2008). students perhaps are reluctant to answer the question because of the topic of the word problem is not what they like. if compared with other factors mentioned, language literacy factors (which is the ability to read and write) and basic mathematics factor are the key role in the process of mathematics solving. larwin (2010) compare language literacy with confidence, teacher’s expectation, and the use of technology in affecting students’ score in a test that is dominated with word problems. language literacy has the biggest effect among other factors. larwin even stresses that students’ low language literacy should be addressed after being known by teacher because this issue may hamper students’ learning in the next stages. schoppek & tulis (2010) stated that basic mathematics competence, compared to motivation, gives more significant impact to solving word problems. beside that, basic mathematics competence and language literacy are two factors that are always involved in every word problem solving process (seifi, haghverdi, & azizmohamadi, 2012). these two factors is contained in each strategy of solving problems and determine whether the strategy is appropriate or not. it is already elaborated that solving word problems is affected by many factors in which two among them (language literacy and basic mathematics competence) are the most significant factor. these two factors are then analyzed to know their effect on students’ ability in solving word problems. some literatures have mentioned the relationship between language literacy and basic mathematics competence and solving word problems. sammons (2011) stated that these three aspects are supporting each other. if one aspect is sufficient, then other aspects will be sufficient and improved, too. language literacy and basic mathematics competence have strong relation with word problem solving (schoppek & tulis, 2010). these two aspects are involved in important points in the process of word problem solving, thus the effect is enormous in determining the success of students in solving word problem. based on vilenius‐tuohimaa, aunola, & nurmi (2008), language literacy is one of the factors that determine the success of someone in solving word problems. through his research, it is found that language literacy and the ability to solve word problems have strong positive correlation. an exemplary language literacyis indicated with exemplary ability in solving word problems. in solving word problems, process of absorbing information from written texts happened (oecd, 2011). this process is needed to design a plan on how to solve particular word problem. language literacy takes important role in this part because language literacy determines whether student has chosen appropriate information from the given word problem. awofala, balogun, & olagunju (2011) strengthened this statement by comparing low language literacy students with high language literacy in solving word problems. the result was that the students who have high language literacy was significantly more excellent in solving mathematics word problems. basic mathematics competence is also an unavoidable criteria in solving word problems (sammons, 2011). the knowledge of mathematical procedure determines how students fatmanissa & sagara, language literacy and mathematics competence effect … 200 process information obtained from word problem accurately and efficiently. besides, geary (2000), in his research that studied the development of mathematics basic competence of an individual from pre-school to adult and its relation with solving word problems, stated that the development of basic mathematics competence of an individu would significantly help his/her development in solving word problems. geary also stated that basic competences such as numerical operation, together with language literacy, affected the ability to solve word problems, regardless the age. method sample of the research was 315 senior high school students of grade 10 in greater jakarta area, indonesia. the participants were taken from one school in each jakarta region, i.e. north, south, west, east, and central jakarta. more detail information of participants was given in table 1. table 1. number of participants in each region region number of participants ( ) north jakarta 56 south jakarta 45 west jakarta 65 east jakarta 90 central jakarta 59 total 315 data was taken using three sets of test given to participants and each test represented particular variable. score for language literacy (ll) test as independent variable 1, score for basic mathematics competence (bm) test as independent variable 2, and score for word problem (wp) test as dependent variable. basic mathematics (bm) test was developed based on four dimensions which were arithmetics, algebra, geometry, and logic. language literacy (ll) test was developed based on three dimensions which were same-meaning words, different-meaning words, and syntax. word problems test was developed with four dimensions which were word problems on algebra, arithmetics, geometry, and logic. each test consisted of 10 questions with the expansion of each dimension could be seen in table 2. table 2. dimension expansion variable dimension question no. bm algebra 3, 4, 5 geometry 6,7 arithmetic 1, 2 logic 9, 10 ll same-meaning words 5, 6, 9, 10 different-meaning words 1, 2, 7 syntax 3, 4, 8 volume 6, no. 2, september 2017 pp 195-206 201 variable dimension question no. wp algebra 4, 5, 6 geometry 7, 8 arithmetic 1, 2, 3 logic 9, 10 beside those dimensions, the test was controlled by varying level of difficulties of the questions. in 30 questions, there are easy, medium, and hard question. the level of difficulty was determined by how complex the steps needed to solve it. besides, for ll test, the level of difficulty was determined by how complex the words and syntax that were in the question. in wp test, the combination between those two elements became the cause of difficulty level. table 3 showed the level of difficulty of each question. table 3. level of difficulty of each question level of difficulty question no. bm ll wp easy 1, 3, 9 1, 5, 7 1, 3, 5, 9 medium 2, 5, 7, 6 2, 3, 6, 9 2, 4, 7, 10 hard 4, 8, 10 4, 8, 10 6, 8 after all 30 questions had been developed and underwent several revisions, the test was given to participants. the participants were given three sets of test with break time in between. the break time was given to reduce the boredom or stress happened during test taking. the test was given with the following order; ll test continued by break time, then continued by bm test and break time, and last test was wp test. participants of the test were not allowed to discuss the question with anyone and use calculator or other calculating tools during the test. the scheme of the test was given in figure 1. figure 1. test scheme after the test, students’ work were evaluated and graded. each correct final answer was scored 1, and 0 for wrong answer. the score was given for each test, thus student who got perfect score was the one who got 10 in each test. results and discussion initial data analysis showed four outliers and these were not considered during further analysis. normality test showed that data distribution was not normal. validity test showed that one question in ll test was not valid and then were not included in analysis. reliability test showed that each test was reliable with alpha-cronbach coefficient of 0.812. ll break time bm break time wp fatmanissa & sagara, language literacy and mathematics competence effect … 202 some participants could answer the questions correctly, but some could not. the information of number of students having correct answer was given in table 4. table 4. number of students having correct answer bm no. students answered correctly ll no. students answered correctly wp no. students answered correctly % % % 1 243 77.14 1 247 78.41 1 257 81.59 2 261 82.86 2 237 75.24 2 260 82.54 3 132 41.90 3 300 95.24 3 105 33.33 4 136 43.17 4 287 91.11 4 157 49.84 5 190 60.32 5 292 92.70 5 139 44.13 6 60 19.05 6 164 52.06 6 60 19.05 7 173 54.92 7 243 77.14 7 106 33.65 8 98 31.11 8 212 67.30 8 205 65.08 9 214 67.94 9 302 95.87 9 247 78.41 10 11 3.49 10 141 44.76 10 226 71.75 mean 151.8 48.19 mean 242.5 76.98 mean 176.2 55.94 in average, each question of ll test was answered correctly by 76.98% students. this was the highest percentage compared to bm test (48.19%) and wp test (55.94%). besides, the average students’ score on bm test was also shown as the highest. the score of students in ll test had the lowest variance. descriptive statistics of each test were given in table 5 below. table 5. descriptive statistics of each test bm ll wp mean 4.819 7.698 5.594 highest score 10 10 10 lowest score 0 2 0 standard deviation 2.353 1.676 2.284 variance 5.537 2.81 5.217 “did basic mathematics competence and language literacy affect students’ ability in solving word problems?” to answer this question, there were three numerical data to be analyzed, which were bm test score as 1st independent variable, ll test score 2nd independent variable, and wp test score as dependent variable. knowing the fact that the data is not normally distributed, wilcoxon test was used. in wilcoxon test, there were two separated tests. the first test checked whether bm variable affected wp variable, while the second test checked whether ll variable affected wp variable. in wilcoxon test, null ( ) and alternative ( ) hypothesis as follow. : relative frequency distribution for both variables were identical; : independent variable made the relative frequency distribution of dependent variable shifted; or independent variable affected the dependent variable. volume 6, no. 2, september 2017 pp 195-206 203 the result of wilcoxon test run by spss was given in table 6. table 6. wilcoxon test result from wilcoxon test, with , it was known that both the first and second test showed rejection to null hypothesis. this means that, with significance level of 5%, basic mathematics competence and language literacy had significantly affected the ability to solve word problems. related to this, some examples of student’s written work were found. for example in the question of, “two kids can dye a wall in 5 hours. how long does it take for 3 kids to dye that wall?” several students’ answers were like given in figure 2. the answer given in figure 2 showed that students did understand the question, however failed to use proper mathematical concept to give correct answer. figure 2. sample of students’ answer some others, show the lack of understanding of question aim. student’s answer given in figure 3 was the example. the question given was “ani sucks 5 times and gulps 4 times to consume a glass of milk. while to consume 2 glasses of milk, she sucks 13 times and gulps 7 times. so, one ani’s gulp equals how many suck?”. the student’s answer showed that he could not understand words representing variables. he simply consecutively put numbers given in the question into mathematical model, and could not complete his work. figure 3. sample of students’ answer bm toward wp ll toward wp n 311 311 test statistic 25,091.000 25,984.000 standard error 1,229.752 1,160.814 sig. .000 .000 fatmanissa & sagara, language literacy and mathematics competence effect … 204 while students in figure 2 failed to give correct answer because of the lack of mathematical concept understanding, students in figure 3 failed because of the lack of understanding words role in representing variables. “if those have effect, which one did give greater effect toward the ability to solve word problems? basic mathematics competence or language literacy?” it had been known that basic mathematics competence (bm) and language literacy (ll) had significantly affected the ability to solve word problems (wp). an analysis to answer the second question was by determining r-square value for each variable. spss output of each variables’ r-square was given in table 7. table 7. r-square value of bm and ll toward wp bm ll r .544 .410 r-square .296 .168 sig. .000 .000 based on table 7, bm variable can explain 29.6% variance of wp variable, while ll variable can explain 16.8% variance of wp variable. from this fact, it can be inferred that bm variable had more effect compared to ll variable. in other words, the ability of student in solving word problem was slightly more determined by basic mathematics competence than by language literacy although both correlations actually considered as low. the fact that basic mathematics competence has more role in affecting the ability to solve word problem is not inline with the initial hypothesis that both variables have equal effect. this leads to a question that several studies (geary, 2000; schoppek & tulis, 2010, larwin, 2010) support the initial hypothesis. it cannot be denied that the referred studies have different contexts compared to this study. the type of sample, for example in geary (2000), has great range of age, has its own inferences compared to this study in which all participants are in the same age. theoretically, language literacy and basic mathematics competence are not the only factors in determining the ability to solve word problems. the existence of other factors can affect the “sharing” of impact towards the ability to solve word problems. for example in schwanebeck (2008) who stated that solving problems cannot be separated from other factors beside language literacy and basic mathematics competence, for example emotional aspect of students. the number of questions or problems given in this study has met or represented the dimensions. however, by adding the number of questions, the variety of question type and level of difficulty might increase thus the data will be more varied. the conclusion given in this study did not answer a more detailed question like, “how do students use his/her language literacy in solving word problems?”. because of that, this study can be expanded into a qualitative approach. volume 6, no. 2, september 2017 pp 195-206 205 conclusion both basic mathematics competence and language literacy affect the ability of students in solving word problems. compared to language literacy, basic mathematics competence has more effect in determining that ability. teachers should pay more attention to how students solve word problems in terms of how they use their mathematics competences (e.g. calculating, using algorithm, etc) and also how they use their language literacy (e.g. understanding the problem, giving meaning to words, etc). in the context of word problems discussion, teachers are expected to discuss why certain information transformed into certain mathematical model. the process of understanding mathematical words that are used in problems and then interpreting them cannot be neglected as language literacy also contributed in this process. besides, word problems are one of the ways in knowing students higher order thinking. if teacher wants to improve students higher order thinking, they can intensify the use of word problems. students are expected not too focused on memorizing formula and rigid procedures. they were expected to pay attention to their mathematics competence without neglecting their ability to make meaning and understanding written information given in the word problems. references awofala, a. o., balogun, t. a., & olagunju, m. a. 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(2012). analisis kesalahan siswa dalam menyelesaikan soal cerita matematika materi sistem persamaan linear dua variabel berdasarkan analisis newman (studi kasus man malang 2 batu). artikel ilmiah universitas negeri malang. sammons, l. (2011). building mathematical comprehension: using literacy strategies to make meaning. huntington beach: shell education schwanebeck, t. (2008). a study of summarization of word problems.nebraska: university of nebraska. seifi, m., haghverdi, m., & azizmohamadi, f. (2012). recognition of students’ difficulties in solving mathematical word problems from the viewpoint of teachers. journal of basic and applied scientific research, 2(3), 2923-2928. schoppek, w., & tulis, m. (2010). enhancing arithmetic and word-problem solving skills efficiently by individualized computer-assisted practice. the journal of educational research, 103(4), 239-252.. sutarni, m. 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(2007). word problem solving tasks in textbooks and their relation to student performance. the journal of educational research, 100(6), 347-360. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p55-76 55 prospective teachers representations in problem solving of special angle trigonometry functions based on the level of ability yayan eryk setiawan* universitas islam malang, indonesia article info abstract article history: received feb 11, 2021 revised aug 3, 2021 accepted aug 6, 2021 one of the materials used as the basis for solving trigonometric function problems is special angle trigonometry. prospective teachers' representation in problem-solving of trigonometric functions with special angles is thought to be influenced by prospective teachers' abilities. therefore, this study aims to analyze the representations used by prospective teachers in problem-solving of special angle trigonometric function based on ability categories. this research is qualitative descriptive. the research subjects are prospective teachers of the mathematics education study program at a university in malang. the data collected in this study are in the form of work results and observation. the research instrument consisted of the problem of the trigonometric function value of the special angle and the interview guide developed by the researcher. the analysis of prospective teacher work results was carried out by classifying the ability categories into low, medium, and high abilities. the work results of each of these categories are classified based on verbal, numeric, image, and algebraic representations. the analysis of the interview transcripts was carried out by coding the words or sentences which aims to determine prospective teachers' understanding of using representations. the results showed that prospective teachers with low ability use a lot of verbal representation, while prospective teachers with medium and high abilities use a lot of image representation in problem-solving of special angle trigonometric function. the implication of the results of this study is to teach special angle trigonometric function material based on appropriate representations. keywords: prospective teachers, representation, special angle, trigonometry this is an open access article under the cc by-sa license. corresponding author: yayan eryk setiawan, departement of mathematics education, universitas islam malang jl. mayjend haryono 193 malang, east java 65144, indonesia email: yayaneryksetiawan@unisma.ac.id how to cite: setiawan, y. e. (2022). prospective teachers representations in problem solving of special angle trigonometry functions based on the level of ability. infinity, 11(1), 55-76. 1. introduction as a prospective teacher, you should have mastered the concepts in trigonometric material. this is because trigonometric material is taught at the senior high school level, https://doi.org/10.22460/infinity.v11i1.p55-76 https://creativecommons.org/licenses/by-sa/4.0/ setiawan, prospective teachers representations in problem solving of special angle trigonometry … 56 where this trigonometric material is important material at the senior high school level (cavanagh, 2008; kamber & takaci, 2018; maknun et al., 2019; nabie et al., 2018; nejad, 2016; siyepu, 2015; tuna, 2013; usman & hussaini, 2017; wongapiwatkul & laosinchai, 2011). this trigonometry material is a prerequisite for vector analysis, calculus, and differential equations (setiawan, 2021). also, in this trigonometric material, there are concepts in mathematics that are important for students to master (nabie et al., 2018). therefore, a prospective teacher must master the concepts in the trigonometric material, to be able to teach correctly about the concepts in the trigonometric material to senior high school students. trigonometry is a part of mathematics that generally discusses triangles (downing, 2009), and specifically discusses trigonometric functions (lial et al., 2016). one of the important materials to study is the trigonometric function of special angles. special angles consist of 30º, 45º, and 60º. the importance of studying the material on trigonometric functions is because this material is used as a basis for determining the value of the trigonometric function for non-acute and negative angles (setiawan, 2021). for example, to find the value of sin 210º is to find the reference angle. the reference angle is defined as an acute angle that is not a quadrant angle in a standard position (lial et al., 2016). since 210º in quadrant iii, the reference angle of 210º = 210º 180º = 30º. so we can determine the value of sin 210º = sin30º = ½ (since 210º is in quadrant iii, the value for sin is negative). so it can be said that this special angle trigonometric function material is important for students to learn, in order to solve problems about the non-acute angle and negative angle trigonometric functions. however, the results showed that prospective teachers still experienced errors in determining the value of the trigonometric function for special angles. this can be seen from the results of research that show that prospective teachers experience errors in drawing cosine function graphs, errors in applying trigonometric comparisons, and errors in writing down trigonometric reduction identities to determine the value of the trigonometric function for special angles (setiawan, 2021). the results also show that prospective teachers consider trigonometric material to be difficult and abstract (nabie et al., 2018), so that conceptual understanding of prospective teachers is low in learning trigonometric material (mustangin & setiawan, 2021). this error is usually solved by introducing the prospective teachers to triangular trigonometry rather than circular trigonometry. however, prospective teachers have difficulties when introduced to triangular trigonometry, where the difficulties of prospective teachers include: (i) connecting triangular images with numerical relationships, (ii) determining trigonometric ratios, and (iii) manipulating symbols involved in trigonometric ratios (wongapiwatkul & laosinchai, 2011). so in general it can be said that prospective teachers still experience errors and difficulties in studying the material of special angle trigonometric functions. one of the factors that influence the errors and difficulties of students in studying the material of special angle trigonometric functions is the representation used to understand the value of the special angle trigonometric function. representations in mathematics are defined as representatives of abstract concepts (bishop, 2000). the results show that students' understanding can be related to their schema structure and flexibility in using different representations (trigueros & martínez-planell, 2010). the results also show that representation can support children's understanding and reasoning in multiplication (barmby et al., 2009). therefore this representation is one of the keys to student success in learning mathematics. students can develop and deepen their understanding of mathematical concepts and relationships as they create, compare, and use various representations, such as physical objects, pictures, charts, graphs, and symbols (joyner & reys, 2000). furthermore, the representation and symbolization of mathematical ideas is the heart of mathematics volume 11, no 1, february 2022, pp. 55-76 57 related to mathematical activities (kaput, 2008; kaput et al., 2008). one must rely on concrete representations to make conclusions (bishop, 2000). so it can be said that representation can affect a person in understanding mathematics, including mistakes and difficulties in learning mathematics. this is because representations can develop and deepen understanding of concepts, reasoning, and problem-solving. representations used by students in solving mathematical problems are also influenced by students' backgrounds. the results showed that the diagrammatic representation helped all students in solving trigonometric problems, but the illustration of trigonometric problems was influenced by the students' backgrounds (cooper & alibali, 2012). one of the backgrounds that influence a person in solving math problems is an ability (clark et al., 2014; perkins et al., 1993; ron, 2001; tishman & andrade, 1995; tishman et al., 1993). ability is defined as the adequacy of a person's capacity to solve problems (clark et al., 2014). the results showed that students who were able to prove had relevant basic knowledge (setiawan, 2020a). the results of other studies show that students and students with low abilities often experience errors in solving problems (mustangin & setiawan, 2021; setiawan, 2020b, 2020c, 2021; setiawan et al., 2020). this means that representation can be influenced by the abilities students have in understanding the material or solving math problems. based on this description, it can be said that there are still errors made by students in determining the value of the special angle trigonometric function. these difficulties and errors are also influenced by the representations used in teaching the material for special angle trigonometric functions. representations used by students in studying a material are also influenced by the abilities possessed by students in learning certain materials. the results of previous research have shown that diagrammatic representations have helped all students in solving trigonometric problems (cooper & alibali, 2012). however, this previous research did not consider the background in the form of students' abilities in solving trigonometric problems. therefore, research is still needed on student representation in solving trigonometric problems based on the ability categories possessed by students. based on this, this study aims to describe the various representations used by students in solving special angle trigonometric function value problems based on ability categories. the theoretical benefit of the results of this study is to develop a theory of student representation in solving the problem of the value of the trigonometric function for special angles based on the ability category. the practical benefit of the results of this study is that it can be used by lecturers or teachers in teaching special angle trigonometric function values based on the level of ability of prospective teachers or students. if the representation used is in accordance with the student's ability level, then the student will better understand the material given. in the end, the representation used in learning can reduce the errors or difficulties of students in studying the material for the value of special angle trigonometric functions. 2. method by following the purpose of this study, namely describing the various representations used by students in determining the value of the trigonometric function of the special angle, this research method is a qualitative descriptive study with a case study approach to 12 research subjects who are students of a mathematics education study program at a university in malang city. the process of selecting subjects in this study consisted of three steps (see figure 1). the first step is to ask 82 prospective teachers of semester 1 of the mathematics education setiawan, prospective teachers representations in problem solving of special angle trigonometry … 58 study program to solve six conceptual understanding questions, one of which is to determine the value from a special angle (see figure 2) by providing a written explanation of the method they use. the second step is to correct student answers using the scoring guidelines developed by the researcher. of the 82 students, the lowest score was 24 and the highest score was 97. from the scores obtained by 82 students, the students' abilities would be categorized into three categories, namely high, medium, and low. the categorization of students' abilities is based on the scores obtained when completing six questions of conceptual understanding. the determination of this category is based on the normative reference guidelines that exist at the islamic university of malang in the 2020/2021 academic year, namely: a value (score 80-100), b value (score 70-79), c value (score 5569), and d value (score < 55). the results of categorizing the abilities of 82 students are obtained in table 1. figure 1. the process of selecting subjects finished determination of research subjects score categorization begin do the test answers low medium high 4 subjects 4 subjects 4 subjects volume 11, no 1, february 2022, pp. 55-76 59 the third step is to classify students' answers from the high, medium, and low categories based on verbal, numeric, image, and algebraic representations (see table 2). from the results of this classification, 4 students were taken from the high, medium, and low categories, so that the research subjects consisted of 12 prospective teachers. these subjects were chosen, because they can provide an explanation of the answers in detail. table 1. results of prospective teachers ability categorization no. score (𝒙) category number of prospective teachers 1 80 ≤ x ≤ 100 high 23 2 55 ≤ x < 80 medium 40 3 x < 55 low 19 total 82 by following this type of research, the data collected in this study consisted of the results of the subject's work and transcripts of interview results. data collection procedures in the form of subject work results are carried out by following the steps for selecting the subject. while the procedure for data collection of interview transcripts was carried out using two steps. the first step is to conduct interviews with research subjects through whatsapp media. during the interview, an audio recording was also conducted. the second step is transcribing word for word so that a transcript of the results of the interviews with the research subjects is obtained. by following the data collected in this study, the research instrument consisted of questions about the value of the trigonometric function for special angles (see figure 2) and interview guidelines developed by the researcher. the instrument in figure 2 has various representations that can be used in finding the value of the trigonometric function of a special angle. for example, using triangles, tables, graphs, circles, or algebraic representations of angular relations. because this research instrument can be completed with various representations, this research instrument is valid to be used to identify the representations used by students. figure 2. research instruments the analysis of student work results is classified based on the representation used by students in determining the value of the trigonometric function for special angles. the representation system consists of an internal representation system and an external representation system (goldin & shteingold, 2001). external representation systems range from conventional mathematical symbol systems (e.g., number symbols, formal algebraic notation, number lines, and cartesian coordinates) to structured learning environments (e.g. involving concrete manipulative material), whereas internal representation systems include student symbolization constructs and assignment of meanings to mathematical notation, students' natural language, visual imagery, and spatial representation, and problem-solving strategies (goldin & shteingold, 2001). this means that the external representation system is in the form of mathematical symbols that have been used by mathematicians which are then introduced to students in learning mathematics, while the internal representation is the setiawan, prospective teachers representations in problem solving of special angle trigonometry … 60 student's construction in learning mathematics. this study uses internal representations, namely constructs used by students in determining the value of the trigonometric function for special angles. the classification of representations in this study is carried out using four representations, namely: verbal, numerical, pictorial, and algebraic representations (friedlander & tabach, 2001). this is because the use the verbal, numeric, image, and algebraic representations have the potential to make the learning process of algebra meaningful and effective (friedlander & tabach, 2001). the framework for the classification of student representations in determining the value of the trigonometric function for special angles can be seen in table 2. table 2. the framework of prospective teachers representation classification types of representation indicators of representation verbal verbal representations are usually used in posing problems and are needed in the interpretation of the final results obtained in solving problems. verbal representations are used to understand the context and to communicate solutions. a student who can describe a situation verbally is not always able to represent it symbolically (scher & goldenberg, 2001). this means that someone's indicator in using verbal representations is that they can explain the final result verbally. verbal representations in this study include rote memorization. numeric numerical representation is representing a geometric model through its numerical features, students can look for patterns, arrange experiments to test conjectures (scher & goldenberg, 2001). this numerical representation emphasizes the use of important numbers to gain an understanding of problem-solving. that is, someone's indicator in using numerical representations emphasizes the use of numbers. image the graphical representation is very effective in providing a clear picture of the function of real number variables. graphics are intuitive and very attractive to students who like a visual approach. someone's indicator in using image representations is to use graphics, use geometric images, or other objects. algebra algebraic representation is effective in presenting mathematical patterns and models (scher & goldenberg, 2001). manipulation of algebraic objects is sometimes the only method of justifying or proving general statements. indicators of students using algebraic representations are using algebra or algebraic manipulation furthermore, the transcript analysis of the interview results was carried out by coding the words or sentences that showed the subject's understanding of the representations he used to determine the value of the trigonometric function. thus it will be known that the subject understands the representation used. through the analysis of the results of the work and the transcripts of the results of these interviews, it is hoped that the representation of students in solving the problem of the value of the trigonometric function for special angles can be identified. volume 11, no 1, february 2022, pp. 55-76 61 3. results and discussion 3.1. results the results of the research in the form of representation classification used by 82 prospective teachers in determining the size of the special angle based on the ability category can be seen in figure 3. from figure 3 it can be seen that prospective teachers with low ability use a lot of verbal representation, while prospective teachers with medium and high abilities use a lot of image representation. furthermore, from each category of this ability level, four prospective teachers will be taken as research subjects to be further analyzed about the representations used by these prospective teachers. figure 3. prospective teachers representation diagram 3.1.1. prospective teachers representation of low ability categories from figure 3 it can be seen that of the 19 prospective teachers who have low abilities, it is obtained that 47% of prospective teachers use verbal representations (i.e. in the form of memorization), 37% of prospective teachers use image representations (consisting of six prospective teachers using triangles and one prospective teacher using graphs), and 16% uses algebraic representations (using angular relations) in determining the value of the acute angle trigonometric function. this means, in the low ability category, prospective teachers still use rote memorization to determine the value of the special angle trigonometric function. furthermore, from the 19 prospective teachers, 4 prospective teachers were selected as the subjects of this study, each of which represented the representation used by the prospective teachers. figure 4. verbal representation of the first subject from figure 4 it can be seen that the subject directly determines the value of the special angle trigonometric function. however, the subject experienced an error in low medium high verbal 9 0 0 image 7 23 16 numeric 0 5 1 algebra 3 12 6 0 5 10 15 20 25 n u m b e r o f p ro sp e ct iv e t e a ch e rs representations setiawan, prospective teachers representations in problem solving of special angle trigonometry … 62 determining the value of the trigonometric function for the function cos 60°. explanation of the subject using this method can be seen from the following interview excerpt. r : try to explain how you solve the problem! s1 : those are special angles, so i immediately determined by using the memory from senior high school, that is, sin 30 is equal to 1/2, and cos 60 is equal to -1/2. from the interview transcript, it can be seen that the subject only memorized in determining the value of the special angle trigonometric function. furthermore, from the seventh prospective teachers who used image representations, two prospective teachers were taken as the second subject (s2) and the third subject (s3), each of which used a triangle representation (see figure 5) and graphs (see figure 6). figure 5. triangular representation of the second subject from figure 5 it can be seen that the subject uses a triangular representation in determining the value of the trigonometric function for special angles. in using this triangular representation, the subject has succeeded in correctly determining the value of the privileged angle trigonometric function. the explanation of the method used by this subject can be seen from the following interview excerpt. r : try to explain how you solve the problem! s2 : … the length of the sides of the base of the triangle, 1 divided by ½ into ½. sin 30 sin front side hypotenuse, equal to ½. for cos 60 to be the same, cos 60 to ½. cos is equal to the side hypotenuse so that ½ divided by 1 equals ½. r : why are you using this method? s2 : because it is easier to use right triangles, namely sindemi (sin = front/hypotenuse, and cosami (cos = side/hypotenuse). from the interview transcript excerpt, it can be seen that the subject was correct in using triangular comparisons to determine the value of the special angle trigonometric function. this method also according to the subject is the easiest way to determine the value of the special angle trigonometric function. volume 11, no 1, february 2022, pp. 55-76 63 figure 6. graphical representations of the third subject from figure 6 it can be seen that the subject still experiences errors in drawing the graph of the trigonometric function for the cos function. due to the error in drawing this graph, the subject experienced an error in determining the value of the special angle trigonometric function. the explanation of the method used by this subject can be seen from the following snippet of the interview transcript. r : try to explain how you solve the problem! s3 : … when i was at school the picture was also described, so i remember it like that. i only remember that explanation in senior high school, sir. because i forgot a bit, i was also given a picture like that. r : why are you using this method? s3 : because that's what i remember and that's what i think is easy to use graphics. from the interview transcript, it can be seen that the subject only remembers the graph of the trigonometric function. because the subject forgot the correct graph, the subject experienced an error in determining the value of the trigonometric function when using a graphical representation. next, one prospective teacher was taken as the fourth subject (s4) out of 3 prospective teachers who used algebraic representations (namely angular relations) in determining the value of the special angle trigonometric function. the results of the fourth subject's work can be seen in figure 7. figure 7. representation using angular relations of the fourth subject from figure 7 it can be seen that the subject uses an algebraic representation in the form of an angular relation. however, the subject did not determine the value of his special angle trigonometric function. therefore, the subject experienced an error in determining the value of the special angle trigonometric function. an explanation of the methods used by the subject can be seen from the following interview transcript excerpt. setiawan, prospective teachers representations in problem solving of special angle trigonometry … 64 r : try to explain how you solve the problem! s4 : … sin 𝟑𝟎𝟎 is equal to cos 60 degrees or cos 60 degrees is equal to sin 30 degrees. r : why are you using this method? s4 : because i know it, sir from the interview transcript, it can be seen that the subject uses only angular relations (without determining the value of the trigonometric function). also, the subject only knows that one way to determine the value of the trigonometric function of a special angle. from the research results, it is found that the representation used by subjects with low abilities is the representation using a triangle. it can be seen that the subject using the triangle representation has used the correct concept. meanwhile, subjects who use verbal representations, graphics, and angular relations tend to make mistakes. this error arose because the subjects only memorized the value of the special angle trigonometric function, memorized the graph of the trigonometric function, and also memorized the angle relation. therefore, learning starts from the concept of a triangle to explain the value of the trigonometric function for special angles. 3.1.2. prospective teachers representation of medium ability categories from figure 3 it can be seen that of the 40 prospective teachers who have the medium ability, 58% of prospective teachers use image representations (22 use triangles and 1 use graphs), 12% of prospective teachers use numerical representations (tables of special angle trigonometric function values), and 30% of prospective teachers use algebraic representations (i.e. angular relations) in determining the value of the acute angle trigonometric function. for each of the types of representation used, one prospective teacher will be selected as a research subject who has medium abilities. so that the research subjects obtained for prospective teachers who have a medium ability category are four prospective teachers. the results of the fifth subject's work (s5) can be seen in figure 8. figure 8. representation using triangles of the fifth subject from figure 8 it can be seen that the subject is correct in determining the value of the trigonometric function for special angles using a triangle representation. an explanation of the method used by this subject can be seen from the following interview excerpt. volume 11, no 1, february 2022, pp. 55-76 65 r : try to explain how you solve the problem! s5 : … i made a right triangle with the other angles 600 and 300. because it comes from an equilateral triangle, i take the side lengths 2 and 1, then i find the other side lengths using pythagoras. after that, i determined the values of sin 30 and cos 60 using the sindemi (sin = front/hypotenuse, and cosami (cos = side/hypotenuse). r : why are you using this method? s5 : because this material is the same as the class xi material, i still remember the old material in senior high school. from the interview transcript excerpt, it can be found that the subject has understood how to determine the value of the trigonometric function of a special angle using a triangular representation. this method is obtained by remembering the material on special angle trigonometry. next is the sixth subject (s6) that uses graphics as can be seen in figure 9. figure 9. graphical representations of the sixth subject from figure 9 it can be seen that the subject can use the graphical representation correctly to determine the value of the special angle trigonometric function. the explanation of the method used by the subject can be seen from the following interview excerpt. p : try to explain how you solve the problem! s6 : … i use a coordinate system. x and y coordinates. when x is 1 and y is 1, then it is in sin 90 because sin starts from 00, the value is zero, then the upward value gets bigger, so when sin is 300 the value is half. for cos, i use the opposite method for sin. when sin 00 equals cos 900, then the value of cos decreases in value. so when cos 600, the value equals sin 300, so cos 600 equals ½. … from the interview transcript, it can be seen that the subject has understood the graph of the sin function and the graph of the cos function. therefore, the subject answered correctly in determining the value of the special angle trigonometric function. the second representation used by prospective teachers is a numerical representation in the form of a table of the values of the special angle trigonometric function. of the 5 prospective teachers who used this representation, 1 prospective teacher was chosen as the seventh subject (s7) in this study. the results of the seventh subject's work can be seen in figure 10. setiawan, prospective teachers representations in problem solving of special angle trigonometry … 66 figure 10. numeric representation of seventh subjects from figure 10 it can be seen that the table made by the subject still has errors, namely the value of cos 60º, while the value of sin 30º is correct. as a result, the subject experienced an error in determining the value of the trig function for cos 60º. the explanation of this method can be seen from the following interview transcript excerpt. r : try to explain how you solve the problem! s7 : i use the method in sir's books, namely i make a table of the values of the trigonometric function of special angles using memorization. then i write down the values of the special angle trigonometric functions in the table. so that the value of sin 300is ½ and the value of cos 600 is ½ the root of three. from the interview transcript excerpt, it can be seen that the subject still uses rote memorization in determining the values of the special angle trigonometric function which is then made a table. as a result, the trigonometric function value is still wrong, namely the value of cos 60º. next is the eighth subject (s8) which uses an algebraic representation in the form of angular relations. the results of the eighth subject's work can be seen in figure 11. figure 11. algebraic representation of the tenth subject from figure 11 it can be seen that the subject uses angle relations. however, the subject is wrong in writing the subtraction form of the cos function. as a result, the subject experienced an error in determining the value of the special angle trigonometric function. the explanation of the subject can be seen from the following interview excerpt. volume 11, no 1, february 2022, pp. 55-76 67 r : try to explain how you solve the problem! s8 : … cos 600like that a, i also reduce it. so cos 90 minus cos 30 then i break it down to be cos 90 multiplied sin 30 minus sin 90 multiplied cos 30. cos 90 is equal to zero multiplied ½ minus sin 90 1 multiplied cos 30 is 1 2 √3. then minus 1 2 √3 like that r : why are you using this method? s8 : because this material is the same as class xi material, i still open old books in senior high school. from the interview transcript, it can be seen that the subject determines the value of the cosine function by describing it. however, the subtraction formula for the cos function is still wrong. this error is caused by only opening material during senior high school, without understanding how the formula is correct. therefore, the subject experienced an error in determining the value of the special angle trigonometric function. from the research results, it is found that the representation used by subjects with medium ability is the representation using the triangle and the representation using the sin and cos function graphs. it can be seen that the subject using the triangle representation and the graphical representation of the sin and cos functions has used the correct concept. meanwhile, there are subjects who use numerical and algebraic representations who still experience errors in determining the value of trigonometric functions. because the subject only memorized the value of the special angle trigonometric function and memorized the formula for the angle relation. therefore, learning for medium category prospective teachers can be done with the concept of triangles and graphs to explain the value of the trigonometric function for special angles. 3.1.3. prospective teachers representation of high ability categories from figure 3 it can be seen that out of 23 prospective teachers who have high abilities, 70% of prospective teachers use image representations (i.e. 15 prospective teachers use triangles and 1 prospective teacher use circles), 4% of prospective teachers use numerical representations (i.e. use tables), and 26% of prospective teachers use algebraic representations (i.e. angular relations). from each of these representations, one student will be selected as the research subject. so that the number of research subjects for students with high ability categories there are 4 prospective teachers. figure 12. triangular representation of the ninth subject setiawan, prospective teachers representations in problem solving of special angle trigonometry … 68 from figure 12 it can be seen that the subject uses the correct triangle representation to determine the value of the special angle trigonometric function. an explanation of the method used by the subject can be seen from the following snippet of the interview transcript. r : try to explain how you solve the problem! s9 : first i made an equilateral triangle by assuming the side length is 2a, then i divided the triangle into two so that two congruent right triangles were obtained. then i determine the value of sin 300, that is, the sindemi is 𝑎 2𝑎 = 1 2 . next, i determine the value of cos 60, that is, cosami is 𝑎 2𝑎 = 1 2 . so we get sin 300 = 1 2 and cos 600 = 1 2 . r : why are you using this method? s9 : because in my opinion, this method is easy to use. from the interview transcript excerpt, it can be seen that the subject already understands how to use triangular representations, namely sindemi (sin = front/hypotenuse) and cosami (cos = side/hypotenuse) in determining the value of the special angle trigonometric function. furthermore, the tenth subject (s10) uses a circular image representation to determine the value of the special angle trigonometric function. the results of the tenth subject's work can be seen in figure 13. figure 13. representation of circles of tenth subject from figure 13 it can be seen that the subject uses a circle to determine the value of the trig function. in this case, the subject managed to answer correctly. the explanation of the subject in using the circle can be seen from the following interview transcript excerpt. r : try to explain how you solve the problem! s10 : … i use a circle to determine the value of the sin and cos functions, where the y-axis is the sin value and the x-axis is the cos value. so i wrote that the result of sin 300 is equal to ½ because the value of ½ is on the straight y-axis at an angle of 300 and the result of cos 600 is ½ because the value of ½ is on the x-axis. r : why are you using this method? s9 : because besides memorizing, i also learned about circles to determine the value of trigonometric functions. from the interview excerpt, it can be seen that the subject has an understanding that the x-axis is the value of the cos function and the y-axis is the sin value. this understanding is by following the concept of the unit circle. where the unit circle has a radius of 1 unit. if we are going to find the value sin𝜃 = 𝑦 𝑟 = 𝑦 1 = 𝑦 and cos 𝜃 = 𝑥 𝑟 = 𝑥 1 = 𝑥. so subject with high abilities can understand the representation of the circle to determine the value of the trigonometric function of special angles. volume 11, no 1, february 2022, pp. 55-76 69 next is the eleventh subject (s11) who uses a numeric representation in the form of a table in determining the value of the special angle trigonometric function. the results of the eleventh subject's work can be seen in figure 14. figure 14. table representation of the eleventh subject from figure 14 it can be seen that the subject determines the values of the trig function correctly so that the subject can determine the value of the functions sin 30º and cos 60º correctly. an explanation of the methods used by the subject can be seen from the following interview transcript excerpt. r : try to explain how you solve the problem! s11 : i'm using a table of values for sin, cos, and tan. so i immediately determined the result of sin 300 = 1 2 and cos600 = 1 2 . r : where did you get your values in the table? s11 : i obtained the values in the table using a right triangle with angles 300, 450, and 600, while the angles 00 and 900 are quadrant angles. from the interview excerpt, it can be seen that the subject obtains the values in the table by using a right triangle. however, the subject did not write down the triangle method, where the subject focused more on obtaining the trigonometric ratio values which were then written in the table. this means that the subject can use the right triangle representation and the subject can also use the numerical representation with a focus on the numerics generated from the right triangle. next is the twelfth subject (s12) which uses algebraic representations in the form of angular relations. the work of the twelfth subject can be seen in figure 15. figure 15. representation of angular relations of the twelfth subject setiawan, prospective teachers representations in problem solving of special angle trigonometry … 70 from figure 15 it can be seen that the subject uses the angular relation, which is the relation between sin and cos. the explanation of the method used by the subject can be seen from the following interview transcript excerpt. r : try to explain how you solve the problem! s12 : i write sin 300 = sin(90 − 60), because sin(90 − 𝛼) = cos 𝛼, we get sin 300 = cos600 = 1 2 . then for cos 600 = cos(90 − 30), because cos(90 − 𝛼) = sin 𝛼, we get cos600 = sin 300 = 1 2 . r : why are you using this method? s12 : because this method can be used to determine the values of acute angles. from the interview transcript excerpt, it can be seen that the subject has a good understanding of the angular relation, namely the relation between the function sin and cos at an acute angle. therefore, the subject managed to answer correctly. from this explanation, it can be concluded that the representations used by highly skilled subjects are representations using triangles, representations using circles, representations using tables, and representations using angular relations. it can be seen that subjects who use triangular representations, circle representations, table representations, and angle relation representations have used the correct concept. there are no students who experience errors in determining the value of trigonometric functions. this is because the subjects have understood from the representations they use to determine the value of the special angle trigonometric function. therefore, learning for high category prospective teachers can be done with the concept of triangles, circles, tables, and angle relations to explain the value of the special angle trigonometric function. from the explanation of the results of this study, in general, a description of the prospective teachers representation in problem-solving the value of the trigonometric function for special angles based on the ability category can be seen in table 3. table 3. description of prospective teachers representations in solving special angle trigonometric function problems ability category type of representation verbal image numeric algebra low, namely prospective teachers who get a concept comprehension test score below 55. the verbal representation used is still based on memorization and there are errors in memorizing the value of the special angle trigonometric function. the image representation that has been successfully used is a right triangle, while the graph representation of the sin and cos functions is still experiencing errors. numerical representations are not used. algebraic representations that use angular relations still have errors. medium, namely prospective teachers who get a concept comprehension test score between 55 to 80. verbal representations in the form of rote are not used. the image representation in the form of a right triangle and the graph of the sin and cos functions is used correctly. the numerical representation in the form of a table of trigonometric function values has an error. the algebraic representation in the form of an angular relation still has errors. volume 11, no 1, february 2022, pp. 55-76 71 ability category type of representation verbal image numeric algebra high, namely prospective teachers who get a concept comprehension test score above 80. the verbal representation in the form of rote is used correctly. the image representation in the form of right triangles and circles is used correctly. the numerical representation in the form of a table of trigonometric function values can be used correctly. algebraic representations in the form of angular relations can be used correctly. 3.2. discussion the results of this study contribute to the representation theory used by prospective teachers in problem-solving the value of the trigonometric function of special angles based on the background category of concept understanding ability. the results showed that prospective teachers who are in the low ability category can only use the right triangle representation to determine the value of the special angle trigonometric function. prospective teachers in the medium ability category can use right triangle representations and graphs. meanwhile, prospective teachers in the high ability category can correctly represent right triangles, unit circles, tables, and angular relations. the results of this study are in accordance with the results of previous studies which show that representation can help students solve math problems (byers, 2010; cooper & alibali, 2012; özsoy, 2018). however, the results of this study expand on the results of previous studies by explaining the representations used by prospective teachers in solving special angle trigonometric function problems based on ability categories. the first is that prospective teachers with low ability categories can only use image representations in the form of right triangles correctly in solving the problem of special angle trigonometric functions. this can be seen when prospective teachers use verbal representations in the form of memorization, image representations in the form of graphics, and algebraic representations in the form of angular relations still experiencing errors. the errors that arise are generally caused by students only memorizing the methods used and not understanding the methods used. previous research results also showed that prospective teachers still experienced errors in drawing trigonometric function graphs (jaelani, 2017; setiawan, 2021). the results of other studies also show that prospective teacher's understanding of concepts in trigonometric material is still lacking (mustangin & setiawan, 2021; nabie et al., 2018; tuna, 2013). however, what is new from the results of this study is that all prospective teachers with low abilities can use the triangle representation correctly so that they can successfully solve the problem of special angle trigonometric functions. therefore, learning material on this special angle trigonometric function should be started by using a right triangle representation. second, prospective teachers with the medium ability category can only use image representations in the form of right triangles and graphs of the sin and cos functions. the results of this study indicate that prospective teachers in this category are successful in using the representation of right triangles and graphs. this is because prospective teachers already understand the concept of right triangles and the graphs used to determine the value of a special angle trigonometric function. meanwhile, prospective teachers who use numerical and algebraic representations still experience errors. this error is caused due to inaccuracy in writing the values of trigonometric functions in tables and inaccuracy in writing the formula for angular relations. previous research results also show that this accuracy is setiawan, prospective teachers representations in problem solving of special angle trigonometry … 72 directly proportional to the abilities possessed (byers, 2009; hästö et al., 2019). the results also show that the result of this inaccuracy is that students experience errors in solving math problems (setiawan, 2020b, 2020c, 2021; setiawan et al., 2020). the results of other studies also show that students can correct wrong answers by reexamining the wrong answers (setiawan, 2020e, 2020f). the results showed that this accuracy is also important in using certain representations in solving mathematical problems. the third is that prospective teachers with high ability categories can use verbal, pictorial, numerical, and algebraic representations in solving special angle trigonometric function problems correctly. this can be seen from the results of research which show that prospective teachers in this category have understood how to use right triangles, unit circles, tables of special angle trigonometric function values, and algebraic relations correctly in determining the value of trigonometric functions. this means that a person's ability can influence the representation they use to solve problems. the results of this study are in accordance with the results of previous studies which show that students who can prove it are students who have relevant knowledge (setiawan, 2020a). even this ability is an important component of thinking disposition (perkins et al., 1993; ron, 2001; setiawan, 2020d; tishman & andrade, 1995; tishman et al., 1993). ability in the context of problemsolving is defined as the adequacy of knowledge that a person has in solving problems. therefore, prospective teachers who have high abilities have knowledge of visual, image, numerical, and algebraic representations in solving special angle trigonometric function problems. so in general it can be said that representation is influenced by the ability in the form of sufficient knowledge capacity in using the representation. for example, low-ability prospective teachers are successful in using the right triangle representation because they have good knowledge about the use of this right triangle representation. therefore it is important to equip prospective teachers with knowledge of various representations when solving the problem of special angle trigonometric functions. one of the methods proposed by the researcher based on the results of this study is to teach various representations according to the prospective teacher's ability level. thus, the main implication of the results of this study is that it contributes to the learning of special angle trigonometric functions using various representations based on ability categories (see figure 16). figure 16. representation in learning trigonometric functions special angle from figure 16 it can be seen that the first representation used in learning the value of a special angle trigonometric function is a right triangle using the acronyms sindemi right triangle angular relation table of trigonometric values graph of sin and cos functions unit circle volume 11, no 1, february 2022, pp. 55-76 73 (sin = front/hypotenuse), cosami (cos = side/hypotenuse), and tandesa (tan = front/side). the results of this study indicate that all prospective teachers of a low, medium, and high ability can use this right triangle representation. therefore, as a basis for learning the value of the trigonometric function for special angles is to use a right triangle. the second representation is the angular relation. although the results of this study indicate that there are prospective teachers with the medium ability category who experience errors in using angular relations. however, many prospective teachers use this angular relation. therefore, it is important to learn about the correct angular relation knowledge in determining the value of the trigonometric function for special angles. the third representation is to use a table of the values of the trigonometric function for special angles. this is because after understanding right triangles and angular relations in determining the value of the trigonometric function for special angles, the resulting values of right triangles are presented in tabular form. the results of this study also showed that prospective teachers with medium abilities used tables a lot. the fourth representation is by using the graph of the sin and cos functions. this is because the values in the table have a relationship if they are placed in a cartesian field. this relationship will form a graph, which is a graph of the sin and cos functions. therefore, this fourth representation is to use the graph of the sin and cos functions in learning the value of the special angle trigonometric function. the fifth representation is using the unit circle. by providing an understanding that the graph of the sin and cos functions is a circular shape with a radius of one unit. therefore, the representation of the unit circle is closely related to the graph of the sin and cos functions. through learning these five representations, it is hoped that prospective teachers can understand the material value of the trigonometric function for special angles properly and correctly. so that the problems or difficulties of prospective teachers in studying the material for the value of trigonometric functions can be reduced. 4. conclusion from the research results, it can be concluded that the background abilities possessed by prospective teachers affect the representation used in determining the value of the special angle trigonometric function. prospective teachers with low abilities can use triangular representations, prospective teachers with medium ability can use triangular and graphical representations, while prospective teachers who have high abilities can use representations of triangles, unit circles, tables of values of special angle trigonometric functions, and angular relations in determining the value of the acute angle trigonometric function. this research is only limited to special angle material which is only based on the background of the ability to understand the concept. the next research can investigate the representation used by students or prospective teachers in determining the value of the trigonometric function for non-acute angles. the results of this study will contribute to an understanding of the methods used by students or prospective teachers in determining non-acute angles, namely whether they tend to use graphical representations, unit circles, tables of trigonometric function values, reference angles, and angle relations. researchers also recommend lecturers or teachers use a sequence of representations in teaching special angle trigonometric functions based on ability. acknowledgements thanks allah swt and thanks to the parents who have educated, guided, and motivated their sons to continue to be enthusiastic in pursuing their goals. the author also setiawan, 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(2011). enhancing conceptual understanding of trigonometry using earth geometry and the great circle. australian senior mathematics journal, 25(1), 54-64. https://doi.org/10.33603/jnpm.v4i2.3386 https://doi.org/10.24815/jdm.v7i1.14495 https://doi.org/10.31980/mosharafa.v9i3.751 https://doi.org/10.25217/numerical.v4i1.839 https://doi.org/10.35706/sjme.v5i1.4531 https://doi.org/10.22342/jme.11.1.9134.77-94 https://doi.org/10.1186/s40594-015-0029-5 https://doi.org/10.1186/s40594-015-0029-5 https://doi.org/10.1080/00405849309543590 https://doi.org/10.1007/s10649-009-9201-5 https://doi.org/10.5430/wje.v3n4p1 https://doi.org/10.9790/5728-1302040104 https://doi.org/10.9790/5728-1302040104 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p255-272 255 infinity characteristics of students' mathematical problem solving abilities in open-ended-based virtual reality game learning surya amami pramuditya*, muchamad subali noto, fuji azzumar universitas swadaya gunung jati, indonesia article info abstract article history: received jan 17, 2022 revised sep 6, 2022 accepted sep 20, 2022 this research is motivated by the problem-solving ability of students is still low, and the teaching materials used cannot be studied independently. the purpose of this study was to determine students' profiles about students' problem-solving abilities seen from indicators of choosing and implementing the problem-solving strategies to solve mathematical and outside mathematics using open-ended-based virtual reality games. a virtual reality game through open-ended-based learning media is made with the concept of moving the situation of learning mathematics in the real world into a virtual world displayed through a computer. this research used a qualitative method with a case study design. the research instrument used is a question of mathematical problem-solving ability through open-ended-based and interview transcripts. the research subjects were six eighth grade junior high school students consisting of two students with high knowledge, two with moderate knowledge, and two with common knowledge. the result showed that students have different mathematical problem-solving abilities from each indicator, namely 1) identifying the adequacy of data for problem-solving, 2) making mathematical models from everyday situations or problems, 3) selecting and implementing strategies to solve math and math problems outside of using virtual reality games through open-ended based. keywords: educational game, mathematics, open ended, virtual reality this is an open access article under the cc by-sa license. corresponding author: surya amami pramuditya, department of mathematics education, universitas swadaya gunung jati jl. pemuda raya no.32, sunyaragi, kesambi, cirebon city, west java 45132, indonesia. email: amamisurya@gmail.com how to cite: pramuditya, s. a., noto, m. s., & azzumar, f. (2022). characteristics of students' mathematical problem solving abilities in open-ended-based virtual reality game learning. infinity, 11(2), 255-272. 1. introduction based on the national education curriculum in indonesia, the learning process is required not only to be oriented to the ability of understanding and knowledge but also the ability of students to think at a high level (high order thinking). according to brookhart (2010), high-level thinking is a combination of several abilities such as combining concepts into new concepts, creative, critical, systematic thinking, which caused someone can solve problems in all aspects. according to abdullah et al. (2015), high level capabilities can be https://doi.org/10.22460/infinity.v11i2.p255-272 https://creativecommons.org/licenses/by-sa/4.0/ pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 256 developed, one of them with problem-solving strategies. therefore, in the learning process, it must be created by looking at indicators that are suitable for mastering high-level abilities. if an evaluation or problem given only memorizing the formula or only applies the formula to the question, then the students will not have good problem-solving abilities. problem-solving according to muir et al. (2008), was one of the goals that must be achieved in mathematics. according to branca (1980), problem-solving was a process that prioritizes strategic steps taken to solve problems and finally find answers to those problems. therefore, the problem-solving ability must be possessed by students to train solving problems inside and outside mathematics (harisman et al., 2017). in solving the problems, students need to know the procedure or stages that are used to find out the right solution. balta and asikainen (2019) said that what students must have in the process of problemsolving is to use their qualitative understanding to get a quantitative solution. thus, the ability of problem-solving can be interpreted as the students’ ability to solve a given problem through steps and the right strategy to get a solution to a given problem. mathematics problem-solving was very important to be owned by students in mathematics learning. according to pehkonen (1997), the ability to solve mathematical problems can develop the cognitive ability and a part of the mathematical application process. valdez and bungihan (2019) revealed that problem-solving ability is not only useful for mathematics but also other science lessons even very important for daily life. however, the students’ mathematical problem-solving ability in indonesia is still low, especially in the step of choosing and using strategies that were appropriate with the given problem. based on the experience of researcher teaching in the ppl (teaching practice program), many students did the task as in the example given only. many students were still confused about the steps of completing it and did not know the strategies they have to use to answer the problem of the task. this was because students have not been trained since elementary and middle school to solve the problem of the task. students prefer to answer questions using practical formulas, thus they do not need to think hardly to answer them. the students from elementary until high school have low problem-solving abilities and there were also difficulties for teachers in teaching them. one of the materials that was considered difficult by students was the cube (solid figure), because it took the imagination of the space and parts of the building. the hardest part in this cube material was that it has not mastered optimally in terms of the nature of solid figure and plane figure, looking for area and volume, and solving routine problems in solid figure (pertiwi et al., 2021; wahyuni et al., 2020). thus, it impacted to students’ difficulty in solving the solid figure material’s problem. based on the interview results, the teaching materials used were still in the form of modul. the modul used cannot be studied independently by students, because they were not accompanied by illustrations supported and did not stimulate students to think openly or open mind. nowadays, there are many approaches that can improve problem-solving ability. one of them was an open-ended approach. the open-ended approach allowed students to choose to improve problem-solving abilities appropriate with the level of ability and interest of their students (hwang et al., 2014; kosyvas, 2016; mann, 2006). the open-ended approach made students think openly, but in the right path of solving problems because students are given open questions. according to hino (2007) learning mathematics by giving open questions was a good approach in improving problem-solving ability. the open-ended approach was an approach that emphasizes students to think openly about a problem of the task. the open-ended approach can develop students' thinking about a problem, because the given problem was open. setiawan and harta (2014) stated the openended approach was a learning approach that gave students the opportunity to solve problems in a variety of ways and more than one correct answer then discussed to compare work volume 11, no 2, september 2022, pp. 255-272 257 infinity results. according to pehkonen (1997) the open-ended approach was a method of using open-ended questions in the classroom to generate discussion activities. to support learning that used the open-ended approach in today's technology was very important. information and communication technology has increasingly developed. the benefits of smartphone have been felt by all members of society including teachers and students. based on interview with teachers at majalengka junior high school, 70% of teachers have a smartphone device and all students in one class have a smartphone. however, the use of smartphone was only limited for gaming and chatting. game that many manufactured in the market was the kind of game that was only for entertainment purposes, such as adventure, strategy of war, violence, and others. through technology will be very useful when combined between students’ skill and students’ cognitive ability. this was in line with the statement of suwanroj et al. (2019) that students' digital competence will emerge when their cognitive ability and skill are combined with technology. to solve the problems above and take advantage of technological improvement, researcher will create learning media in the form of real-world simulation games on space building material. according to pramuditya et al. (2017) the game was an entertainment media that used by someone to eliminate physical happiness and spend free time. in addition, based on previous research conducted by pramuditya, noto and purwono (2018), learning media in the form of rpg (role playing game) educational games for mathematics lessons have been made valid and practical. however, it is limited only to the materials giving and simple evaluation tools not to be made based on real-world situations that allow users to think open-ended with the material and questions given. hence, the learning media in the form of educational games can be used to overcome the difficulties and boredom of students in mathematics and can be used in classroom learning. educational game that is designed was a game in the form of virtual reality (vr). sahulata et al. (2016) stated that vr generally presents a visual experience that can be felt directly by its users displayed through a computer or through a media viewer such as google cardboard glasses. vr can be made to make the users felt the learning situation in the game directly, thus it made the students gain experience and new knowledge that useful for improving their mathematical problem-solving abilities. this study was conducted to answer the research questions; (1) how are the characteristics of the high ability students in problem-solving ability using virtual reality game through open-ended based? (2) how are the characteristics of the moderate ability students in problem-solving ability using virtual reality game through open-ended based?, (3) how are the characteristics of the low ability students in problem-solving ability using virtual reality game through open-ended based? 2. method the research method used is qualitative research, in which the researcher acts as the main instrument in this research and the data collected is not in the form of numbers but the results of interviews towards the research subjects continuously. qualitative research is done to examine the problems faced deeply by the research subjects, therefore, it can explore the selection and the implementation of problem-solvingstrategies used. as creswell (2014) stated that qualitative research is interpretive research, it means that researchers are involved in ongoing and continuously experience with the participants. the research design used was a case study of 3 students at majalengka junior high school. the case that is examined in this research is the ability of students to solve mathematical problems that is seen from the selection and implementation of problemsolving strategies used with the help of educational games. algozzine and hancock (2017) pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 258 argued that case study research is a research in exploring a system that is bound or cases that occur within a certain time from deep and reliable data sources. the example of an open-ended question for the problem-solvingability used (see table 1). table 1. an example of an open-ended problem-solving question open-ended problem-solving question questions display in the game there are 3 known objects, namely: block bricks where height = 2 cm width = 4 cm, and length = 7 cm, rubik, a cube in which the ribcage is 2 cm long acrylic with prism shaped. then mr. roni scaled the objects above, the following results of the scales are: scales a has perfect balance because the left side has 1 blockshaped object and the right side has two prism-shaped objects and one cube-shaped object. scales b also has perfect balance because the left side has 1 prism shaped object and the right side has 3 cube shaped objects. scales c on the left has 2 blockshaped objects and 1 cube-shaped object. therefore, in order to make scale c balanced, which objects that might be arranged on the right side? mr. roni has 3 types of items like in the game, then mr. roni scaled it. to make scale c becomes balanced, the total number of students who were the subjects of the study were 6 students of majalengka junior high school. the selection of these 6 students is based on the teacher's consideration by observing the students’ basic mathematics ability. the selected students consisted of 2 students with high basic mathematical ability, 2 students with moderate basic mathematical ability, and 2 students with low basic mathematical ability. the initials of the research subjects can be seen in the following table 2. volume 11, no 2, september 2022, pp. 255-272 259 infinity 3. result and discussion 3.1. result table 2 presents one of the answers of high ability students on the indicator identifying the data adequacy for problem-solving using virtual reality games with openended based. table 2. analysis of students’ high ability answers to indicator 1 students’ answers analysis of answers students can identify the data that is known in the question completely: students write the volume of box 1 and volume of box 2 correctly. students also write down the data that is known from the exact size of the aquarium such aquarium height = height of box 1 even though the size of the aquarium is no specified directly in the form of numbers. students also know what problems are asked in question number 1. although what is asked is not directly written, but it can be seen from the final answers of students who show how much to take water using the box provided that is 2 times using box 1. while for aspects of data sufficiency, students understand that to get the final results, it is required aquarium height and length data. students search for insufficient data first, that is aquarium height = box height 1 = 4 cm and aquarium length = 8 cm this research is motivated by the importance of mathematical problem-solving ability in mathematics learning, but the students' problem-solving ability is still low and the teaching materials used cannot be studied independently. the purpose of this study was to determine students’ profiles about students' problem-solving abilities seen from indicators of choosing and implementing problem-solving strategies to solve mathematical problems and/or outside mathematics using virtual reality games through open-ended based. virtual reality game through open-ended based is a learning media that is made with the concept of moving the situation of learning mathematics in the real world into a virtual world that is displayed through a computer. this research used qualitative method with case study design. the research instrument used is question of mathematical problem-solving ability through open-ended based and interview transcripts. the research subjects were 6 eighth grade junior high school students consisting of 2 high ability students, 2 moderate ability students, and 2 low ability students. data analysis in this research was carried out with the data reduction stage, the data presentation stage, and the conclusion drawing stage. the result showed that the indicators of identifying the data adequacy for problem-solving, the high ability and the pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 260 moderate ability students were able to solve the mathematical problem well, but the low ability students are not able to solve mathematical problem well. moreover, based on the results of interviews, it is obtained that students can mention all the known data correctly including the height of the aquarium. to understand the problem, students understand what is asked about the question even though it is not written on the answer sheet but students can answer according to what was asked on the question 2 times using box 1. it can be seen from the following interview script: researcher : please state what is known in the question? researcher : what is the answer? student : use box 1 twice, but don't know whether it’s right or wrong. students : volume of box 1 is 64 cm3, volume of box 2 is 8 cm3, aquarium height is the same as box 1 height, aquarium width is 4cm, and aquarium length is twice its height. in addition, students also know that the data in question number 1 is not sufficiently visible when students say they must look for the height and length of the aquarium. one of the moderate ability students’ answers on indicator identifying the data adequacy for problem-solving using virtual reality game through open-ended based (see table 3). table 3. analysis of students' moderate ability answer on indicator 1 students’ answers analysis of answers students can identify the data that is known in the question but there is one point missed. students write volume in box 1 and volume in box 2 correctly. students also write the data that is known from the size of the aquarium that is the length and width of the aquarium but do not write the height of the aquarium = height of box 1. students also know what the problem is asked in question number 1. even though what is asked is not directly written, but it can be seen from the students’ final answer which show how much to take water using the box provided that is 2 times using box 1. while for the aspect of data sufficiency, students understand that to get the final result, it needs aquarium height data. when the student look for aquarium volumes, student seems to find the height of the aquarium first, that is the height of the aquarium = height of box 1 = 4 cm moreover, based on the results of interviews, students can mention all the known data correctly including the height of the aquarium. to understand the problem, students understand what is asked about the question even though it is not written on the answer sheet but students can answer according to what was asked. when they are asked what is the answers to question number 1, students answer with box 1, 2 times. seen from the following interview script: volume 11, no 2, september 2022, pp. 255-272 261 infinity researcher : please mention what is known on the question? student : volume of box 1 is 64 cm3, the volume of box 2 is 8 cm3, the width of the aquarium is 4 cm, and the length of the aquarium is 2 times higher. researcher : what is the answer? student : use box 1 twice in addition, students also know that the data in question number 1 has not been seen enough when students say it is not enough, because they have to find the height of the aquarium first. one of the low ability students' answers on indicators identifying the data adequacy for problem-solving using virtual reality games with through open-ended based (see table 4). table 4. analysis of low ability students’ answers to indicator 1 students’ answers analysis of answers students can identify the data that is known in the question but there is one point missed. students write volume in box 1 and volume in box 2 correctly. students also write down the data that is known from the size of the aquarium that is the length and width of the aquarium but students do not write the height of the aquarium = height of box 1. students do not know what the problem asked in question number 1, it is seen that it is not written what is known and from the final answer also shows the aquarium volume is inaccurate, the aquarium volume = 2t2 x 4 cm3. while for the aspect of data sufficiency, students do not understand that to get the final result, aquarium height data is needed. when students look for aquarium volume, students do not write height data and do not look for aquarium height. moreover, based on the results of interviews, it is obtained that students cannot mention the data that is known correctly. students did not mention the known data regarding aquarium height. students also do not know the data known about the height of the aquarium seen from the students’ conversation stated that it is difficult to find height value. for the data adequacy in the questions, students knew that the data is not enough but the reason is because they can't find the volume of the aquarium. in addition, students cannot solve the problems correctly until they finish. when they are asked what the answers is to question number 1, students feel stuck when looked for the aquarium volume. therefore students know the data that is asked, but do not know the data adequacy requirements and cannot mention the data that is known completely and correctly. one of students’ answers with high ability on indicators creating mathematical models of a situation or daily problem and solving them using virtual reality game with through open-ended based (see table 5). pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 262 table 5. analysis of high ability student answers to indicator 2 students’ answers analysis of answers students write the mathematical model of the question’s problem correctly, that is 𝑉.𝐶𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 𝑉.𝑅𝑢𝑏𝑖𝑘 , therefore, students get the maximum score of rubik in 1 cardboard box correctly which is 25 rubik. another mathematical model written correctly by students is to fill in how many rubik in the box and the remaining rubik should be added to the cardboard. in addition, students are able to complete the mathematical model they have made until writing the final answer correctly. furthermore, based on the results of the interview it was found that students understood the problem from this question. students mention the mathematical model of the question correctly that is the volume of cardboard divided by the volume of rubik = maximum rubik in 1 cardboard ( 𝑉.𝐶𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 𝑉.𝑅𝑢𝑏𝑖𝑘 = 𝑚𝑎𝑘𝑠. 𝑟𝑢𝑏𝑖𝑘). hence, students get the maximum score of rubik in 1 cardboard box correctly which is 25 rubik. in addition, students are able to complete the mathematical models that they have made, it can be seen from students' conversations that they can solve until they get the final results. table 6 shows one of students' moderate ability answers on indicators creating mathematical models of a situation or daily problem and solving them using virtual reality games through open-ended based. table 6. analysis of moderate ability students’ answers to indicator 2 students’ answers analysis of answers students write the mathematical model of the question’s problem correctly, that is 𝑉.𝐶𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 𝑉.𝑅𝑢𝑏𝑖𝑘 , therefore, students get the maximum score of rubik in 1 cardboard box correctly which is 25 rubik. another mathematical model written correctly by students is to fill in how many rubik in the box and the remaining rubik should be added to volume 11, no 2, september 2022, pp. 255-272 263 infinity students’ answers analysis of answers the cardboard. in addition, students are able to complete the mathematical model they have made until writing the final answer correctly. based on the results of the interview, it was found that students understood the problem from this question. students mention the mathematical model of the question’s problem correctly, that is the volume of cardboard divided by the volume of rubik = maximum rubik in 1 cardboard ( 𝑉.𝐶𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 𝑉.𝑅𝑢𝑏𝑖𝑘 = 𝑚𝑎𝑘𝑠. 𝑟𝑢𝑏𝑖𝑘). hence, students get a maximum score of rubik in 1 cardboard correctly which is 25 rubik. in addition, students are able to complete the mathematical model that they have made, it can be seen from the students' conversation that they can solve until they get the final result. one of the low ability students' answers on indicators to make mathematical models of a situation or daily problems and solving them using virtual reality games through open-ended based (see table 7). table 7. analysis of low ability students’ answers to indicator 2 students’ answers analysis of answers students write a mathematical model of the question’s problem but it is not quite right. students write mathematical models to find the remaining rubik in 3 boxes. although in the final answer the students write down the lack of rubik in each cardboard but the maximum rubik and the number of rubik that must be entered into each cardboard are not written down. pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 264 meanwhile, the results of interviews obtained that students understand the problem from this question. students mention the mathematical model of the question’s problem correctly, that is the volume of cardboard divided by the volume of rubik = maximum rubik in 1 cardboard ( 𝑉.𝐶𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 𝑉.𝑅𝑢𝑏𝑖𝑘 = 𝑚𝑎𝑘𝑠. 𝑟𝑢𝑏𝑖𝑘). therefore, students get a maximum score of rubik in 1 cardboard correctly which is 25 rubik. in addition, students are able to complete the mathematical model that they have made, it can be seen from the students' conversation that they can solve until they get the final result. one of the answers of high ability students on indicators of choosing and implementing strategies to solve mathematical problems and or outside mathematics using virtual reality games through open-ended based (see table 8). table 8. analysis of high ability student answers to indicator 3 students’ answers analysis of answers student answered the questions by collecting data on scales a and scales b, therefore to obtain the prism volume that is value prism volume = 24 cm3. then students use the volume of prism to obtain objects that must be in the scale c. thus the student has chosen the right way or strategy to answer the problem in the question that is the strategy of organizing data. in the final answer to the question’s problem, students write the correct answer that is on the right side of the c scale, 15 cubes are needed hence the scales are balanced. therefore, students are able to solve the problem using the strategy. besides, based on the results of interviews, students answer the questions by collecting data on scales a and scales b to obtain the prism volume value that is prism volume = 24 cm3. therefore, the strategy used by students is to process or organize data from existing data. in the conversation, it appears that students collect the known data from scales a and b afterwards, looking for prism volumes based on these data. then we get the objects needed, so, the scale c is balanced. thus, students have chosen the right way or strategy to answer the problem in the question. in the final answer to the question’s problem, students said the correct answer is that on the right side of the scale c, 15 rubik are needed hence the scale is balanced. volume 11, no 2, september 2022, pp. 255-272 265 infinity table 9 shows one of the answers of moderate ability students on indicators of choosing and implementing strategies to solve mathematical problems and or outside mathematics using virtual reality games through open-ended based. table 9. analysis of students' answers to the ability being on indicator 3 students’ answers analysis of answers students answer the questions by looking for patterns from scale a and scale b thus they get the right pattern. then students substitute the patterns of the scales a and b, namely 1 beam = 2 prisms + 1 cube and 1 prism = 3 cubes into the c scale. hence using the pattern, the student writes the final answer correctly that is to the right of the c scale we need 15 objects cube shaped thus the scales are balanced. meanwhile, based on the results of interviews obtained that students answer the questions by collecting data on scales a and scales b thus obtain the items needed on the scale c to be balanced. therefore the strategy used by students is to look for patterns from known data. in the conversation it appears that students look for patterns from the scales a and b. after knowing the pattern enters into the scales c. then objects obtained for the scales c. thus, the student has chosen the right way or strategy to answer the problem in the question. in the final answer to the question problem, students said the correct answer is that on the right side of the scale c, 15 cubes are needed thus the balance is balanced. therefore, students are able to solve question’s problem using these strategies. one of the answers of students with low ability on indicators to choose and implement strategies to solve mathematical problems and or outside mathematics using virtual reality games through open-ended based (see table 10). table 10. analysis of low ability student answers to indicator 3 students’ answers analysis of answers students answer the questions by looking for patterns from scale a and scale b thus they get the right pattern. then students substitute the pattern of the scale a, namely 1 beam = 2 prisms + 1 cube into the scale c. hence, using the pattern, students write the final answer correctly, that is to the right of pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 266 students’ answers analysis of answers the scale c, it takes 15 cube-shaped objects thus the scales are balanced. moreover, based on the results of the interview it was found that students answered the questions by looking for a balanced pattern of scales a and b thus obtain the items needed on the c balance to be balanced. thus the strategy used by students was to find patterns from known data. in conversation shows that, the students look for patterns of scales a and b. after knowing the pattern, then obtained any object for scales c. thus students have chosen the right way or strategy to answer the problem in the question. in the final answer to the question problem, students said the correct answer that is to the right of the scale c requires 4 prisms and 3 cubes thus the scales are balanced. therefore students are able to solve problem problems using these strategies. 3.2. discussion students have good or not good mathematical problem-solvingabilities, there are indicators or aspects to assess them. there are many kinds of indicators for assessing students' mathematical problem-solving abilities. the questions made by researchers, namely: 1) identifying the adequacy of data for problem-solving, 2) making mathematical models of daily situations or problems and how to solving them, 3) choosing and implementing strategies to solve mathematical problems and/or outside mathematics using virtual reality games through open-ended based (widodo et al., 2021). based on data taken from the results of students' answers and interviews, it can be seen the differences in the characteristics of students in solving problem question using virtual reality games through open-ended based. 3.2.1. characteristics of students' low problem-solving abilities using virtual reality games through open-ended based when students are given a problem-solving question that is presented in virtual reality games through open-ended-based, students are only able to know the data adequacy requirements and the data being asked. but students were not able to mention data that is known to be complete and correct. therefore, the two low-ability students have not been able to identify the adequacy of the data for problem-solving in the game. the problem-solving was a difficult process for students, one of which is at the problem understanding step and planning a problem. but when problems that contain indicators make a mathematical model of a situation or daily problem and solve it, students are able to make a mathematical model of a situation or daily problem and solve it. it can be seen that all students can write mathematical models of problem questions in the game correctly. this is because students are helped by illustrating problems in the game, thus it is volume 11, no 2, september 2022, pp. 255-272 267 infinity easier to make mathematical forms of the problem (harisman et al., 2021; hutajulu et al., 2022; kariadinata, 2021). this is in line with pramuditya, noto and syaefullah (2018) that the mathematics rpg educational game is interesting, fun, and educated, because of illustrations. students can also solve problem questions using models or plans that they have previously made. this is in line with prihandika and saputro (2021) states that students are able to draw up a completion plan or make a mathematical model and implements a completion plan. while the questions that contain indicators choose and implement strategies to solve mathematical problems and/or outside mathematics that all students studied are able to choose and implement strategies to solve mathematical problems and/or outside mathematics from the questions in the game. it can be seen that all students use a way of solving it by choosing the right problem-solving strategy. all students also get the correct final answer by using a solution to the strategy they have made before. this is in line with karlimah et al. (2021) which stated that in making a settlement plan in this case choosing and implementing a problem-solving strategy, look for the relationship between the information provided and the unknown exactly hence it is possible to obtain unknown results. 3.2.2. characteristics of students' moderate problem-solving abilities using virtual reality games through open-ended based based on the data taken from the results of students' answers and interviews, it can be seen that moderate-able students are able to identify the adequacy of the data for problemsolving in the game. it can be seen that students can understand the problem in the game well. this is because students feel the illustrations in the game help in remembering what is known, therefore they can understand the problem question. based on research by akbar et al. (2018) which stated that students more often finish directly and feel no need to write it down, because they feel they are wasting time. but in this study, students write the data that is known and the steps to solve them, which are not written only the data that is asked. students are also able to know the adequacy of the data as a condition for answering questions. it can be seen that students can write mathematical models of problem questions that exist in the game correctly. this is because students are helped by illustrating problems in the game, hence it is easier to make mathematical forms of the problem. this is in line with pramuditya, noto and syaefullah (2018) that the mathematics rpg educational game is interesting, fun, and educational because of illustrated illustrations. students can also solve problem problems using models or plans that they have previously made. this is in line with research prihandika and saputro (2021) states that students are able to draw up a completion plan or make a mathematical model and implements a completion plan. that all students studied are able to choose and implement strategies to solve mathematical problems and / or outside mathematics from the problems in the game. it can be seen that all students use a way of solving it by choosing the right problem-solving strategy. all students also get the correct final answer by using a solution to the strategy they have made before. this is in line with karlimah et al. (2021) which stated that in making a settlement plan in this case choosing and implementing a problem-solving strategy, look for the relationship between the information provided and the unknown exactly hence it is possible to obtain unknown results. pramuditya, noto, & azzumar, characteristics of students' mathematical problem solving … 268 3.2.3. characteristics of students' high problem-solving abilities using virtual reality games through open-ended based based on data taken from the results of students' answers and interviews, it can be seen that high-ability students have been able to identify the adequacy of data for problemsolving in the game. it can be proven that students can understand the problem in the game as well. this is because students feel the illustrations in the game help in remembering what is known. therefore they can understand the problem question. based on research by akbar et al. (2018) which stated that students more often finish directly and feel no need to write it down because they feel they are wasting time. however, in this study, students write the data that is known and the steps to solve them, which are not written only the data that is asked. students are also able to know the adequacy of the data as a condition for answering questions. it can be seen that students can write mathematical models of question’s problems that exist in the game correctly. this is because students are helped by illustrating problems in the game so that it is easier to make mathematical forms of the problem. this is in line with pramuditya, noto and syaefullah (2018) that the mathematics rpg educational game is interesting, fun, and educational because of illustrated illustrations. students can also solve problem problems using models or plans that they have previously made. this is in line with research prihandika and saputro (2021) states that students can draw up a complete plan or make a mathematical model and implements a completion plan. that all students studied can choose and implement strategies to solve mathematical problems and/or outside mathematics from the problems in the game. it can be seen that all students use a way of solving it by choosing the right problem-solving strategy. all students also get the correct final answer by using a solution to the strategy they have made before. this is in line with karlimah et al. (2021) which stated that in making a settlement plan in this case choosing and implementing a problem-solving strategy, look for the relationship between the information provided and the unknown exactly hence it is possible to obtain unknown results. muir and beswick (2005) stated that the process of thinking in combining some knowledge in problem-solving is an important factor in whether students can solve a problem. every student who has different abilities have different problem-solving thinking processes. according to muir et al. (2008) that someone’s ability to understand a mathematical problem structure is an important element in problem-solving. the following findings are presented about the characteristics of students ranging from low, moderate, and high abilities found during this study (see table 11). table 11. characteristic of students' problem-solving abilities no low ability students moderate ability students high ability students 1 guessing when choosing answers in the game. answering questions with a strategy that students can then choose the correct answer choose the answer then adjust the solution strategy according to the answer choice 2 students are only able to choose one problem-solving strategy students are only able to choose one problemsolving strategy students can choose more than one problem-solving strategy in the game volume 11, no 2, september 2022, pp. 255-272 269 infinity no low ability students moderate ability students high ability students 3 students are only able to work on problems with one completion strategy students are only able to work on problems with one completion strategy students can answer questions in a variety of ways 4 some steps are not yet right in the problem-solving process students can answer correctly at each stage of completion students can answer correctly at each stage of completion 5 students are not able to interpret known data from game illustrations students can interpret known data from game illustrations able to interpret known data from game illustrations 4. conclusion based on the analysis and discussion of the mathematical problem-solving of 6 students of majalengka junior high school can be concluded that students have mathematical problem-solving abilities different from each indicator. for indicators identifying the adequacy of data for problem-solving, high-ability students and moderateability students can solve mathematical problem-solving well. students with low ability have not been able to solve mathematical problem-solving well. the indicators make a mathematical model of a situation or daily problem and solve it that students with high, moderate, and low abilities are able to solve mathematical problems well. while there are indicators for choosing and implementing strategies to solve mathematical problems and/or outside mathematics for in-game problems, those students with high, moderate, and low abilities can solve mathematical problem-solving well. references abdullah, a. h., abidin, n. l. z., & ali, m. 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(2021). development of mathematical problem solving tests on geometry for junior high school students. jurnal elemen, 7(1), 221-231. https://doi.org/10.29408/jel.v7i1.2973 https://doi.org/10.29408/jel.v7i1.2973 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p165-178 165 mathematical probability: learner's misconception in a selected south african school winston hendricks, babawande emmanuel olawale* university of fort hare, south africa article info abstract article history: received nov 13, 2022 revised jan 31, 2023 accepted feb 22, 2023 mathematics plays an essential role in developing human thought, particularly in developing problem-solving and reasoning. while mathematics has become a problem-solving tool in various fields, including science, it has distinct qualities known as probability and, more specifically, probability theory. for most learners, the probability is difficult to learn and conceptualize. hence, the present study investigates learners’ misconceptions in the teaching and learning of probability in a selected school in the eastern cape province, south africa. underpinned by a post-positivist paradigm, the study employed a quantitative research approach and a survey design in which data were gathered from mathematics learners from grades 10-12. findings revealed that although the frequency of misconceptions varied across grade levels, it was difficult to describe how misconceptions about probability changed. as such, while learners progressed through the grades, some misconceptions faded with age, others remained stable, and others grew in power. the findings also revealed that the types of probability misconceptions did not differ significantly by gender, and male learners tend to have more misconceptions about probability than female learners. keywords: experiment, influence, misconception, outcomes, performance, probability theory this is an open access article under the cc by-sa license. corresponding author: babawande emmanuel olawale, school of further and continuing education, faculty of education university of fort hare university of fort hare alice campus, ring road, alice 5700, south africa. email: bolawale@ufh.ac.za how to cite: hendricks, w., & olawale, b. e. (2023). mathematical probability: learner's misconception in a selected south african school. infinity, 12(1), 165-178. 1. introduction student misconceptions have been a concern for many educators, researchers, and mathematics teachers. as a result, an increasing number of researches in mathematics education focus on students’ misconceptions in various mathematical domains (mut, 2003; santos-trigo, 2020; stylianides et al., 2016). several studies that deal with probability and probabilistic thinking have often been categorised into two types. while the first type focuses on how people think, the second focuses on influencing how people think, both of which are investigated by psychologists and mathematics educators, respectively (mut, 2003; https://doi.org/10.22460/infinity.v12i1.p165-178 https://creativecommons.org/licenses/by-sa/4.0/ hendricks & olawale, mathematical probability: learner's misconception … 166 shaughnessy, 1992). there is also substantial evidence that traditional probability instruction, which consists mostly of formal definitions, rules, and procedures, does not eliminate misconceptions about probability (khazanov, 2005; mutara & makonye, 2016; shaughnessy, 1992). the above assertion is true, given that misconceptions can coexist perfectly with correct conceptions, thereby interfering with learners’ ability to apply them regularly and confidently (khazanov & gourgey, 2009; mutara & makonye, 2016). as a result, while learners may master probability rules and processes and can calculate accurate answers on mathematics tests, these same learners often misunderstand essential principles and concepts, frequently disregarding the rules when making decisions regarding uncertain situations. however, numerous studies from various theoretical vantage points appeared to support the idea that students frequently hold beliefs that impede their understanding of ideas in probability (ang & shahrill, 2014; batanero et al., 2016). as a result, it is widely agreed upon that representativeness, equiprobability bias, beliefs, and human control are a few frequent ways of thinking that prevent learners from learning about probability (ang & shahrill, 2014). for instance, the representativeness misconception concerns students’ erroneous belief that samples that match the population distribution are more likely to be accurate than those that do not (kahneman & tversky, 1973). as such, students with this misconception, for instance, will believe that a series of coin tosses with roughly equal numbers of heads and tails is more likely than a series with significantly more tails than heads (ang & shahrill, 2014; kahneman & tversky, 1973). however, the probability for both series is the same. regarding equiprobability bias, students who hold this false belief typically believe that random events are equally likely by their very nature (ang & shahrill, 2014; khazanov & prado, 2010). or, to put it another way, they see the odds of achieving diverse results as equally likely occurrences. according to lecoutre (1992), the propensity of learners to see various experiment results as equally likely is equiprobability. for instance, pupils who believe in the equiprobability bias mistakenly believe that all possible sums are equally likely when two dice are rolled. however, students are unaware that the sum of the two dice is more likely to be 6 or 7 than 2 or 12. in terms of beliefs, many young learners believe that a force outside of their control determines the final result of an event. sometimes this power is god, other forces like the wind, and other times its luck or wishes (truran, 1994). regarding human control, nicolson (2005) argues that some learners believe that the outcomes depend on how one throws or manipulates these various tools. it has long been understood that one of the most crucial educational objectives of stochastic instruction is to rectify students' misconceptions about probability (ang & shahrill, 2014; gallagher, 2023). shaughnessy, a prominent researcher in the field of probability and statistics education, argues that one of the critical objectives of stochastic instruction should be to show students "how misconceptions of probability can lead to erroneous decisions" (shaughnessy, 1992, p. 482). thus, addressing students' incorrect intuitions and assumptions requires a significant shift in focus from merely supplying formulas, rules, and calculational procedures when teaching probability for conceptual understanding (khazanov & gourgey, 2009). while probability provides a tool for modelling and producing reality, misconceptions about probability can influence people's judgment in crucial situations, such as investing, jury verdicts, and medical tests (sharna et al., 2021). given the importance of probability, new mathematics curricula for schools are being established in countries worldwide. for example, since gaining democracy in 1994, south africa has implemented several educational reforms (olawale et al., 2021), particularly in the mathematics curriculum, with the probability topic becoming compulsory for the first volume 12, no 1, february 2023, pp. 165-178 167 time in grades 10 to 12 in 2012 (khazanov & prado, 2010; makhubele, 2015). the topic of probability in the south african curriculum assessment policy statements (caps) entails knowing how to determine the likelihood of events occurring, and this topic is ranked sixth in importance in the mathematics curriculum for further education and training (fet) (department of basic education, 2011). while topics such as theoretical and experimental probability, dependent and independent events, simple and compound events, and the generalisation of the fundamental counting principle are covered, the probability is given an 18 percent weighting in grades 10, 11, and 12 mathematics curricula due to its relevance and importance (mutara & makonye, 2016). however, despite the importance of probability and its weighting in the south african curriculum, one of the major challenges faced by high school teachers saddled with the responsibilities of teaching this topic include incoherent probability content knowledge which became an immediate issue after the reintroduction and compulsion of this topic (chernoff, 2012). it could also be argued that most south african mathematics teachers face numerous difficulties in teaching probability because teaching probability for conceptual understanding requires a considerable shift in focus from just providing formulas, rules, and methods for computations to addressing students' erroneous intuitions and prejudices (khazanov & gourgey, 2009; khazanov & prado, 2010; sharma, 2006). in addition, there were also insufficient teaching and learning support materials available to deal with this new issue (mutara & makonye, 2016). in light of these circumstances, we investigated learners’ errors and misconceptions related to the solutions of probability problems amongst learners in grades 10-12. thus, the main objective of this study was to investigate how the five common misconceptions of mathematical probability differ in relation to grade level and gender among high school learners. 2. method this study is guided by a post-positivist paradigm. according to creswell (2014), post-positivists are deterministic, reductionists interested in identifying the reasons that impact specific outcomes. this paradigm shifts away from the solely objective perspective taken by the logical positivists and is concerned with the subjectivity of reality. a postpositivist paradigm was found suitable for this study because it prioritises creating numerical measures of observations and researching human behaviour. given that paradigm selection determines the research approach, this study employed a quantitative research approach. to investigate phenomena and their interactions in a methodical way, quantitative research methods use numbers and anything that can be measured to explain, predict, and control a phenomenon; the quantitative approach seeks to provide answers to queries about correlations among quantifiable variables (creswell, 2014; mohajan, 2020). for this study, three groups of learners were investigated which are five learners in grade 10 (ages 15-16), 12 learners in grade 11 (ages 16-17), and seven learners in grade 12 (ages 17-18), making a total of 24 learners (see table 1). the statistical age standard per grade is the grade number + 6 according to the south african schools act, 84 of 1996 (department of basic education, 2011). the convenience sampling technique was considered appropriate for the study given that the sample represented a range of learners from different socio-economic and cultural backgrounds. for the present study, female learners constituted 54% of the sample size, while male learners were only 46% (see table 2). hendricks & olawale, mathematical probability: learner's misconception … 168 table 1. learner’s distribution according to grade level grade number(s) percentage grade 10 5 21% grade 11 12 50% grade 12 7 29% total 24 100 table 2. learners’ distribution according to gender grade number(s) percentage females 13 54% males 11 46% total 24 100 the probabilistic misconception test (pmt) was developed to collect data for the study. the pmt test consisted of five well-known probability questions administered to the participants. the test was related to five different types of probability misconceptions such as: a. simple and compound events: for example, “the letters in the word “cicek” are written one by one on the cards, and then these cards are placed in a bag. what is the probability of getting the letter “c” from this box at random?” (i̇lgün, 2013; mut, 2003). b. representativeness: for example, “say you flip an ordinary quarter several times in successions with h representing a head coming up and t representing a tail. the notation ht means that in two successive flips, a head occurred, followed by a tail. if you flip a quarter 5 times in succession, which of the following sequences are you most likely to observe” (i̇lgün, 2013; mut, 2003). c. positive and negative recency effects: for example, “when tossing a coin, there are two possible outcomes: either heads or tails. adu flipped a fair coin three times, and in all cases, tails came up. adu intends to flip the coin again. what is the chance of getting heads at the fourth time?” (i̇lgün, 2013; mut, 2003). d. effect of sample size: for instance, “a doctor keeps the records of newborn babies. according to his records, the probability of which of both gender (male & female) is higher?” (i̇lgün, 2013; mut, 2003). e. equiprobability bias: e.g., “there are six fair dies, each of which an ordinary cube with one face is painted white, and the other faces painted black. if these dies are tossed, which would be more likely?” (i̇lgün, 2013; mut, 2003). as such, one question each was raised based on the different types making a total of five (5) questions. the items in the pmt test were gathered from relevant literature. two mathematics teachers from the selected school and one lecturer from the mathematics education department revised and controlled the question in terms of mathematical structure to ensure the instrument's content validity. the data collected were analysed descriptively. the levels of all the independent variables utilised in this study were used to compute the frequency of dependent variables. frequency tables were used to tabulate the dependent variables in relation to the independent variables. the university's ethics committee approved this study, and formal approval letters were sent to the participating schools' students, lecturers, and principals to request their consent. volume 12, no 1, february 2023, pp. 165-178 169 3. result and discussion the present study sought to investigate learners’ misconceptions with regard to probability based on grade level and gender. 3.1. types of misconceptions in relation to grade level 3.1.1. misconceptions in relation to simple and compound events for this study, the first question in relation to “simple and compound events” with respect to grade levels was presented as shown in table 3. table 3. number and percentages of learners' responses to question 1 by grade level answers grade levels total grade 10 grade 11 grade 12 incorrect answers 1 2 0 3 20% 17% 0% 13% correct answers 2 40% 10 83% 7 100% 19 79% misconceptions 2 40% 0 0% 0 0% 2 8% table 3 shows that 79 percent of learners answered the question correctly, and it could be concluded that learners better understand the simple and compound events in probability. in grades 11 and 12, the percentages of correct answers were higher. however, only 8 percent of learners had misconceptions in relation to this topic. from the above table, one could conclude that there was an effect of grade level on this misconception type as learners in grades 11 and 12 had not reported any form of misconception which could be because of their exposure to this topic during their entrance into the further education and training (fet) phase, that is, grade level 10. 3.1.2. misconceptions in relation to representativeness table 4. number and percentages of learners' responses to question 2 by grade level answers grade levels total grade 10 grade 11 grade 12 incorrect answers 2 7 4 13 40% 58% 57% 54% correct answers 0 0% 3 25% 2 29% 5 21% misconceptions 3 60% 2 17% 1 14% 6 25% in the second question, which investigated the misconceptions in relation to representativeness, 25 percent of the learners had misconceptions in this question. however, misconceptions varied across the grade level. as shown in table 4, a larger percentage of learners in grade 10 showed misconception at 60 percent, while 17 percent and 14 percent of misconception were recorded as grade 11 and 12, respectively. this finding is similar to that of ang and shahrill (2014), who argued that due to the false belief that samples that hendricks & olawale, mathematical probability: learner's misconception … 170 match the population distribution are more likely to occur than samples that do not, learners frequently hold misconceptions about samples that are representative of the population. this is mostly due to the widespread perception that outcomes or consequences are determined by natural forces that impact an event’s course of action (ang & shahrill, 2014). from the above findings, although students showed a significant number of misconceptions about representativeness, these misconceptions become less common as students move up the grade levels. 3.1.3. misconception in relation to positive and negative recency effects table 5. number and percentages of learners' responses to question 3 by grade level answers grade levels total grade 10 grade 11 grade 12 incorrect answers 2 6 5 13 40% 50% 71% 54% correct answers 1 20% 1 8% 0 0% 2 8% misconceptions 2 40% 5 42% 2 29% 9 38% in this type of misconception, learners assume that independent events' outcome depends on the previous outcomes. learners believe that the probability of obtaining a head is increased on the next toss after a run of five tails with a fair coin. as such, the result in table 5 shows that only 8% of the learners were able to provide correct answers to the 3rd question. however, 38 percent of the learners had a main misconception, while 54 percent provided an incorrect answer. thus, in terms of grade level, positive and negative recency effect misconception types did not change across grade levels, as shown in table 5, given that 40 percent demonstrated this misconception at grade level 10. in comparison, 42 percent and 29 percent showed this misconception in grades 11 and 12, respectively. this stresses the importance of effective teaching of probability in schools. thus, mut (2003) argues that there are two approaches to effectively teaching probability. as such, while some learners believe they must estimate a specific conclusion, others think they must evaluate the probability of a series of outcomes. even for the same issue, conflicts may result in discrepancies between the two approaches. 3.1.4. misconception in relation to effect of sample size table 6. number and percentages of learners' responses to question 4 by grade level answers grade levels total grade 10 grade 11 grade 12 incorrect answers 3 3 0 6 60% 25% 0% 25% correct answers 1 20% 1 8% 0 0% 2 8% misconceptions 1 20% 8 67% 7 100% 16 67% volume 12, no 1, february 2023, pp. 165-178 171 in question 4, only 8 percent of the learners answered the question correctly, whereas 67 percent had a misconception about this question. this percentage is high, showing that learners are confused between ratio and proportion, as well as probability subjects. as shown in table 6, none of the learners in grade 12 solved the question correctly. as such, the frequency of the misconception increased with grade level. with high expectations that this concept would be easier for learners in the higher-grade level, the causes of this misconception type are, therefore, difficult and complex to explain. therefore, kaplar et al. (2021) argue that learners often commit this type of error because they think that the probability of the judged sample statistic is independent of the sample size. as such, they become insensitive to sample size. in fact, a lot of learners think that the sample size has no bearing on how closely the sample statistic and population parameter resemble each other. as a result, they tend to commit errors. the findings of this study refute those of other studies, which revealed that most learners often answered probability questions that are related to the effect of sample size correctly (kang & park, 2019; kaplar et al., 2021; kustos & zelkowski, 2013). 3.1.5. misconception in relation to equiprobability bias table 7. number and percentages of learners' responses to question 5 by grade level answers grade levels total grade 10 grade 11 grade 12 incorrect answers 1 3 1 5 20% 25% 14% 21% correct answers 4 80% 8 67% 4 57% 16 66% misconceptions 0 0% 1 8% 2 29% 3 13% in table 7, it is clear that learners do not have much difficulties solving the question, as 66 percent answered it correctly. as such, only 13 percent of the learners had misconceptions. however, the frequency of equiprobability bias misconception increases from grade level 10 to grade level 12. on the other hand, the frequencies of correct answers decrease across grade levels. although the question seems easy, learners in high-grade levels, such as grades 11 and 12, had difficulty in solving this question correctly. surprisingly, the percentage of the correct answer for this question was the highest in grade 10. one could conclude that the equiprobability bias amongst these learners is because they see various experiment results as equally plausible. as a result, learners who suffer from equiprobability bias, for instance, often believe that when two dice are rolled, all possible sums are equally likely. the fact that the sum of 6 for the two dice is more likely than the sum of 2 is not apparent to them. this finding is consistent with that of gauvrit and morsanyi (2014), who argued that equiprobability bias is not exclusive to any class or grade. as a result, learners are prone to this misconception because of their belief that all outcomes have the same probability, and in such situations, the base set is always neglected. hence, researchers such as (ang & shahrill, 2014; gauvrit & morsanyi, 2014; kaplar et al., 2021) adds that learners who hold this false belief typically believe that random events are equally likely by their very nature. or, to put it another way, they see the odds of achieving diverse results as equally likely occurrences (gauvrit & morsanyi, 2014). hence, a need for teachers and researchers to better understand the topics which are perceived as important so that they hendricks & olawale, mathematical probability: learner's misconception … 172 may better evaluate the conceptions of probability that learners have at different ages and how these conceptions can be changed. 3.2. types of misconceptions in relation to gender table 8. number and percentages of learners' responses to questions 1-5 based on grade level questions gender gl 10 gl 11 gl 12 i c m i c m i c m question 1 female n % 2 100% 7 100% 4 100% male n % 1 33% 2 67% 2 40% 3 60% 3 100% question 2 female n % 2 100% 3 43% 2 28.5% 2 28,5% 3 75% 1 25% male n % 3 100% 4 80% 1 20% 1 33.3% 1 33.3% 1 33.3% question 3 female n % 1 50% 1 50% 3 43% 1 14% 3 43% 2 50% 2 50% male n % 1 33.3% 1 33.3% 1 33.3% 3 60% 2 40% 3 100% question 4 female n % 1 50% 1 50% 2 29% 1 14% 4 57% 4 100% male n % 2 67% 1 33% 1 20% 4 80% 3 100% question 5 female n % 2 100% 1 14% 5 72% 1 14% 1 25% 3 75% male n % 1 33% 2 67% 1 25% 3 75% 1 33% 2 67% 3.2.1. misconceptions on simple and compound events in relation to gender question 1 examines the “simple and compound events” misconception. the misconceptions were more frequent among females than males in grade 10 as there were more correct answers from male learners. this finding is consistent with that of mut (2003), who investigated students’ probabilistic misconceptions and found that the percentage of females who had the simple and compound event misconception was higher than males. this may be explained by the fact that the majority of learners who had this kind of misconception were unable to distinguish between these two types of events (i̇lgün, 2013; mut, 2003). kennis (2006) also added that this misconception may have its roots in the fact that some learners fail to take into account the sequence in which the outcomes of a compound event will occur, which leads them to determine the sample size for this event wrongly. however, in grades 11 and 12, there were no misconceptions from both genders. this finding is in line with that of i̇lgün (2013), who examined the reasons underlying probabilistic misconceptions in relation to gender, findings revealed that, in terms of simple and compound events, males and females do not differ from each other. i̇lgün (2013) claimed that the underlying reason for no significant difference could be attributed to the fact that both genders have been exposed to the idea at the higher grades and now have a sufficient grasp of the concept. in conclusion, one could argue that misconceptions in relation to simple and compound events varied across gender at the lower grade. 3.2.2. misconceptions on representativeness in relation to gender while question 2 investigated misconceptions in relation to gender, findings in table 8 revealed that this misconception was frequent amongst male learners in grades 10 and 12. volume 12, no 1, february 2023, pp. 165-178 173 in contrast, female learners tend to have this type of misconception in grade 11. thus, one could conclude that the misconception regarding representativeness varies across gender. this finding corroborates that of mut (2003) who argued that although misconception type in probability in relation to representativeness may not vary across gender in different grade level, female learners had more tendencies in misconceptions than their male counterparts. however, kustos (2010), argued that regardless of gender, insensitivity to sample size impacts on predictive accuracy, inappropriate confidence in prediction based on false input data, misunderstandings of chance, the illusion of validity, and misconceptions of regression are some of the ways that learners maintain a representative misconception. 3.2.3. misconceptions on positive and negative recency effects in relation to gender in this study, the third question examined the misconception f positive and negative recency effects concerning gender. while females had a high misconception rate in question 3, male learners deferred a little in grades 10 and 11. as shown in table 8, in grade levels 10, 11, and 12, the misconception was stronger among female learners. for grade 12 learners, while females showed a high percentage of misconception of 50%, male learners had no misconceptions with respect to the question. one could conclude that positive and negative recency effects were more frequent amongst female than male learners. the finding of this study is in line with mut (2003), who investigated the distribution of misconception type ‘positive and negative regency effect’ with respect to gender at all grade levels and found out that the positive-negative regency effect was more frequent among female learners than male (kustos, 2010; mut, 2003). according to the data gathered, mut (2003) also discovered that the misconception type for the positive-negative regency effect remained constant across grade levels. however, it was shown that learners were less likely to fall victim to this fallacy, also known as the “gambler fallacy,” in higher grade levels where probability is taught than in lower grade levels. this result emphasizes how crucial it is to teach probability adequately in the classroom. 3.2.4. misconceptions on the effect of sample size in relation to gender the distribution of the misconception-type effect of sample size in relation to gender at all grade levels is shown in table 8. question 4 examined the effect of sample size as a misconception type. in grade 10, the misconception was higher amongst female learners than male learners. however, in grade 11, the frequency was higher amongst male learners than female learners. lastly, in grade 12, both genders reported a high frequency of misconception. in conclusion, although misconception varied across gender, it appears in table 8 that the misconception type was more frequent amongst male than female learners. this result is comparable to that of kennis (2006), who looked at probabilistic misconception across age and gender and discovered that females outperform males in tasks, involving the effect of sample size. this is due to the fact that knowledge of equal fractions, percentages, or fractions outweighs the use of common sense amongst female learners. however, both genders are more inclined to ignore sample space while making a probabilistic decision (kennis, 2006). 3.2.5. misconceptions on equiprobability bias in relation to gender the distribution of the equiprobability bias misconception in relation to gender at all three-grade levels is shown in table 8. in grade 10, while there were no misconceptions by participating learners, the frequency of this misconception in grade 11 was higher among hendricks & olawale, mathematical probability: learner's misconception … 174 females than males. however, in grade 12, males showed a high level of misconception than females. based on the above frequency shown in table 8, it could be stated that equiprobability bias was frequent among male learners. the finding of this study refutes that of mut (2003), who stated that the frequency of equiprobability bias among females is often higher than among males across all grade levels. the findings further revealed that males had much more tendency to have this misconception than females. in the literature, we were unable to locate any recent and relevant studies that looked at gender-related misconceptions about equiprobability bias. as a result, we were unable to compare the current study's findings further. however, we decided to indirectly compare the findings of this study with research on the gender gap in mathematical achievement, some in probability. although each area had mixed results, that is, while some studies such as (bottia et al., 2015; fortin et al., 2015; li et al., 2018; marcenaro–gutierrez et al., 2018) showed that females had a higher achievement in mathematics and probability, others such as (bottia et al., 2015; innabi & dodeen, 2018; niederle & vesterlund, 2010) showed that males had higher achievement, and some found no significant difference (guo et al., 2015; reilly et al., 2015). 4. conclusion the study's findings indicate that the prevalent assumption about intuition's stability is incorrect. to put it another way, the frequency of misconceptions varied by grade level. it was difficult to describe how misconceptions about probability changed. as learners progressed through the grades, some misconceptions faded with age, others remained stable, and others grew in power. however, it should be noted that, in comparison to previous grades, learners in grades 10 and 12 show minimal tendency against misconceptions. according to the study, the curriculum program contains probability subjects at all three grade levels. also, another goal of the study was to see how different types of misconceptions differ by gender. according to the findings, describing the change in the type of misconception in probability with respect to gender is fairly tough and complex. despite the fact that the types of probability misconceptions did not differ significantly by gender, male learners tended to have more misconceptions about probability than female learners. thus, given that probability as a subject necessitates a method of thinking that is not solely based on technical information and actions that lead to solutions, mathematics teachers should strive to encourage learners to develop new intuitions when teaching probability. furthermore, probability instruction should enable the learners to experience conflicts between their intuition and specific sorts of reasoning in stochastic settings. lastly, these probability misconceptions should be taken into account by programs/policymakers in the development of mathematics curricula in schools. also, teacher training colleges should also incorporate several ways of teaching probability and statistics in classrooms to aspiring mathematics teachers. however, while the sample size for this study comprises a very small number with three grade levels at the further and education and training phase, further studies may consider a big sample size and other educational phases such as the senior phase. also, while this study relies solely on quantitative data, other studies may consider using a mixed methods approach as this may provide more accurate and robust information. volume 12, no 1, february 2023, pp. 165-178 175 references ang, l. h., & shahrill, m. 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(1994). children’s understandings of random generators. in c. beesey & d. rasmussen (eds.), mathematics without limits (pp. 356-362). proceedings of the 31st annual conference of the mathematical association of victoria. https://doi.org/10.5951/tcm.12.2.0083 https://doi.org/10.1257/jep.24.2.129 https://doi.org/10.20853/36-3-4681 https://doi.org/10.20853/36-3-4681 https://doi.org/10.1037/edu0000012 https://doi.org/10.1007/978-3-030-15789-0_129 https://doi.org/10.29333/iejme/170 https://doi.org/10.15663/wje.v26i2.881 https://doi.org/10.1007/978-94-6300-561-6_9 https://doi.org/10.1007/978-94-6300-561-6_9 hendricks & olawale, mathematical probability: learner's misconception … 178 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 10, no. 2, september 2021 e–issn 2460-9285 https://doi.org/10.22460/infinity.v10i2.p301-318 301 mathematics learning through character education based on integrated thematic learning: a development of learning materials yenny suzana1, sabaruddin*1, suesthi maharani2, zainal abidin1 1institut agama islam negeri langsa, indonesia 2institut agama islam negeri salatiga, indonesia article info abstract article history: received apr 26, 2021 revised july 13, 2021 accepted july 18, 2021 this research is to develop mathematics teaching materials that are integrated with elementary school thematic learning. the purpose of the development is to obtain mathematics teaching materials that prioritize the local wisdom of the acehnese people. this teaching material is helpful for fifth-grade elementary school students in which there are character values for learning mathematics. this study uses a qualitative descriptive research method with the addie model development research design. this research focuses on analyzing elementary school teachers who face the problem of integrated mathematics-based character education in thematic learning, then making initial designs and developing character education-based mathematics teaching materials that are integrated into the learning theme. the results showed that character education-based mathematics teaching materials were compatible with elementary students' thematic learning. mathematics teaching materials focused on solving math problems for elementary students, integrating character values in mathematics with various themes in thematic learning by integrating each mathematics material into themes. the mathematics teaching materials developed were designed with various activities related to daily activities with straightforward language to be understood and made into a mathematical model. keywords: character, material development, mathematics learning, thematic copyright © 2021 ikip siliwangi. all rights reserved. corresponding author: sabaruddin, department of mathematics education, faculty of tarbiyah and teacher training science, institut agama islam negeri langsa jl. meurandeh, langsa, aceh 24415, indonesia email: sabaruddin@iainlangsa.ac.id how to cite: suzana, y., sabaruddin, s., maharani, s., & abidin, z. (2021). mathematics learning through character education based on integrated thematic learning: a development of learning materials. infinity, 10(2), 301318. 1. introduction primary school mathematics learning is the initial introduction to numerization and integrating abstract knowledge into real objects (tzanakaki et al., 2014). however, in practice, mathematics learning in primary schools encountered various difficulties in reallife applications (banda & kubina jr., 2009; dennis et al., 2016). most of the problems https://doi.org/10.22460/infinity.v10i2.p301-318 suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 302 encountered were related to students' ability to translate sentences into mathematical models (boonen et al., 2016). the application of mathematics in life is not applied in the learning process in the class (nasrum & herlina, 2019; setiyani, putri, & prakarsa, 2019; yunianta, putri, & kusuma, 2019). mathematics books focus on solving mathematics analytically and are heavy for students (nurharyanto & retnawati, 2020). the problem-solving presented does not relate to problems in the real world, so mathematics is less attractive to primary school students (khodeir et al., 2017; saputra et al., 2018). various studies in implementing mathematics learning out state that schools' learning process has not paid attention to students' needs and potential for improving students' abilities (wijaya et al., 2019). learning tends to be theoretical, the role of the teacher is very dominant (teacher-centered), and the communication style is in one direction (zain et al., 2012). as a result, the learning process is limited to the transfer of knowledge that is less related to environmental learning. students are unable to apply key concepts of knowledge to solve problems in everyday life. as a result, this condition creates a reluctance to read and write in indonesian primary school students. handal and bobis’s (2004) research on teacher understanding and practice towards thematic approaches can link mathematics with real life. however, in reality, thematic learning has not been fully implemented in class. in this study, instructional, curriculum, and organizational factors were considered by teachers as obstacles in the application of thematic approaches to teaching mathematics. broadly, the material presented in the mathematics book is not entirely thematically applicable. the completeness of the facilities and the learning environment is also influential in applying them to primary schools' mathematics learning (haylock & thangata, 2007). the thematic approach is one of the learning strategies to create active, meaningful, and exciting learning (chumdari et al., 2018; narti et al., 2016). it also provides a framework for developing correlated concepts, a more stable learning model, and suitable students with different abilities (handal & bobis, 2004). at present, primary schools' educational practice phenomenon shows a high trend that only focuses on one subject. learning is only aimed at instructional impact. the evaluation system is involved in reproducing information. they have not exceptionally motivated students (nahrowi, 2019). the teacher comes to class with a book and then asks students to complete the book's exercise before explaining the lesson. most teachers still use lesson plans at the planning stage without paying attention to students' abilities. teachers are still less effective in using media to involve students in learning actively. equitable education can be carried out. the education system must serve all student ages to enjoy education at least in the essential skills needed, namely reading, writing, and arithmetic (john, 2015). according to bier et al. (2016) character education requires young people to judge what is right, pay close attention to what is right, then do what is right, even in the face of external pressure and temptations from within. learning activities in elementary schools are arranged based on the 2013 curriculum which implements integrated thematic learning with a scientific approach. the thematic model is applied to the first three classes (grades 1, 2, and 3), while the fragmented model is applied to the next class (classes 4, 5, and 6). the thematic model is a learning model that uses themes to connect several subjects to provide meaningful experiences to students (chumdari et al., 2018). it is also interpreted as a learning model that departs from a specific theme as a center of interest in understanding other symptoms and concepts, both from various subjects and one subject (john, 2015). this learning model starts with great ideas, essential questions, or problems in reality and has contextual meaning for students (handal & bobis, 2004). students use skills and knowledge from various subjects simultaneously to answer questions or solve problems. thus, the thematic learning model is a learning model volume 10, no 2, september 2021, pp. 301-318 303 designed based on specific themes to provide students with meaningful experiences (ekowati, utami, & kusumaningtyas, 2018; narti et al., 2016). the application of integrated thematic learning resulted in changes in learning, changes in manuals for teachers and students. thematic or integrated learning departs from the idea that students gain the best knowledge when learning in a coherent overall context because they can relate what they learn to the real world (wangid et al., 2014). thematic is one type of integrated learning model as said that integrated learning includes three types: the connected, thematic or webbed type, and integrated (islam & suparman, 2019). the connected type integrates subject matter from a particular discipline. webbed or thematic types develop material from the specific subject matter or several subject matter or scientific disciplines (sunhaji, 2013). the integrated type combines material from several subjects or disciplines. the development of the learning model begins with determining a specific theme and is accompanied by sub-themes' development by paying attention to the interrelationships between subjects. themes are chosen based on negotiations between teachers and students or discussions between teachers. after the themes are determined, sub-themes are formulated. furthermore, student learning activities are designed based on sub-themes. information was obtained from various sources by interviewing several teachers from different schools, that thematic learning helps teachers improve the application of character in learning. however, the problem with mathematics is that to have not met the standards for fulfilling basic mathematical concepts. meanwhile, mathematics lessons in the previous curriculum had too high a mathematical concept so that character values could not be applied in mathematics lessons (sabaruddin et al., 2020). burns (2012) distinguish integrated curricula into three categories: multidisciplinary, interdisciplinary, and transdisciplinary approaches. teachers integrate sub-disciplines in subjects in a multidisciplinary approach, such as reading, writing, and speaking in languages. integrate history, geography, economics, and politics into interdisciplinary social science programs. in an interdisciplinary approach, teachers compile curricula from across general disciplines. for example, students learn to make wind and rain machines while learning a language simultaneously. students learn about a particular theme concerning several related subjects. the teacher uses overlapping material on several subjects simultaneously. in a transdisciplinary approach, teachers and students discuss specific themes that are broad and cross-subject. in this study, thematic learning is included in an interdisciplinary approach. the teacher presents particular themes that are learned from diverse and related subjects. based on this description, it can be concluded that the application of thematic learning models in primary schools is a valuable effort to achieve student learning outcomes, both in the aspects of knowledge, skills, and values and attitudes/characters. therefore, this study develops mathematics teaching materials that are integrated with thematic learning for fourth grade elementary school students in langsa city. mathematics teaching materials in accordance with the theme of the local wisdom of the acehnese people. in addition, this mathematics teaching material can be useful for students because in it there are character values for learning mathematics. this mathematics teaching material containing local wisdom can be recommended as a solution to overcome the problems faced during the covid-19 pandemic. 2. method the method used a descriptive qualitative with a development research design. the development model used the addie model. the addie model consists of 5 stages, namely analysis, design, development, implementation, evaluations. the development in this suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 304 research has reached the stages of analysis, design and development. this study analyzes the importance of developing mathematics teaching materials combined with thematic learning, planning development designs and developing mathematics teaching materials with a character education approach to thematic grade 5 elementary schools. implementation and evaluation are reserved for further studies and recommendations from the research that has been carried out, the implementation and evaluation studies will be continued in the next period. 2.1. preleminary research preliminary research activities are carried out to collect information before the research is carried out, including literature studies and field studies. literature studies are used to find the concepts, scope, supporting conditions, and the most appropriate steps to develop products. preliminary research in the form of literature studies is carried out to analyze more depth and find relevant research literature to solve the problems found. 2.2. materials development planning the product development process includes the process of designing and compiling mathematics teaching materials that contain local wisdom of the acehnese people and in which there are character values in elementary school thematic learning in langsa city which were developed on nine themes in fifth grade. researchers compiled the steps of teaching materials as follows: following. first, choosing ki and kd for 2013 curriculum subjects, developing mathematics materials that instill values of logical thinking, critical, hard work, curiosity, independence, honesty, democracy and self-confidence which include the theme. second, determine the material and formulate learning indicators in accordance with the theme. third, determine the place of research trials for the sake of implementation and evaluation of teaching materials that have been developed. 2.3. initial product development initial product development is carried out by selecting material included in character education-based mathematics teaching materials. the researchers developed the theme in fifth grade, determined the learning process by the learning material, and compiled a learning evaluation to measure whether the indicators can achieve the expected learning outcomes. 3. results and discussion 3.1. results 3.1.1. preliminary study results of teaching materials thematic study teaching materials that are developing are mathematics teaching materials for grade 5 primary schools. the current curriculum that is currently in effect fully applies thematic learning so that all teaching materials are included in a series of themes. assessed thematically for grade one to grade three, primary school is reasonable and very appropriate because there are many student behavior developments. however, it is different with thematic for grades four, five, and six. most teachers consider it difficult and feel deficient in mathematics teaching material, so they need to return to mathematics lessons like the previous curriculum as stated by halimah, s.pd (pseudonym), one of the principals of sd negeri in langsa city in an interview: volume 10, no 2, september 2021, pp. 301-318 305 "...in grade 5 we still use thematic, but for mathematics, we use special subject teachers, because in thematic we are worried that our students will not understand mathematical concepts...." the policy will set mathematics lessons to be considered because of concerns from various parties about students' mathematics quality. usually, on the final exam, questions will be given with a high standard. it is feared that students will miss mathematical concepts as a whole if only by participating in thematic learning as expressed by mr. surya (pseudonym), a classroom teacher at a public primary school: "...it is straightforward to teach during our thematic lessons, let alone many learning activities directly related to daily life. however, we are worried about maths lessons, math topics are concise, and there are some materials in math textbooks not included in thematic lessons..." during the implementation of the 2013 curriculum, there were many developments and advances in character building among students. with various efforts in its application, revisions were made from various sessions, and a curriculum that was friendly to the internalization of moral values was obtained in an integrated manner. the combination of cognitive, affective, and psychomotor is effortless to apply in thematic learning; it is just that the method needs to be improved so that all elements are met and can be accepted by all groups. many teachers' integrated thematic view has begun to seem understood, where their weak and strong sides can slowly be detected. as discussed with mrs. juli (pseudonym), sdit teacher in langsa city, as follows: "...at the beginning of the implementation the 2013 curriculum, we were confused because all the lessons were combined in one book. we see that the subject matter is complicated and low. it is not the same as in the previous lessons. we finally got socialization several times to understand that an integrated thematic is useful as an application of learning concepts in everyday life by emphasizing the internalization of character values to students..." from the observations made in several schools, mathematics lessons converted to themes have been carried out by primary teachers in langsa city. not all schools have implemented the thematic curriculum since the beginning of the 2013 curriculum; only a few schools are pilot studies, others are voluntary in their application. however, in this curriculum, the ministry made many modifications and revisions based on input from schools that became the pilot studies. thematic lessons for grades four, five, and six are believed by teachers to be applicable and integrated with students' character values. however, integrating with mathematics lessons have difficulties in practice. following are the results of the initial discussion with mrs. husna (pseudonym), sdit teacher in langsa city: "...with our thematic books, it is easy to teach and create activities related to everyday life. character values are also easy as students are easy to work with, respect each other, discipline, understand and love the environment and several other characters. nevertheless, we cannot be sure in maths lessons whether they are adequate and arrive at a math concept equivalent to their class because we saw that some questions at the national level were much higher and deeper. so sometimes we keep using the old math textbooks for math lessons..." based on several discussions that have been carried out with primary school teachers and principals in langsa city, preliminary information was obtained that thematic learning suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 306 in primary schools is good and can be integrated with students' character values. however, mathematics still needs to be explicitly developed and in-depth, like teaching materials that can combine themes and math lessons. to produce teaching materials that meet thematic and sufficient elements of mathematical concepts. there is no concern among teachers about the academic achievement of their students, especially in mathematics. moreover, to prepare students for the national level examinations and several other tests carried out correctly. two steps were taken at the preliminary study stage, namely information gathering, and planning. in the information-gathering step, activities are carried out by field surveys and literature studies. the field survey was conducted to obtain information about the state and availability of teaching materials used by teachers and students in the mathematics learning process. activities carried out at this stage include field observations and interviews with mathematics teachers. the results of the field survey obtained information that the learning process is still teacher-centered. the teacher begins the lesson by directly providing the material, which does not allow students to understand the material. in addition to the learning process, what is observed from this observation stage is that the teaching materials used in learning are only textbooks provided by the government and worksheets from private publishers. in addition to observation, other preliminary study activities are interviews with mathematics teachers. the results show that the teacher has never given a test that demonstrates mathematical reasoning and communication skills. the teacher has never carried out thematic learning. private teaching mathematics textbooks use teaching materials from the government and private publishers. literature studies in this research include thematic studies, studies of mathematical reasoning skills, studies of mathematical communication skills, and study of teaching material development models. 3.1.2. study of mathematics teaching materials mathematics was implemented before the 2013 curriculum and is still being applied in many schools even though it implements thematic learning. mathematics lessons in primary schools face many challenges in the learning process, such as many classroom teachers who do not master mathematics well, as was expressed by mrs. sugiarti (pseudonym), a teacher at the public primary school in langsa city: "...we miss certain material because it is difficult to explain, knowing that i am not an alumnus of mathematics education or pgsd. i used to study spg, and then in 2007, i continued to s1 historically. however, i often work as a classroom teacher, and i have no problem with other subjects, but i find it difficult" it is not only the teaching materials that are difficult for the teacher, but also many other difficulties, such as being difficult to apply daily. in the mathematics textbook, no character value content will be achieved. mathematics teaching material tends to be very complicated. many concepts must be understood by teachers and students but are not applicable in everyday life. the following is the statement of a teacher in an interview with mrs. rosnidar (pseudonym), a public mi teacher in langsa city: "...we often try to relate mathematics to life. however, it is not easy to give an example. i also see that no character value will be achieved in mathematics teaching materials—furthermore, more focus on finding answers to the textbook's questions. so i do not dare to spend time applying character in maths lessons..." volume 10, no 2, september 2021, pp. 301-318 307 the same thing was also expressed by mr. alamsyah (pseudonym), a private primary school teacher in langsa city. "...i have not applied many aspects to mathematics in my life, especially in teaching. i am worried that i will not be able to finish the existing syllabus. if i teach by doing activities outside the classroom, it will take much time, and i also rarely use the time to study mathematics to do according to my daily life. some materials are difficult to miss because they are difficult to explain to students..." based on initial conversations with primary school teachers in langsa city, it can be explained that mathematics lessons need to be developed so far. in line with the 2013 curriculum and its revision emphasizes students' character and competence, modifications are made to integrate lessons with character so that lessons can be applied to daily life and can foster good attitudes towards students, especially in mathematics lessons. the development of friendly and easy teaching materials for children is also the dream of parents, as stated by ibu susi (pseudonym), one of the parents of primary school students in langsa city: "... we hope that there will be teaching materials that are easy to explain because when we do children's assignments, we do not bother asking people anymore. in the meantime, we have much material that we cannot help to explain. in thematic lessons, it is straightforward and understandable, but i am worried that later on the children will master math teaching materials during the final tests and exams ... " mathematics feels alive and benefits students if what they learn is directly related to daily practice. with the perceived benefits of learning mathematics, students will be more enthusiastic and enthusiastic about learning, making it easier for teachers in the learning process. mathematics is expected to contribute to improving students' understanding of everyday applications. in addition to increasing student competence, it is also expected that they have good character. mathematics also plays an essential role in improving student character. therefore, the development of thematic-based mathematics teaching materials is significant. because it will stand in the middle of both, lessons with mathematical concepts are deliberate on various themes that meet the elements of character values that will be grown. the character values that are grown are integrated with mathematics lessons with the various activities presented. the results of discussions with several teachers and school principals can be used as the basis for the development of this teaching material is essential. several vital aspects must be fulfilled, such as competence and character values to be achieved. in the fgd, there were also some problems in general in schools at this time. in the learning process, many teachers were not good at math. there were lots of mathematics books, so that teachers were chasing material. they were not applicable in everyday life and had nothing to do with character values. in comparison, thematic problems are obtained for grades four, five, and six. in this class, mathematics should be presented in detail and in-depth, but in a very light thematic, and it is feared that it will become a problem during the national exam. in this case, the fgd expects that there will be a modification of teaching materials that meet the two elements sufficiently in the teaching material and can be applied in life and integrated with students' character values. suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 308 3.1.3. teaching material development thematic mathematics learning module grade5 elementary school based on character educationbelow contains material exposure, illustrations, examples of solving practice questions, practice questions, and student activities packaged attractively to become more enjoyable for students. this module's distinctive feature is that its content is contextual and realistic. it is close to the students in terms of language, and the pictures also include several teachings of culture and good character. it is hoped that the mathematics teaching material in the form of a thematic mathematics learning module class 5 sd based on character education can be accepted and used by students to explore abilities and get to know the natural surroundings well. so, the author humbly accepts suggestions and criticisms that can build and hopes that the writer will improve the preparation of this book. this teaching material focuses on finding common ground from thematic and mathematics lessons to correctly improve character and understanding of mathematical concepts. so that students can find the purpose of learning mathematics through applications in everyday life. some essential aspects in this research are the book's attractive image design, the use of language, the students' experience-based question, the explanation of the objectives and benefits of learning, the character values achieved, and the problem-solving cases moral messages for students. 3.1.3.1. book cover mathematics textbooks are designed with mathematical characteristics and show an impression of seriousness in understanding mathematical concepts. in this study, the cover is designed in a light and straightforward form so that it does not impress mathematics on purpose, but mathematics will be applied lightly, even accidentally. the figure 1 are some differences of the book cover before being developed and after being developed. (a) (b) figure 1. cover design before and after development volume 10, no 2, september 2021, pp. 301-318 309 based on figure 1, the development can be seen. the 5th-grade mathematics textbook shows the numeration of objects and messages only through less visible writing and coloring. the development carried out in this research is to provide a more visible inheritance and an image as if it provides an important message to protect the environment. the development is related to the theme so that the cover image of the book shows the theme being studied. the identity of the book can be seen from the mathematics textbook that is clear and understandable. in contrast, in the thematic book, the book's identity based on the theme studied does not contain explanations of mathematics material. in the teaching materials developed, it is clear that the themes being studied and the mathematics lessons are obvious. the message should give an impression to students, and the goals should be better in behavior and advance in mathematics ability. 3.1.3.2. preface and book instructions the development of teaching materials is explained by the use and uniqueness of the book's contents through the introduction. clear instructions that make it easier for book users to understand the purpose of using this teaching material. this teaching material can also be used personally by students at home without teacher guidance because it is designed with easy language and examples. for parents who assisted their children in learning mathematics during the pandemic, this book can be easily understood and do not need to worry about their children's mathematics achievements. mathematics textbooks do not explain instructions for use, and many teaching materials are difficult and less applicable in everyday life. in contrast, the guidebook theme is also less related to complex mathematics teaching material. 3.1.3.3. material presentation this mathematics teaching material starts with natural pictures according to the theme, examples of themes of caring for the surrounding environment, and sub-themes of objects of natural surroundings. the pictures presented are often observed by students in everyday life. students easily observe the mathematical values such as how many are entered and the geometry positions so that mathematical concepts can be explained correctly and efficiently. after students can accept it, an explanation is presented with examples of nature cases by modeling mathematics. the example presented provides a moral message to students regarding the theme being presented, such as caring for the surrounding environment, fostering an attitude of love for the environment, and keeping the environment clean with mathematics (see figure 2). suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 310 (a) (b) figure 2. description of the presentation of the material based on figure 2, it can be seen the differences in the presentation of teaching materials. in figure 2a, the mathematics material is presented directly with a rigid and mathematical image. whereas in figure 2b, teaching materials have been developed with presentation through images related to caring for the surrounding environment and subthemes of natural products. pictures and descriptions of stories show that students must care for their environment with mathematical and fun activities so that learning becomes fun. 3.1.3.4. student worksheet after presenting the teaching material, the teaching material is given space for students to carry out worksheet activities or student worksheets to determine whether students already understand the material presented and can actualize it in real examples. in mathematics books, in general, examples of problems are presented with textual solutions with mathematical concepts. in developing this teaching material, student worksheets are presented with examples of activities related to the theme and entering the character values to be achieved (see figure 3). volume 10, no 2, september 2021, pp. 301-318 311 (a) (b) figure 3. students’ worksheet based on figure 3, it can be seen the differences in student worksheets. figure 3a shows the student worksheets in the class 5 mathematics book. the student worksheets can be seen with questions related to the teaching material. meanwhile, there is no application in life, and no moral message is delivered. whereas in figure 3b, the development carried out is by providing examples of student work with questions related to the theme being studied and applying mathematics in everyday life. 3.1.3.5. group workspace the development of this teaching material also pays attention to group workspaces. it provides work steps and a place to provide reports on mathematical activities carried out by the theme being studied. whereas in the previous mathematics books, there was no room for discussion and no problems that students had to solve collectively (see figure 4). figure 4. student collaboration sheet with the team suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 312 based on figure 4, it can be seen that in the development of mathematics teaching materials, students are also given space to carry out discussions in groups. in preparation for the discussion, students are also given several instructions on working and steps for problemsolving. in this way, some character values can be applied, such as collaborating with friends, cooperating, respecting opinions, and having a sense of responsibility towards tasks and teams. then it is hoped that students can share experiences and provide explanations to friends who do not understand. 3.1.3.6. formative test development is carried out up to the formative test. formative test questions usually meet the elements of indicators and competency standards to be achieved. in the formulation of the questions, an analysis stage of the students 'thinking is developed to make it easier to map students' competencies and achievements. in the math book, formative test questions are arranged with no space to solve them. whereas in the development of this teaching material, questions were arranged with several levels of student thinking processes ranging from low to high levels and given space to solve these questions. the questions are presented by paying attention to the character elements to be achieved (see figure 5). (a) (b) figure 5. examples of formative test questions based on figure 5, the differences in mathematics textbooks and the development of teaching materials can be seen. teaching materials modified in figure 5b provide space for students to answer directly. the questions presented invite students to think creatively by modeling mathematics from stories that contain character values. each pillar presented is related to the theme being studied to absorb and solve problems easily. figure 6 describes student worksheets with patterns of helping each other solve problems related to everyday life themes. volume 10, no 2, september 2021, pp. 301-318 313 (a) (b) figure 6. examples of student worksheets in test form 3.2. discussion based on field data, it is found that thematic learning applied in schools helps and makes it easier for teachers to apply character values that have a specific relationship between the subject matter and the students' natural experiences. in line with ekowati et al. (2018) state that the main characteristic of thematic learning in primary school combines lessons with a contextual framework and deepening character values. as hasrawati (2016) expressed, thematic learning also provides direct experience for students to absorb lessons related to life and learning to be interesting to follow. thus, increasing learning motivation and facilitating classroom management. it was obtained information that the teacher was ready to enact the new law. many also welcomed thematic learning and were ready to implement it even though various ways were taken to get instructions. this finding is in line with the findings of wangid et al. (2014) that primary school teachers' readiness in implementing thematic learning in the 2013 curriculum is perfect. based on this percentage, the teacher's readiness from behavioral readiness is also very good, although teachers need to carry out training from various sources independently. moreover, it is hoped that the readiness of the teacher can increase student achievement through thematic learning (riwanti & hidayati, 2019). the development of the thematic learning module based on character education for class v elementary school has had a good impact on students where students can slowly get used to having the desired attitude or character education values. experimental where the implementation of integrated learning is beneficial and the student response to it is very good. chumdari et al. (2018) concluded in their research that the thematic learning model in primary schools could be done well, but students' learning activities are less than optimal. it is due to conventional suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 314 teacher-centered methods that emphasize the active role of teachers rather than students. it needs to be improved by promoting innovative learning models that allow students to carry out learning activities, such as object/environment manipulation, discussion activities, and group activities with fellow students. thematic teaching materials can improve students' abilities and character. as expressed by utami and mustadi (2017), teaching materials developed based on thematic can provide convenience and improve abilities and improve student character, tie learning motivation, and increase student achievement. the development of teaching materials is essential to do in order to adapt to the conditions and characteristics of students, and development will also provide innovations and strategies that facilitate the absorption of learning. the development of teaching materials and teaching aids is carried out based on necessary modifications so that what is already there is to be better and useful and right on target in the application in life (wulandari & prasetyaningrum, 2018). mathematics lessons need development, considering that the thematic concepts are still low and mathematics is too high. it is necessary to combine both of them to produce suitable teaching materials. the development of teaching materials related to direct experience has a very positive impact on mathematics (putri, hasratuddin, & syahputra, 2019). improving character and motivating students are ways to solve various mathematical problems. the preparation and development of textbooks would help teachers and students learn the material in a particular lesson (chumdari et al., 2018). textbooks play a critical role in the improvement and content of thematic and mathematics curricula. all forms of theoretical interpretation can be applied in the development of teaching materials. in terms of teacher background, this study found much diversity, such as education level, the field of study, teaching experience, and teacher certification. teacher background and education level are very influential on teacher interaction skills in learning. it is in line with iswadi and richardo (2018) research findings that the background level of education and teaching experience of a teacher can affect teachers' professional ability in learning interactions. the teacher background problem is also in line with the mutakin (2013), which states that "teacher competence and teacher background together positively influence the performance of primary school teachers." firdaus's findings (2014) showed that teachers' level of education, training, and teaching experience could be used to predict teacher professionalism in teaching. rakib et al. (2016) also argued that "training partially positively and significantly affects teacher professionalism. teaching experience partially has a positive and significant effect on teacher professionalism, and training and teaching experience simultaneously have a positive and significant effect on teacher professionalism". the sad thing is that many teachers do not get the opportunity to improve their competence through training. because so far, the government has provided minimal opportunities for funding for teacher training. based on the preliminary observations from several primary schools in langsa city, there are still many times that need to be considered. more than 30% of primary schools do not have adequate facilities and infrastructure so that it is challenging to implement thematic learning. the environment is a slum, the learning room lacks facilities, inadequate learning media, materials for experiments are not available, the library is incomplete, and even books are not available. there are no laboratories available at the school. it is a major obstacle in the learning process, especially in implementing thematic learning. similar findings have also been highlighted by kurniawan's (2017) research that the importance of planning from school leaders to plan facilities and infrastructure. this study also concludes that "one of the efforts that can be made to improve learning effectiveness is by improving facilities and infrastructure for increasing teacher competence." volume 10, no 2, september 2021, pp. 301-318 315 the development of mathematics teaching materials is needed because teachers have several obstacles in using thematic books. as the results of research conducted by kurbaita et al. (2013), problems such as the teacher have not been able to identify the basic competencies of several subjects that can be integrated into one learning theme. also, thematic books that have been well prepared also experience problems in their use due to many unsupportive things such as facilities and infrastructure, school environment, teacher training, and learning media. sukiniarti (2014) showed that the obstacles experienced include "most teachers consider it more challenging to develop thematic lesson plans than lesson plans in the field of study. especially in determining methods and preparing evaluation questions; all teachers find it difficult to determine the media of each predetermined theme; most of the teachers to determine the theme with the right method, still have to discuss it with fellow teachers. the development of teaching materials that occurs is combining integrated thematic mathematics lessons in primary schools. mathematics lessons need to be modified so that mathematics lessons do not feel challenging for students; by providing examples and applications in everyday life, students easily accept mathematics. it is in line with the research conducted by narti et al. (2016) and ahsani (2020) that thematic learning plays an essential role in increasing student attention, learning activities, and understanding of the material. as learning is more student-centered, it provides students with hands-on experience. the current concept offers a flexible range of topics. besides, learning outcomes can be developed according to student interests and needs. thematic learning plays an important role in increasing students' attention, learning activities, and understanding of the material. as learning is more student-centered, it provides students with hands-on experience. the current concept offers a flexible range of topics. also, learning outcomes can be developed according to student interests and needs. teaching materials that have been developed include cover designs that attract students to open books, complete mathematics material so that all mathematical content can be learned by students but by combining them with them. each student teaching material is invited to carry out learning activities according to the theme being carried out. in addition to modification in terms of student worksheet material, it is also modified. up to the formative exam, it is attractive to make students not bored and not find it difficult to solve them (putri et al., 2019). the results show that learning materials based on a realistic mathematics education approach are important things that need to be considered to maximize student mathematics learning achievement. thus, it is expected that mathematics teachers are looking for mathematics. learning uses teaching materials based on a realistic mathematics education approach. 4. conclusion based on the results of the development, it is obtained that mathematics teaching materials contain local wisdom of the acehnese people and in which there are character values of logical thinking, critical, hard work, curiosity, independence, honesty, democracy and self-confidence. this mathematics teaching material was developed for fifth grade students in thematic learning at elementary schools. there are character values like learning mathematics. student worksheets are available in the teaching materials for students to solve problems, there is also space for students to make reports on their work in groups. mathematics teaching materials containing local wisdom are easily understood by students and are appropriate to be recommended as teaching materials during a pandemic to make students strong and hardworking. suzana, sabaruddin, maharani, & abidin, mathematics learning through character education … 316 references ahsani, e. l. f. 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(2012). student-centred learning in mathematics constructivism in the classroom. journal of international education research (jier), 8(4), 319-328. https://doi.org/10.19030/jier.v8i4.7277 https://doi.org/10.23960/mtk/v8i2.pp168-181 https://doi.org/10.1088/1742-6596/1028/1/012093 https://doi.org/10.22460/infinity.v8i2.p143-156 https://doi.org/10.21009/pip.282.6 https://doi.org/10.1111/1467-9604.12069 https://doi.org/10.21831/jpk.v7i1.15492 https://doi.org/10.21831/jpe.v2i2.2717 https://doi.org/10.22342/jme.10.3.7798.357-364 https://doi.org/10.15575/psy.v5i2.2977 https://doi.org/10.22460/infinity.v8i1.p43-56 https://doi.org/10.19030/jier.v8i4.7277 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p211-222 211 the argument and demonstration exemplified in a mathematical dialogue luisa morales maure1,3*, marcos campos nava2, orlando garcía marimón1, jaime gutiérrez1 1universidad de panamá, panama 2autonomous university of the state of hidalgo, mexico 3member of the national system of researchers sni-i, senacyt-panamá, panama article info abstract article history: received jun 6, 2022 revised aug 15, 2022 accepted aug 16, 2022 the teaching-learning process is analyzed in a course for a group of professors who were taught subjects on calculus, to study the episodes of problemsolving in them, focused on the identification of patterns and argumentation using counterexamples. the explanation and the argument in the classroom can be used together so that the argument (issued as a counterexample) supports the explanation (conjecture). developing the mathematics class so that the above occurs is a form of interaction and how to encourage students to move from explanation to argumentation (placing a hybrid system). furthermore, both forms of reasoning can influence dialogue protocols and strategies. in this work, the dialogue model is described as a tool to address the problem that arises when working with students. keywords: argument, conjecture, counterexample, dialogue this is an open access article under the cc by-sa license. corresponding author: luisa morales maure, universidad de panamá. panama transístmica, panamá, panama. email: luisa.morales@up.ac.pa how to cite: maure, l. m., nava, m. c., marimón, o. g., & gutiérrez, j. (2022). the argument and demonstration exemplified in a mathematical dialogue. infinity, 11(2), 211-222. 1. introduction since ancient times, explanation and argumentation are resources used to improve learning. for example, when socrates and his young disciple teeteto discussed the meaning of science, socrates questioned the conjecture of his disciples in this dialogue, when teeteto expo exposed by plato (2003), “what can be learned with theodore, such as geometry and the other arts you have mentioned, are so many other sciences and even all the arts, whether that of a shoemaker or any other trade, are nothing but science”. socrates, with a counterexample, answered his disciple stating that when you ask about what science is, it is to make a fool of yourself by giving an answer to the name of a science. this is to answer about the object of science, and not about science itself, which is what the question refers to, with a rebuttal that helps change teeteto's answer. here we see how the teacher induces his disciple to "conjecture" to modify his answers, allowing him to develop a better-structured thought. https://doi.org/10.22460/infinity.v11i2.p211-222 https://creativecommons.org/licenses/by-sa/4.0/ maure, nava, marimón, & gutiérrez, the argument and demonstration exemplified … 212 there are multiple definitions of what is a guess in mathematics, and most do not make it very clear how it is structured? will it be possible to build a conjecture with the help of certain elements or how is it built? there is no clear idea or established manual that followed by mathematicians to structure a guess. however, it is intended to identify some forms presented by researchers in this area. an approach presented in the books is the idea of using particular cases to seek regularities and establish a conjecture such as canadas et al. (2008). although some authors today highlight the difficulties in separating these two in practice, conjecture and counterexample (garcía & morales, 2013; ibañes, 2001; marrades & gutiérrez, 2000; stenning & monaghan, 2005), we strive to focus our research in the inductive reasoning process. the authors claim about inductive reasoning, which is a cognitive process that allows to advance knowledge by obtaining more information than the initial data with which the process begins. human thought then takes a stance that produces affirmations and reaches conclusions based on particular cases and identifying patterns. in other words, in order to structure a conjecture, inductive reasoning must be developed to lead into possible generalization. in any calculus course, the teacher intervenes in the planting of ideas (socratic approach) by asking his students to propose their argument. students learn to listen sympathetically to the ideas of other peers and to contribute their own. then, they have to learn to criticize the defects that appear in the development of discussions and accept the corrections that are made to them; establishing a sufficiently broad theoretical scenario for the approach of course members’ projects. in general terms, reasoning involves extracting references from principles and evidence from which individuals draw new conclusions or evaluate conclusions based on what is already known (johnson-laird & byrne, 1993). there are two main types of reasoning, deductive reasoning and inductive reasoning. deductive reasoning refers to the process of reasoning from a set of general premises to arrive at a valid logical conclusion, while inductive reasoning is the process of reasoning from specific premises or observations to arrive at a general conclusion or rule. general. thus, deductive reasoning draws conclusive conclusions from given information, while inductive reasoning adds information (klauer, 2001). this educational recommendation addresses only inductive mathematical reasoning for students in calculus courses. mathematical induction contains information about all instances of a class (for example, the class of all positive integers) and thus can draw conclusions with certainty, whereas students' inductive reasoning generally refers to a certain instance. therefore, the conclusions it draws are not necessarily applicable to all possible situations (sternberg & gardner, 1983). in many cases, however, inductive reasoning is valid and provides an important foundation for understanding mathematical laws. both regularity and unity are the basis for the generation of concepts and categories, which play an important role in our daily lives (klauer & phye, 1994). our research focuses on the inductive reasoning required for learning programs in higher mathematics (mat-121) and for most intelligence tests (eg, analogies, classifications, series completion problems, arrays). neubert and binko (1992) relate inductive reasoning in mathematics to the search for patterns and relationships between numbers and figures. this idea goes back to the work of polya (1967), who defined inductive reasoning as one that allows us to obtain scientific knowledge. polya (1967) also believes that inductive reasoning in mathematics education is a method of discovering properties of phenomena and logically discovering laws. inductive reasoning as a method consists of four steps: experience with specific cases, formation of conjectures, testing of conjectures, and verification of new specific cases (polya, 1967). based on these steps, cañadas (2002) developed a system consisting of secondary school volume 11, no 2, september 2022, pp. 211-222 213 students' thinking actions to solve proof and inductive reasoning problems related to the justification of a statement where inductive reasoning appears. although in most elementary mathematical problems students are tasked with discovering patterns of relationships or characteristics between the various given elements of the problem, the stimulation of thinking skills is not cleanly pursued. these skills are often seen as byproducts of what is taught in traditional curriculum definitions. for different topics (sánchez et al., 2021). as a result, most students do not master basic mathematical concepts and have difficulty solving problems (godino et al., 2007; godino et al., 2011; mallart et al., 2018; maure et al., 2018; morales-maure et al., 2022). the latter has been demonstrated in many studies, especially international research assessments such as the oecd-pisa 2018 (schleicher, 2019) and timss 2019 (martin et al., 2020). 2. method the experience described in this work was taken from a calculus course whose participants were math teachers. this was done with the intention of encouraging problemsolving episodes, as well as the use of examples and counterexamples to encourage argumentation with their students. taking into account that, as future mathematics teachers, they should receive training with processes similar to those that they are expected to develop in their classes. first, a diagnostic test on previous knowledge was carried out and then the document “is argumentation an obstacle? invitation to a debate by nicolas balacheff was read. in this document, the author arguments on the thesis that men live immersed in a context of arguments (balacheff, 1999). with this in mind, we must say that all the argumentation is part of men’s daily world. there is no conversation, discussion, or opinion in which there is no effort of conviction, because not all individuals think the same way. many of the analyses developed as spontaneous, informal and intuitive. therefore, the purpose of an argument is, above all, to increase attachment to a point of view submitted to an audience (students, teachers). however, it does not demonstrate the veracity of a conclusion as that belongs to the field of scientific demonstration. 3. result and discussion 3.1. the proposed mathematics inductive reasoning framework in the mathematical sense, a conjecture can be built by looking for patterns or regularities in the classroom, which help to promote an environment that contributes to the development of fundamental processes of mathematical thinking such as the search of patterns, use of multiple representations and communication of mathematical ideas. this concurs with (benitez, 2006; castro et al., 2021) who claims that this mathematical sense serves in the learning of students as an axis in structuring their reasoning processes. however, finding numerical, geometric or algebraic patterns should not be considered as easy activities for students. therefore, it should be gradually encouraged by the teacher, to maximize the possibility of favorable cases for the formulation of a generality. following the idea of developing guesswork by students, in this first calculation course, the teacher presented and then explained a problem on the board. in this example, students designed the exponential equation where it is supposed that a single bacterium that starts dividing every hour. after an hour, we have 2 bacteria. after two hours, we have 22, which equals 4 bacteria. after three hours, we have 23 which equals 8 bacteria, and so on (see figure 1). the population of bacteria is modeled after t hours and, by means of the maure, nava, marimón, & gutiérrez, the argument and demonstration exemplified … 214 developed heuristics, led them to work the exponential equation that represents a growth of the population of bacteria f(t)=2t. figure 1. exponential bacterial growth the equation proposed by the teacher during the class implies the understanding of the duplication of previous events. however, the lack of mathematical content development in the explanation limited students to argue about the search for solutions. students often have difficulty recognizing data, graphs or figures because their identification requires the mastery of special conceptions of the subjects involved. the second activity developed in the same session aims to conjecture the recognition of squares built and plotted in different positions and hidden in other figures and to advance the use of some of the properties that characterize them (summaries). (see figure 2) this approach is an invitation to episodes compatible with the problem-solving approach, this time the teacher asked: how many rectangles are in the figure shown? figure 2. image made up of several divisions the objective that the teacher pursues in asking this question is to identify the possible heuristics that students use when addressing this type of problem, and to conjecture some possible solutions, following the steps indicated by schoenfeld (2016). students may also be asked questions that aim to identify each of the squares, giving them as data the number of squares that the model hides (see figure 2). the first step taken was the particular cases to observe the behavior and thus express an equation with the pattern of movement of the numbers. volume 11, no 2, september 2022, pp. 211-222 215 table 1. breakdown of rectangles found in figure 2 2 x 3 2 x 4 2 x 5 area amount area amount area amount factor #1 factor #2 factor #1 factor #2 factor #1 factor #2 1x1 3 x 2 =6 1x1 4 x 2 =8 1x1 5 x 2 =10 1x2 2 x 2 =4 1x2 3 x 2 =6 1x2 4 x 2 =8 1x3 1 x 2 =2 1x3 2 x 2 =4 1x3 3 x 2 =6 2x1 3 x 1 =3 1x4 1 x 2 =2 1x4 2 x 2 =4 2x2 2 x 1 =2 2x1 4 x 1 =4 1x5 1 x 2 =2 2x3 1 x 1 =1 2x2 3 x 1 =3 2x1 5 x 1 =5 2x3 2 x 1 =2 2x2 4 x 1 =4 2x4 1 x 1 =1 2x3 3 x 1 =3 2x4 2 x 1 =2 2x5 1 x 1 =1 it is observed that the #1 factor decreases very differently from the #2 factor (see table 1). the first guess was to maintain a fixed constant as shown below. ∑ 𝑖(𝑛) 𝑚 𝑖=1 + ∑ 𝑖(𝑛 − 1) 𝑛 𝑖=1 + … + ∑ 𝑖(1) 𝑛 𝑖=1 but, when giving values to it, did not show the pattern of results. so, it was assumed that the guess was wrong and the students realized that it was simply a double summation, which is commonly presented when you have values classified into separate groups. suppose we have k groups of values, and in each group, there are n values. ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 here values were given back to m and n, the proposed guess complied with the initial conditions that decreased both factors m and n to 1 (the factors). students induced the development of the double summation as follows: ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = ∑ ∑(𝑚𝑛 − 𝑖𝑚 − 𝑗𝑛 + 𝑖𝑗) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 =∑ ∑ (𝑚𝑛)𝑛−1𝑖=0 𝑚−1 𝑗=0 − ∑ ∑ (𝑖𝑚) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 − ∑ ∑ (𝑗𝑛) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 + ∑ ∑ (𝑖𝑗) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = 𝑚𝑛 ∑ (1) 𝑚−1 𝑗=0 ∑(1) − 𝑚 𝑛−1 𝑖=0 ∑ (1) 𝑚−1 𝑗=0 ∑(𝑖) 𝑛−1 𝑖=0 − 𝑛 ∑ (𝑗) 𝑚−1 𝑗=0 ∑(1) 𝑛−1 𝑖=0 + ∑ (𝑗) 𝑚−1 𝑗=0 ∑(𝑖) 𝑛−1 𝑖=0 = 𝑚𝑛𝑚𝑛 + 𝑚𝑚 𝑛(1 − 𝑛) 2 + 𝑛𝑛 𝑚(1 − 𝑚) 2 + 𝑚(𝑚 − 1) 2 𝑛(𝑛 − 1) 2 maure, nava, marimón, & gutiérrez, the argument and demonstration exemplified … 216 = 𝑚2𝑛2 + 𝑚2𝑛−𝑚2𝑛2 2 + 𝑚𝑛2 − 𝑚2𝑛2 2 + 𝑚2𝑛2 − 𝑚𝑛2 − 𝑚2𝑛 + 𝑚𝑛 4 = 4𝑚2𝑛2 + 2𝑚2𝑛−2𝑚2𝑛2 + 2𝑚𝑛2 − 2𝑚2𝑛2 + 𝑚2𝑛2 − 𝑚𝑛2 − 𝑚2𝑛 + 𝑚𝑛 4 = 𝑚2𝑛 + 𝑚𝑛2 + 𝑚2𝑛2 + 𝑚𝑛 4 = 𝑚𝑛(𝑚 + 𝑛 + 𝑚𝑛 + 1) 4 = 𝑚𝑛(𝑚 + 1)(𝑛 + 1) 4 therefore, the summation represents the number of rectangles that are formed to the following formula: ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = 𝑚𝑛(𝑚 + 1)(𝑛 + 1) 4 continuing with the narration of the episode, the students then wondered how many squares are formed into a rectangular figure of n*m area? to present their arguments, the students highlighted the relationships found as results of explorations on behaviors that remain fixed (particular situations) observing facts as osorio (2002) claims. there they found particular cases to observe what information it produced. figure 3. case 2 by 3 and case 2 by 4 figure 4. expanding case development according to the area volume 11, no 2, september 2022, pp. 211-222 217 figure 3 offered a guide in which each result students realized there was a decrease in the construction of values (see figure 4) and suggested a summation where there is a fixed constant proposing as a guess: ∑ 𝑖(𝑛) 𝑚 𝑖=1 + ∑ 𝑖(𝑛 − 1) 𝑛 𝑖=1 + … + ∑ 𝑖(1) 𝑛 𝑖=1 but this summary had an impact on two other summaries where n also varied. then, a heuristic was proposed that included two summations and that both values would decrease by changing the initial idea. thus, you have a double summation, ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 once presented, the idea was taken in a particular case where m is 4 and n is 2, to check if we got the given products that are 4*2 + 4*1 + 3*2 + 3*1 + 2*2 + 2*1 + 1*2 + 1*1. indeed, the proposed guess complies with the initial conditions of decreased both m and n numbers to 1 (values). and finally, it develops to this double summation: ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = ∑ ∑(𝑚𝑛 − 𝑖𝑚 − 𝑗𝑛 + 𝑖𝑗) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 =∑ ∑ (𝑚𝑛)𝑛−1𝑖=0 𝑚−1 𝑗=0 − ∑ ∑ (𝑖𝑚) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 − ∑ ∑ (𝑗𝑛) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 + ∑ ∑ (𝑖𝑗) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = 𝑚𝑛 ∑ (1) 𝑚−1 𝑗=0 ∑(1) − 𝑚 𝑛−1 𝑖=0 ∑ (1) 𝑚−1 𝑗=0 ∑(𝑖) 𝑛−1 𝑖=0 − 𝑛 ∑ (𝑗) 𝑚−1 𝑗=0 ∑(1) 𝑛−1 𝑖=0 + ∑ (𝑗) 𝑚−1 𝑗=0 ∑(𝑖) 𝑛−1 𝑖=0 = 𝑚𝑛𝑚𝑛 + 𝑚𝑚 𝑛(1 − 𝑛) 2 + 𝑛𝑛 𝑚(1 − 𝑚) 2 + 𝑚(𝑚 − 1) 2 𝑛(𝑛 − 1) 2 = 𝑚2𝑛2 + 𝑚2𝑛−𝑚2𝑛2 2 + 𝑚𝑛2 − 𝑚2𝑛2 2 + 𝑚2𝑛2 − 𝑚𝑛2 − 𝑚2𝑛 + 𝑚𝑛 4 = 4𝑚2𝑛2 + 2𝑚2𝑛−2𝑚2𝑛2 + 2𝑚𝑛2 − 2𝑚2𝑛2 + 𝑚2𝑛2 − 𝑚𝑛2 − 𝑚2𝑛 + 𝑚𝑛 4 = 𝑚2𝑛 + 𝑚𝑛2 + 𝑚2𝑛2 + 𝑚𝑛 4 = 𝑚𝑛(𝑚 + 𝑛 + 𝑚𝑛 + 1) 4 = 𝑚𝑛(𝑚 + 1)(𝑛 + 1) 4 then, ∑ ∑(𝑚 − 𝑗) (𝑛 − 𝑖) 𝑛−1 𝑖=0 𝑚−1 𝑗=0 = 𝑚𝑛(𝑚 + 1)(𝑛 + 1) 4 thus, mathematical development establishes relationships that lead it 𝑚𝑛(𝑚+1)(𝑛+1) 4 to indicate how many rectangles are formed in a rectangular area m*n. maure, nava, marimón, & gutiérrez, the argument and demonstration exemplified … 218 3.2. discussion by bringing more elements to the discussion, it can be observed that the typical textbooks available for calculus content courses are designed specifically for future engineers or mathematicians, who provide solutions to the problems they pose and provide practical examples. this puts course students in the position of being math consumers, rather than doers and creators. according to association of mathematics teacher educators [amte] (2017), wellprepared beginner teachers "strive to position students as authors of ideas, students who discuss, explain and justify their reasoning using various representations and tools" (p. 16). thus, it is argued that textbooks in content courses should be focused on student development as the author of mathematical ideas in the course. consequently, an alternative is not to use a standard textbook in the course but have developed their own series of texts for math students (i.e. engineers and possible mathematicians) (beam et al., 2019a, 2019b). on the other hand, it is noted that the curriculum carried out many activities that do not propose the use of manipulable, whether physical as cardboard or virtual tokens, such as those that can be developed with dynamic geometry. the decision to use them is based on whether they can contribute substantially to students' ability to represent information or understand and think about a problem with the help of teaching resources that use various representation systems (literal, symbolic and concrete) as argued by yáñez et al. (2013). in teaching various geometric contents through observation and manipulation to help students, there is a guide for the teacher, who presents and defines them. underlying this mode of teaching, there is also the belief that the learning occurs not only by simple observation but also through external information provided to a person. contrastingly, from the perspective of the didactics of mathematics, apprenticeships should appear progressively to the extent students are exposed to problem-solving episodes (i.e., various representation systems), in which they need to identify patterns by drawing up guesses and, to a large extent, find ways to justify such conjectures. in addition, the need for understanding mathematics teachers in constructing mathematical objects and their meanings can be done by formulating conjectures and counter-examples so that later they can make appropriate didactic transpositions in their classes. this process can also be used to help students independently organize their mathematical thinking. in the traditional classroom, the teacher focuses on following the textbook and curriculum contents to the letter, not leading to episodes like those described in this experience. in the classroom with a traditional approach, mathematical knowledge is presented as something finished that makes it impossible to ask questions that make students reflect when an assertion is true or not. motivating to validate or invalidate all mathematical ideas that arise in the learning process. teacher training is something that should not be overlooked and must always be taken into account within the didactics of mathematics. therefore, there is a need to work to influence the practice of teachers and to be able to train them in the use of teaching resources such as the construction of conjectures and argumentation through counterexamples, which will allow them to expand their mathematical discourse. mathematical discovery has been addressed as a methodology to be included in educational practices. such a discovery has a close relationship with maieutic. in the sense, the game that is established between the teacher and the student to find the "truth". a truth in mathematics requires a constant cautious reassessment of its purposes, which in this case is intended to change naive thoughts and, for others, it is better developed to structure a mathematical thought. volume 11, no 2, september 2022, pp. 211-222 219 4. conclusion based on this experience, we reflected on the different logical theories and epistemologies in the research processes so that the thematic approaches and developments of our research projects are also at the forefront; besides being viable, solid, and unfold with the best possible structure. in general, urgent changes are needed in traditional educational practices where it can be incorporated into conjecture and counterexample, so that teachers can help their students change their naive thoughts toward structuring appropriate mathematical thinking. thus, students may have tools to be somehow competitive, critical and analytical in a society that underpins communication, that in many respects appear in mathematical language. it is necessary to explore into other works on how to structure mathematical thinking at different educational levels, as teachers are required to incorporate such resources into their practice. acknowledgements this research is financed by the contract for merit id no. 192-2021 of the projects entitled skills and knowledge of primary and secondary teachers for the teaching of mathematics in hybrid modality. the researcher luisa morales maure is a member of sini by the national secretariat of science, technology, and innovation (senacyt) and the authors are members of the research group in mathematics education – giem-21, attached to the vice-rector for research and postgraduate studies of the university of panama (up). references association of mathematics teacher educators [amte]. 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(2013). caracterización del conocimiento matemático para la enseñanza de los números racionales. avances de investigación en educación matemática(4), 47-64. https://doi.org/10.1007/s13394-021-00367-w https://doi.org/10.1177/002205741619600202 https://doi.org/10.1037/0096-3445.112.1.80 https://doi.org/10.1037/0096-3445.112.1.80 maure, nava, marimón, & gutiérrez, the argument and demonstration exemplified … 222 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p41-54 41 implementation of online learning and its impact on learning achievements of mathematics education students simon m. panjaitan, agusmanto j. b. hutauruk*, christina sitepu, sanggam p. gultom, parlindungan sitorus, melati riani marbun, cahyana hotmauli sinaga universitas hkbp nommensen, indonesia article info abstract article history: received jan 17, 2023 revised feb 21, 2023 accepted feb 26, 2023 implementing online learning in a higher education environment requires an analysis of learning outcomes and their impact on students, lecturers, and the institutions that administer the learning. the implementation of online learning shows that the learning process takes place and the readiness of lecturers and students to implement online learning. the impact given by online learning is in the form of gpa and student perceptions of the learning process they experience. based on the research results, various findings were obtained, including that mastery of online learning lms was not optimal, learning outcomes in the form of gpa were relatively high, students' perceptions of online learning were still low, especially in terms of lecturer readiness and students' confidence in their competence after participating in learning. keywords: learning achievement, mathematics education, online learning this is an open access article under the cc by-sa license. corresponding author: agusmanto j. b. hutauruk, department of mathematics education, universitas hkbp nommensen jln. sutomo no.4a, perintis, medan city, north sumatra 20232, indonesia. email: a7hutauruk@gmail.com how to cite: panjaitan, s. m., hutauruk, a. j. b., sitepu, c., gultom, s. p., sitorus, p., marbun, m. r., & sinaga, c. h. (2023). implementation of online learning and its impact on learning achievements of mathematics education students. infinity, 12(1), 41-54. 1. introduction many studies show that it takes time to restore the various impacts caused by covid-19 which takes about more than a decade to return to normal (djalante et al., 2020). interactions between humans are starting to be limited, a very large impact is felt in human life in various countries, including the republic of indonesia. this also has an impact on the education sector which is very broad ranging from learning at the elementary level to learning at the tertiary level. various policies in the field of education were issued by the government, namely prevention and handling in the educational environment, prevention in education units, and education policies during an emergency period for the spread of the corona virus disease-19. https://doi.org/10.22460/infinity.v12i1.p41-54 https://creativecommons.org/licenses/by-sa/4.0/ panjaitan et al., implementation of online learning and its impact on learning achievements … 42 learn from home has begun to be implemented in almost all regions in indonesia. online/online learning is implemented starting from the elementary level to higher education. all people involved in the education sector due to circumstances are forced to adapt and carry out the learning process from home, all schools and universities are prohibited from conducting direct or face-to-face learning, the learning process as a whole and simultaneously is required to carry out distance learning by utilizing technological advances (faozi et al., 2020; hidayat et al., 2023). online and offline learning is a topic that is widely discussed. many parties compare the two learning systems and look for the best learning. there are those who think online is better, but there are also those who think offline is the best. both methods have their advantages and disadvantages. during this pandemic, the face of education changed from face-to-face to online. this online learning is intended to prevent gathering and excessive interaction of students which can cause this cluster of covid transmission to appear. online learning is learning that is carried out by utilizing technology through virtual applications and using the internet, where the process of sending learning materials is not limited to time and place using various technologies in an open, flexible and distributed learning environment (hidayat et al., 2022; kusumaningrum & wijayanto, 2020). online learning is expected to be able to overcome the limitations of space which has been a weakness of conventional learning models (annur & hermansyah, 2020). online learning methods are divided into 2 types: synchronous and asynchronous. synchronous learning is an interaction between educators and students directly with audio or video conferencing through learning media that is connected to a network, while asynchronous learning is an indirect learning interaction with the distribution of teaching materials by educators through online learning media so that students can access it anytime and anywhere (fadila et al., 2021). in the online learning process, of course, you have to pay attention to the whole learning dimension. there are six learning elements that must be present in online learning, namely (1) connectivity makes it easier for students to communicate and interact with each other in the learning process; (2) flexibility, meaning that learning can be done anywhere and anytime; (3) interactive, where learning allows each participant to interact with each other; (4) collaboration, online learning is supported by online communication and discussion facilities between learning participants; (5) open, where student material and abilities can be accessed in various sources and mutually support the breadth of material, and (6) motivation, where the online learning process is fun and can be enjoyed by students. like face-to-face learning, online mathematics learning must also have mathematics learning standards (annur & hermansyah, 2020) as (1) the teacher must bring up meaningful math assignments, (2) in discourse, the teacher must play a responsive role in asking, listening and observing, (3) the teacher facilitates students who are active and interactive in listening, responding, asking questions, exploring and discussing, ( 4) the teacher encourages students to use devices such as models, technological devices, writing tools, visual and oral (presentations), in the context of enhancing learning of mathematics, (5) the teacher must establish a learning atmosphere that fosters the development of mathematical power, (6) the involvement of the teacher in the analysis of teaching and learning processes. online mathematics learning must also be supported by the current curriculum, which is known as kurikulum merdeka (abidah et al., 2020). implementation of online learning has been carried out in various places in learning mathematics. studies regarding the implementation of online learning in mathematics learning were carried out by several researchers, including (aini, 2021; azhari & fajri, 2022; fadila et al., 2021; kusumaningrum & wijayanto, 2020; nofriyandi & andrian, 2022; restian, 2020; simanjuntak et al., 2021; suripah & susanti, 2022; sutriyani, 2020; volume 12, no 1, february 2023, pp. 41-54 43 syarifuddin et al., 2021). in the learning process at several teaching and education faculties, online learning uses various applications (aini, 2021; manik, 2021). however, in practice, there are several obstacles faced by students and lecturers in the learning process, for example the applications used by lecturers often have problems or system errors occur, inadequate student devices, especially for materials with large capacity, lecturers who cannot directly deliver material so that many students do not understand the material, and so on (annur & hermansyah, 2020; hutauruk, 2020; iskandar et al., 2021). vice.com noted that a survey of 3,353 students who took part in online learning conducted by the ugm center for innovation and academic studies during the covid-19 pandemic found that 66.9% of the research subject students showed a good category in understanding lecture material, and 33.1% of students in bad category (azhari & fajri, 2022; nofriyandi & andrian, 2022; restian, 2020; suripah & susanti, 2022). this shows that students are not fully ready to take part in the online learning process, lecturer resources are not ready to manage distance learning, distance learning policies are not yet supported by an appropriate curriculum, facilities and infrastructure are not sufficiently supported, and internet networks are inadequate in some areas/ region. the lack of readiness of human resources in distance learning will also affect learning outcomes. this requires lecturers to carry out quality online learning and remain oriented towards achieving the ultimate achievement/objective of the learning process by paying attention/considering the ability of students to access the lecture material provided. in addition to problems related to the implementation of online learning, another challenge is regarding student learning outcomes after the implementation of online learning. several studies were conducted regarding the problems faced by students and the learning outcomes they got after participating in online learning (anim & mapilindo, 2020; annur & hermansyah, 2020; fadila et al., 2021; hutauruk, 2020; iskandar et al., 2021; manik, 2021; simarmata, 2022; sutriyani, 2020). this raised curiosity about the impact of online learning on student learning outcomes which was followed up through several studies (kusumaningrum & wijayanto, 2020; manik, 2021; syarifuddin et al., 2021). student learning outcomes are learning outcomes obtained by students after participating in a series of learning in the courses they take (widodo et al., 2020). learning outcomes can be carried out through evaluation of lecture results (hendriana et al., 2022). in online learning, evaluation of online lectures is carried out to determine the level of effectiveness of implementing online lectures or in other words to find out the extent to which predetermined learning objectives have been achieved. the learning outcomes for each subject in the mathematics education study program are arranged in accordance with the learning outcomes that have been determined by the study program. there are four aspects of learning outcomes in the mathematics education study program, namely learning outcomes from the aspects of attitude, knowledge, general skills and specific skills. these four aspects of learning achievement are described in each learning achievement per subject in the study program. the measurement of learning achievement is used to measure the learning achievement of graduates, which is an indicator of the success of the learning process that is expected from the implementation of the educational curriculum. fulfillment of learning outcomes can be measured to evaluate the process and results of meeting expected competency standards as well as evaluating the learning process of students, lecturers, and study programs. measurement of graduate learning outcomes carried out at the end of the study can provide information on the fulfillment of learning outcomes for students during their study period as well as an evaluation of graduate learning achievements. in this study, student learning outcomes were measured by learning achievement indicators for each subject that had been prepared by the lecturer concerned. in addition, it will also measure https://www.vice.com/id panjaitan et al., implementation of online learning and its impact on learning achievements … 44 how students perceive online learning and the learning achievements they have obtained in these courses. studies on student perceptions of online learning and the difficulties faced by students have been carried out before (anim & mapilindo, 2020; annur & hermansyah, 2020; manik, 2021; siregar et al., 2021). based on the problems mentioned, the researchers are interested in conducting research on the topic of implementing online learning and its impact on student learning outcomes in mathematics education study programs. 2. method this research is a quantitative descriptive research, to explain how online learning is implemented in the mathematics education department of fkip uhn. quantitative descriptive analysis also measures the impact that the implementation of online learning has on student learning outcomes in the mathematics education department of fkip uhn.the population of this study were all students and lecturers of the mathematics education department of fkip uhn, with the sample consisting of 15 permanent lecturers and 100 active students from various subjects in the mathematics education department of fkip uhn. the selection of 15 lecturers and 100 students was carried out by purposive sampling, where the lecturers and students were lecturers and students participating in online learning. descriptive data analysis techniques were used in this study to explain how online learning is implemented and its impact on student learning outcomes in the mathematics education department of fkip uhn. there are several research instruments used in this research that are used to collect data related to the implementation of online learning and its impact on student learning outcomes. before the research instrument is used, the research instrument validation has been carried out to the instrument validator to ensure the feasibility of the research instrument. instruments for implementing online learning include: a. questionnaire, contains several statements that measure how the implementation of online learning is carried out by lecturers and students of mathematics education study programs. questionnaire statements are prepared based on the realities that occur in the field, which will explain how the online learning process occurs. this questionnaire will be filled in by lecturers and students; b. the observation sheet, which is an observation sheet for the lesson plan, will see the suitability between the results of the questionnaire and the lesson plan that has been prepared; c. interviews are confirmation activities for lecturers and students regarding the implementation of online learning that is taking place. learning achievement instrument include: a. student achievement index (ip) reports which are the results during the online learning process; b. the observation sheet, observation sheet, containing observation sheets on test results, to see the suitability between the test results and the learning outcomes in the lesson plan; c. interviews are confirmation activity to lecturers and students regarding student learning outcomes. volume 12, no 1, february 2023, pp. 41-54 45 3. result and discussion the research was carried out in the even semester of 2021/2022, with the research sample consisting of 120 students and 15 lecturers from the mathematics education department who attended and carried out online learning at fkip uhn. there are two variables that are the focus of the study in this study, namely online learning and learning outcomes. the indicators for studying the implementation of online learning in this research will be seen from two aspects, namely (1) the external aspect consists of the type and mastery of the learning application/lms used, and (2) the internal aspect consists of ongoing learning practices and the readiness of students and lecturers in implementing online learning. indicators of student learning achievement will be reviewed from two aspects (1) achievement of learning achievement for each subject obtained through student achievement index (ip) scores after participating in online learning, and (2) student perceptions of learning achievement that have been obtained after participating in online learning. 3.1. implementation of online learning online learning at fkip uhn uses the virtual meeting method using various applications including zoom, google meet and whatsapp. this learning is integrated in one learning management system, namely google classroom. based on the results of the questionnaire, 100% of the lms types used in online learning in the mathematics education department of fkip uhn use google classroom (see figure 1). figure 1. type of lms data related to lms mastery was measured using a questionnaire containing several statements related to lms operating mastery, including (1) lms login and logout, (2) uploading and downloading documents on the lms, and (3) knowing and operating the various features available on the lms. in terms of lms mastery, of the 15 lecturers, only 6 lecturers had lms operational mastery above the average mastery, while 9 other lecturers had lms mastery below the average mastery. then for students, out of 100 students, 44 students have lms operational mastery above the average mastery, and 56 students have lms mastery below the average mastery (see figure 2). panjaitan et al., implementation of online learning and its impact on learning achievements … 46 figure 2. percentage of lms mastery the percentage of using lms in online learning for lecturers and students is described in detail according to the indicators for measuring lms mastery shown in the figure 3. figure 3. mastery of lms according to indicators on lecturers and students based on figure 3, it can be seen that there is 89.33% mastery of lms login and logout for lecturers, while only 79.83% for students. furthermore, for lecturers, 86.66% mastery is for uploading documents and 89.33% mastery is for downloading documents on the lms. while for students, 75.17% mastery for uploading documents and 75.33% mastery for downloading documents on the lms. regarding mastery of recognizing features in the lms, lecturers mastered the introduction of lms features by 72%, while students mastered the introduction of lms features by 65.83%. and in operating the lms, lecturers have mastery of 69.33% in operating the features on the lms, and students have mastery of 66.33% in operating the features on the lms. the second indicator related to the implementation of online learning in the mathematics education department can be seen from ongoing online learning practices and the readiness of students and lecturers to carry out online learning. this indicator was measured using a questionnaire and interviews conducted with 15 lecturers and 100 students as respondents. the questionnaires and interviews conducted were guided by several subindicators, namely (1) the availability of lesson plans for courses, (2) the suitability of the number of lecture meetings, namely 16 meetings including uts and uas, (3) the completeness of lecture material for each subject, (4) the accuracy of study duration according to the course credit load, with the rule that the duration of lectures is 50 minutes per credit, (5) there is learning in the form of assignments or projects, (6) uts and uas are implemented in each subject and (7) available course assessment rubrics. volume 12, no 1, february 2023, pp. 41-54 47 the data obtained from the questionnaire regarding the practice and readiness of online learning is described as follows (see figure 4). figure 4. online learning practice and readiness based on figure 4, it can be explained that in the sub-indicator of availability of rps and lecture contracts, out of 15 subject lecturers who carry out online learning, only 86.67% have rps and lecture contracts in the courses they teach. in the sub-indicator of suitability for the number of meetings, all lecturers (100%) carry out online lectures according to the specified number of meetings, namely 16 meetings including uts and uas. in the subindicator of the completeness of lecture material, it was found that only 66.67% of the completeness of lecture material in online learning could be completed. furthermore, the sub-indicator for the accuracy of lecture duration obtained the lowest results, namely only 26.67% of online learning lectures were carried out in accordance with the specified lecture duration. then in the assignment/project sub-indicator, 93.99% of online learning lectures apply task-based learning or lecture projects. regarding the implementation of uts/uas, all online learning courses (100%) carry out uts and uas, and on the sub-indicator of the availability of an assessment rubric, there are 73.33% of courses that have an online learning lecture assessment rubric that is implemented. 3.2. student learning outcomes one indicator of the student learning achievement variable for the implementation of online learning at the mathematics education department is the student achievement index after participating in online learning. the student achievement index in this study is the student achievement index for the 2020 and 2021 batches who have participated in online learning since the beginning of their studies at the mathematics education department. student achievement index data is obtained from an academic information system that displays student achievement indexes for all courses attended during online learning. the data is described as follows (see figure 5). panjaitan et al., implementation of online learning and its impact on learning achievements … 48 figure 5. achievement index for online learning students based on the achievement index data of students participating in online learning (see figure 5), overall out of 100 students, 81% have a grade point average between 3.51 to 4.00, 12% have a grade point index 3.01 to 3.51, and 1% respectively for grade point average 2.51 to 3.00 and grade point average between 1.51 to 2.00. separately, there are 49 students in class 2020, 44 of whom have a gpa of 3.51 to 4.00, there are 4 students who have an gpa of 3.10 to 3.50 and 1 person who has an gpa of 1.51 to 2.00. there are 51 students in class 2021, 42 of whom have a gpa of 3.51 to 4.00, 8 people have an gpa of 3.01 to 3.50 and 1 person has an gpa of 2.50 to 3.00. student perceptions of the process and achievements of online learning were obtained using questionnaires and questionnaires, and were strengthened by the results of interviews with respondents from several students. student perceptions of the online learning process are measured based on nine sub-indicators, namely (1) achievement of understanding of course material, (2) smooth learning process, (3) time efficiency during the learning process, (4) interactions during the learning process, (5) the lecturer's attention to student needs/questions, (6) fees for online learning, (7) the level of obstacles encountered, (8) quantity of online learning assignments/projects and (9) the quality of online learning assignments/projects. meanwhile, students' perceptions of online learning achievements consist of three sub-indicators, namely (1) the final score describes competence, (2) confidence in the competencies possessed, and (3) the success of online learning. student perceptions of the online learning process are described as follows (see figure 6). volume 12, no 1, february 2023, pp. 41-54 49 figure 6. student perceptions of the online learning based on figure 6, it is explained that students have a perception of achieving understanding of the course material they are following during online learning only at 59.4%, and the smoothness of the online learning process they are participating at is also 59.4%. regarding time efficiency during learning, students stated an efficiency of 78.6%. regarding the interaction that was built between students and lecturers during online learning, students had a view regarding the interactions that occurred at 57.6%, and the attention given by lecturers to students' needs and questions during learning was at 82%. regarding the costs and obstacles experienced by students in participating in the online learning process, data was obtained that the costs for the learning process were 63.8%, as well as the level of obstacles experienced by students when learning online was 60.8%. finally, regarding assignments or projects during online learning, data on the quantity of assignments/projects during online learning was obtained by 61% and the quality of assignments/projects was 59.2%. student perceptions of online learning outcomes are described as follows (see figure 7). figure 7. student perceptions of online learning outcomes regarding the achievement of the final score whether the final score describes the competencies possessed by students, 38% of students believe that the final score describes panjaitan et al., implementation of online learning and its impact on learning achievements … 50 the competencies possessed, 32% of students doubt that their final score describes the competencies they have, and 30% of students are not sure that the final score describe their competence. furthermore, regarding confidence in competence, 34% of students are confident in the quality of the competencies they have, 30% are unsure of the quality of the competencies they have, and 36% are unsure of the quality of the competencies they have after participating in online learning. regarding their opinion on the success of online learning that they have gone through, 34% of students think that online learning has been successful, 26% of students have doubts about the success of online learning, and 40% of students think that online learning is not successful. based on the research results that have been obtained, the implementation of online learning requires an lms that can be accessed and understood by users, both lecturers and students both in use and assessment (dahlstrom et al., 2014; sahin et al., 2021; watson & watson, 2007). the pandemic has caused higher education to be able to adapt in the learning process using it and lms that support online teaching and learning activities (garcíamorales et al., 2021; mishra et al., 2020). identifying obstacles in using all lms features and considering readiness to take part in online learning is one of the important factors to support smooth online learning, so that it becomes one of the aspects of lms selection and organizing online learning (hidayat et al., 2022; irfan et al., 2020; octaberlina & muslimin, 2020; tang et al., 2021). in addition, pedagogical competence in learning also needs to be a concern in organizing online learning (zhao et al., 2021). regarding student perceptions of online learning and online learning outcomes, students consider that careful preparation is needed to ensure that online learning produces satisfactory outcomes according to learning objectives (adnan & anwar, 2020; almusharraf & khahro, 2020; bestiantono et al., 2020). it is necessary to pay attention to the control and evaluation of learning in order to maintain the quality and outputs and outcomes of ongoing online learning (elmunsyah et al., 2020; giatman et al., 2020; mishra et al., 2020; pertiwi et al., 2021; szopiński & bachnik, 2022). 4. conclusion based on the results of this study, the following conclusions were obtained: (1) online learning for mathematics education study programs uses google classroom as the only lms used; (2) mastery of the lms used by the majority is still below average, both for lecturers and students, with the level of mastery of lecturers being higher than that of students; (3) the lowest lecturer mastery of lms is the use of features in the lms. the lowest student lms mastery is an introduction to the features of the lms; (4) regarding the practice and readiness of lecturers' learning for online learning, the accuracy of lecture duration is the lowest, while the highest is the implementation of the number of meetings and the implementation of uts/uas; (5) learning achievement as 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(2021). digital competence in higher education research: a systematic literature review. computers & education, 168, 104212. https://doi.org/10.1016/j.compedu.2021.104212 https://doi.org/10.22460/infinity.v11i1.p17-32 https://doi.org/10.53299/jagomipa.v1i1.16 https://doi.org/10.1016/j.techfore.2021.121203 https://doi.org/10.1016/j.compedu.2021.104211 https://doi.org/10.1007/s11528-007-0023-y https://doi.org/10.1088/1742-6596/1657/1/012092 https://doi.org/10.1016/j.compedu.2021.104212 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p25-34 25 analysis of students’ mathematical abstraction ability by using discursive approach integrated peer instruction of structure algebra ii elsa komala universitas suryakancana, jl. pasir gede raya, bojongherang, cianjur, west java, indonesia elsakomala@gmail.com received: august 06, 2017 ; accepted: august 24, 2017 abstract this research is subject to describe students' mathematical abstraction ability using the discursive approach of peer instruction integration, to get an idea about the problems faced by students and to study about the interventions needed by students to overcome the problems. this research using the quantitative descriptive approach with pre-experimental design: the one-shot case study design, which is done to the students in the course of the structure of algebra ii. based on the data analysis, result of the research shows that ability of students' mathematical abstraction using the discursive approach of peer instruction integration in the course of the structure of algebra ii does not exceed 85% of ideal criteria determined. obstacles faced by students such as cognitive obstacles, genetic and psychological obstacles, didactic obstacles, epistemological obstacles. interventions that need to be given such as by providing reinforcement to the mastery of prerequisite material, conditioning through drill, practice, and exercise, providing scaffolding, raises students awareness of what they learn, convergent interventions in the form of a closed-ended investigative question. keywords: mathematical abstraction, discursive approach, peer instruction abstrak tujuan penelitian ini untuk mendeskripsikan kemampuan abstraksi matematis mahasiswa menggunakan pendekatan diskursif integrasi peer instruction, memperoleh gambaran tentang permasalahan yang dihadapi mahasiswa dan mengkaji tentang intervensi-intervensi yang diperlukan mahasiswa guna mengatasi permasalahannya. penelitian ini menggunakan pendekatan deskriptif kuantitatif dengan desain pre experimental: the one-shot case study design, terhadap mahasiswa pada mata kuliah struktur aljabar ii. berdasarkan analisis data, menunjukkan bahwa kemampuan abstraksi matematis mahasiswa menggunakan pendekatan diskursif integrasi peer instruction pada mata kuliah struktur aljabar ii tidak melebihi 85% dari kriteria ideal yang ditetapkan. hambatan mahasiswa dalam melakukan abstraksi matematis diantaranya hambatan kognitif, hambatan genetis dan psikologis, hambatan didaktis, hambatan epistemologi. intervensi-intervensi yang perlu diberikan kepada mahasiswa guna mengatasi permasalahan tersebut diantaranya dengan memberikan penguatan terhadap penguasaan materi pra syarat, melakukan pembiasaan melalui kegiatan drill, practice, dan exercise, memberikan scaffolding, memunculkan kesadaran pada diri mahasiswa mengenai apa yang dipelajarinya, intervensi konvergen berupa pemberikan pertanyaan investigasi yang bersifat tertutup. kata kunci: abstraksi matematis, pendekatan diskursif, peer instruction. how to cite: komala, e. (2018). analysis of students’ mathematical abstraction ability by using discursive approach integrated peer instruction of structure algebra ii. infinity, 7 (1), 25-34 doi:10.22460/infinity.v7i1.p25-34 komala, analysis of students’ mathematical abstraction ability … 26 introduction for the college level, the mathematical material is increasingly difficult to learn. artigue (1998) argue that mathematics learning is often rated negatively by students and they have considerable difficulty with some mathematical processes such as reasoning, non-routine problem solving and proving. according to tall (2008), the change from elementary thinking to advanced mathematical thinking involves a significant transition, is from describing to defining, from convincing to proofing logically based on a definition. the transition process is a problem for students herlina (2015). the purpose of education in indonesia requires that students, especially prospective teachers are required to have multi-ability, one of the skills that need to be mastered in mathematics is the ability of abstraction. according to nurhasanah, kusumah, & sabandar (2017) learning mathematics is a complicated process because the objects in it are so abstract, therefore the mathematical concept can not be transferred to the students directly (directly convey the general form which seems like a piece of information), but it must give through a process so that learning will feel more meaningful and lasting, the process is called the process of abstraction of a concept. besides, nurhasanah et al (2017) also expressed that the essence of mathematics is an abstraction and abstract concept. therefore, students need advanced mathematical thinking. one of the subjects that require advanced mathematical thinking is the structure of algebra i and ii. in addition, when students are asked to explain the results of their calculations, they experience confusion in representing and abstracting it. often we see and hear the expression about the number of students whose activities are less thinking. they only learn but their way of learning is limited to hearing the information of his lecturer then not trying to understand the material being taught. especially on courses such as structure algebra i and ii. the course of structure of algebra ii (ring theory) is a continuation course of the structure of algebra i (group theory) which has introduced a structural mathematical concept in general that has been discovered or studied previously. based on observations in the field when the course structure of algebra the generalization process of the basic concept was not all students are able to master it. the same thing also express by nurlaelah & sumarmo (2009) that the course of algebra structure is a subject that contains abstract concepts because the nature of the course is like that then students often have difficulty in learning it. how to introduce the concept with the announcement of the previous concept can be said process of abstraction. building mathematical concepts independently by students is fundamental to mathematics learning. students are given the widest opportunity to build and construct his own knowledge. because the process of abstraction is the way of the emergence of a concept, meaning it is very important in learning mathematics, so the ability of mathematical abstraction becomes a capability that must be owned by students to study the course of algebraic structure. according to tall (2008) that abstraction is the process of describing a particular situation to the realm of a concept that can be thought (thinkable concept) through a construction. such thoughtful concepts can then be used at more complex and complex thinking levels. according to him, the process of abstraction can occur in some circumstances, but there are three circumstances that usually bring up the process of abstraction in the process of learning mathematics. the first state can arise when the individual focuses his attention on the characteristics of the objects he or she observes, then gives the name through a process of classifying by category into several groups. volume 7, no. 1, february 2018 pp 25-34 27 problems that arise in learning structure algebra ii is usually, students difficult to perform an empirical abstraction for example students have difficulty imagining the properties of objects or even generalized numbers because the learning experience is not the same. this can lead to weak visualization of previous concepts so that students feel difficulty when understanding the new concept. likewise difficulties occur in theoretical abstraction, because the experience of learning and the process of illustration of real objects is not used as a reference, but based on previous theories. this creates an obstacle if previous concepts lack understanding which results in the formation of a new concept of theoretical abstraction. this is a challenge for lecturers to solve the problem. one possible solution is to use a particular learning approach. so based on the experience of previous lecture lesson that is structure algebra i, which is the prerequisite course for taking the course of structure algebra ii when the student exam only reveals the material they have memorized it, without being able and understand in doing the abstraction both empirical and theoretical. one of the learning that has potential to develop student activeness and able to improve abstraction ability in learning is discursive approach of peer instruction integration. the discursive approach focuses on communication in the form of debates, logical reasons in writing, and mathematical communication so that this approach views students in the classroom as learning societies that interact with each other. in peer instruction is interspersed with concept questions (crouch & mazur, 2001) and involves student activeness in learning (fagen, crouch, & mazur, 2002). students are given the opportunity to think in solving the conceptual question, then discuss with their peers. in addition, in learning is expected to optimize the concept ability through thinking and discussing with colleagues. accordingly, peer instruction lessons are more effective than classroom discussions (nicol, & boyle, 2003). a rich learning environment with peer discussions can develop critical thinking skills and deep mastery of concepts in students (anderson, howe, soden, halliday, & low, 2001). based on the background that has been described previously, then the formulation of this research problem is to know: 1) how the ability of mathematical abstraction of students using discursive approach of peer instruction integration in the course of structure algebra ii; 2) what problems faced by students in solving the problem of mathematical abstraction in the course of structure algebra ii; and 3) what interventions should be given to the students to overcome the problems faced in solving the mathematical abstraction problem in the course of structure algebra ii. ability of mathematical abstraction the ability of mathematical abstraction is the ability of thinking that connects mathematical concepts into new concepts with the generalization process. according to piaget (bermejo & diaz, 2007), abstraction is divided into 2 types, namely empirical abstraction and reflective abstraction. while mitchelmore & white (2007) distinguish abstractions into empirical abstractions and theoretical abstractions. peer instruction peer instruction is a lesson interspersed with short conceptual questions designed to express misunderstandings and to engage students to be active in learning. komala, analysis of students’ mathematical abstraction ability … 28 discursive approach one approach to learning which views language, communication, discourse, and thinking is not a separate object of theoretical reflection is a discursive approach. so in education, this discursive approach has the intention to use essay writing, discussion and debate communication forum in the field of mathematics in the classroom. method this research uses the quantitative descriptive approach with the experimental method. the experimental design used in this research is pre-experimental design type one-shot case study. in this design, there is a group treated (treatment), and then observed the results. treatment in this research is a discursive approach to peer instruction integration, as well as an independent variable in research. meanwhile, the results observed in this study are students' mathematical abstraction abilities, which is also a dependent variable in the study. the research paradigm is described as follows: x o information: x = discursive approach of peer instruction integration o = ability of mathematical abstraction results and discussion description of learning with discusive approach peer instruction integration table 1. learning phase with discursive approach peer instruction integration phase learning activities lectur student problem orientation in peer a. the lecturer raises the problem through student worksheet done by the student b. the lecturer proposes a conceptual test that relates to the learning material presented a. together with his group's friends, the students ask the concept questions (peer instruction) which will only be answered "yes" and "no" by the lecturer. b. students formulate the issue ses-uai with the material being studied hypothesized in peer (discussion) lecturer guides students to make hypothesis of completion a. students individually think for answers to the given concept tests that are in the student worksheet. b. together with his student group the students hypothesized the problems that would later be commented on by other groups volume 7, no. 1, february 2018 pp 25-34 29 phase learning activities lectur student test the hypothesis in peer (essay writing) a. lecturers guide students to experiment to test their hypotheses. b. lecturer guides students in analyzing data a. students perform experiments in accordance with the student worksheet. b. students conduct, observe and record carefully the results of discussions that have been made with his group friends. c. students analyze thoughtprovoking data and discuss the hypotheses they make with their group mates. d. students also discuss the answers to the mathematical abstraction concept test given at the beginning with their group mates presentation of data in peer (communication forum) the lecturer guides the students to present the results a. the students presented the results, which were then responded by other groups. b. conveys the concept test answers (mathematical abstraction abilities) provided at the beginning of the lesson feedback (discussion) a. the lecturer provides reinforcement of the results as well as provides confirmation of the concept test (mathematical abstraction abilities) given at the beginning. b. the lecturer gives an example of the problem. c. lecturers provide evaluation questions a. students pay attention to the strengthening of the lecturers as well as to revise the concept test results (mathematical abstraction abilities) that have been done at the beginning of the lesson. b. students pay attention to lecturers. c. students work on evaluation questions. peer conclusion evaluating the conclusions and the results of discussions that have been made by the students in accordance with the material presented ogether with his student friends the students made a conclusion of the results, then expressed in the class and responded to other groups komala, analysis of students’ mathematical abstraction ability … 30 description of data ability mathematical ability based on the results of data analysis abstract mathematical abilities obtained from the test instrument in the form of a set of midterm exam and final exam problems using the z test for one sample, obtained the value of z score = -5.715 and the critical z value = -1,645. in the right-handed testing criterion, the value of z score is in the reception area h0 because z score < z critical so that h0 is received (not enough evidence to reject h0). this shows that at 95% confidence level there is not enough evidence to suggest that students' mathematical proofing ability using the discursive approach of peer instruction integration in the course of structure algebra ii significantly exceeds 85% of ideal criteria. in other words, the data obtained is not sufficient to prove that the proposed research hypothesis is true. the description of the data on midterm exam and final exam values indicates an increase in the value obtained by students during midterm exam and final exam. the average increase is 0.14 or is in a low category. in other words, although the students' mathematical abstraction ability has not reached 85% of the ideal criteria set, but by using the discursive approach the integration of peer instruction abstraction ability develops quite well from time to time, so that if this approach is continuously applied and developed then abstraction ability mathematical students can continue to grow. description of the achievement of students' mathematical abstraction abilities based on measured ability indicators that are 1) abstraction reflective, in the form of integration and problem formulation and problem transformation into symbol form; 2) the empirical abstraction, ie making generalizations, the formation of mathematical concepts related to other concepts, the formation of further mathematical objects and the formalization of mathematical objects; and 3) theoretical abstraction, the process of manipulating symbols. . table 2. percentage of mathematical ability achievement ability indicator percentage (%) information indicator 1 (reflective abstraction): integrate and problem formulation and problem transformation into symbol form 38.9 almost half indicator 2 (empirical abstractions): to generalize, the formation of mathematical concepts related to other concepts, the formation of further mathematical objects and the formalization of mathematical objects 65.7 mostly indicator 3 (theoretical abstraction): able to manipulate symbols 33.2 nearly half based on the data from table 2. it is known that from 35 students, in indicator 1 almost half (38.9%) students have been able to integrate and problem formulation and problem transformation into symbol form (reflective abstraction), indicator 2 mostly (65.7 %) has been able to generalize, the formation of mathematical concepts related to other concepts, the formation of further mathematical objects and the formalization of mathematical objects (empirical abstractions). nearly half (33.3%) in indicator 3 students have been able to manipulate symbols (theoretical abstraction). this shows an increase in the percentage of mathematical abstraction ability achievement from the previous year. however, the abilities volume 7, no. 1, february 2018 pp 25-34 31 of mathematical abstraction still need to be developed and improved especially on the first and third indicators relating to performing reflective abstractions and theoretical abstractions. description of mathematical abstraction problems data about problems faced by students in performing mathematical abstraction obtained from the answer sheet of midterm exam and final exam workmanship that is then processed and analyzed descriptively and reinforced with an open questionnaire that is filled after the work on final exam. descriptive analysis is done by describing or describing the data that has been collected as it is without intending to make generalizations. the process of data analysis is done by categorizing errors of student mathematical proofing based on similar errors and making the percentage of answers in each category called obstacles. in general there are four categories of problems faced by students in performing mathematical abstractions which are called obstacles in the course of algebra structure ii that is the problem is in line with the opinion of tall (2008): 1) cognitive obstacles, occurs when students have difficulty in learning; 2) genetic and psychological obstacles, occurring as a result of a student's personal development; 3) didactic obstacles, due to the teacher's teaching nature; 4) the epistemological obstacles, occurs because of the nature of the mathematical concept itself. table 3. percentage of obstacles in mathematical abstraction obstacles percentage (%) information cognitive 77.1 most genetic and psychological 25.7 almost half didactic 51.4 most epistemological 65.7 most by the table. 3 found that from 35 students, most of them (77.1%) students experience cognitive obstacles, almost half (25.7%) students experience genetic and psychological obstacles, most (51.4%) of students have didactic obstacles, and most (65.7%) students experience epistemological obstacles. the data shows that the problems and obstacles faced by students in performing mathematical abstractions in the course of structure algebra ii are very complex so that some interventions given by the lecturers to the students are needed to overcome the problem. interventions to be assumed in course structure algebra ii after knowing the problems faced by students in doing mathematical abstraction in the course of the structure algebra ii, the researchers tried to formulate the interventions that need to be given to overcome the problems faced by students in doing mathematical abstraction in the course of structure algebra ii. the formulation of these interventions is based on the research conducted by the researcher on the problems that occur based on the perspective of learning theory that has been formulated by experts who adapted to the characteristics of the ability and habits of student learning and thinking. based on the result of the research done, the researcher resulted from the suspicion that the problems that occur in relation to students' mathematical abstraction ability are caused by 1) the lack of mastery of pre-condition materials such as group theory, set, number system, binary operation characteristics, 2) lack of exercise intensity in mathematical abstraction; 3) too many definitions and theorems to be studied so that students are confused in determining komala, analysis of students’ mathematical abstraction ability … 32 which definitions or theorems should be used to work on and prove abstraction on the given problem. the researchers conducted a study based on the perspective of learning theory that has been formulated by experts who adapted to the characteristics of skills and habits of learning and thinking students semester v academic year 2016/2017 mathematics education fkip unsur. given the characteristics of students who have started to work (honorary teachers) and active in various activities and student organizations that nota bene can not se t the time betwen lectures with other activities, then most of the student activities are busy by other activities besides study so that the time of study and student independence in learning is very limited. in addition, the abstract characteristics of algebra structure ii material also require the lecturer's dominance in delivering the material in the lecture. based on the above allegations and assessments, in formulating interventions to overcome the problems that occur, researchers are guided by the theory of learning behaviorism and constructivism, including the theory of reinforcement of bf skinner, pavlov's conditioning theory, and social interaction learning theory from vygotsky who emphasizes the giving of scaffolding, the theory of metacognition that emphasizes the learner the ability to look at yourself so that what he did can be controlled optimally. the interventions that can be formulated from the results of the research that has been done by researchers are: 1) provide reinforcement to the mastery of prerequisite material, 2) perform conditioning through drill, practice, and exercise, and 3) provide scaffolding in the form of deductive proof instructions by including definitions or theorems for thinking of mathematical abstractions; 4) raises students awareness of what they learn (metacognition), 5) convergent interventions in the form of a closed-ended investigative question and leads to problem solving given at the beginning of the lesson, as students look for patterns and generalize concepts; and 6) devise a didactic design, which takes into account the characteristics and nature of the concept. some studies of abstraction abilities that have been done before are still theoretical (ferarri, 2003; goodson-espy, 2005). it means that the study is still a description of the emergence of theories and critiques about the abstraction not yet on the stage of learning in the classroom. in addition, research that has been studied by dindyal (2007) reveals that abstraction is done on the topic of geometry. also in line with the research (dindyal, 2007) separately undertook intensive abstraction research, using a quantitative approach to examine the abstraction process in junior high school students. so that researcher do research about bastraksi ability in material of algebra structure which done by student by using megasar approach disursif integration of peer instruction. conclusion based on results and discussion, then the conclusion of the results of this study is (1) the ability of students' mathematical abstraction using the discursive approach of peer instruction integration in the course of algebra structure ii does not exceed 85% of ideal criteria set; (2) problems faced by students in performing mathematical abstraction commonly called obstacles including the cognitive obstacles, occurs when students have difficulty in the learning process; genetic and psychological obstacles occur as a result of a student's personal development; didactic obstacles, occur because of the nature of teaching from the teacher; epistemological obstacles occur because of the nature of the mathematical concepts themselves; (3) interventions that need to be given to the students to overcome the problems such as by providing reinforcement to the mastery of prerequisite material, conditioning volume 7, no. 1, february 2018 pp 25-34 33 through drill, practice, and exercise, providing scaffolding in the form of abstraction instructions by including the definition or theorems for mathematical abstraction, raises students' awareness of what they learn (metacognition), convergent interventions in the form of a closed-ended investigative question and leads to problem solving given at the beginning of the lesson, as students look for patterns and generalize concepts and design didactic, which takes into account the characteristics and properties of the concept. acknowledgments thus the author can convey the results of research contained in this article. on this occasion, the authors would like to express the appreciation and gratitude to all the parties that have the big impact on the success and completion of this research. the author realized that this article is far from perfect, therefore constructive criticism and suggestions are desirable for the perfection of this article. hopefully, this research will be useful for the reader. references anderson, t., howe, c., soden, r., halliday, j., & low, j. 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(2008). the transition to formal thinking in mathematics. mathematics education research journal, 20(2), 5-24.. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p367-380 367 factors that affect students’ mathematics performance at higher education in riau province during the covid-19 pandemic nofriyandi*, dedek andrian universitas islam riau, indonesia article info abstract article history: received sep 23, 2021 revised sep 27, 2022 accepted sep 29, 2022 this study aims to determine the factors influencing the students' mathematics performance in higher education in riau province. the research population was all students in higher education in riau province majoring in mathematics. the samples of this research were some students in higher education majoring in mathematics education which were taken randomly with a proportional random sampling approach. the instrument in this research was a questionnaire developed and validated by experts and distributed to mathematics education students. the data analysis used in this research was path analysis. the results showed that there was an effect of learning motivation on mathematics performance, there was no effect of learning interest on mathematics performance, with a t value of 1.28, (12) there was no effect of self-efficacy on mathematics performance, with t values of 1.38, there was an effect of self-regulated on mathematics performance, with t values of 2.23, and (15) there was no effect of learning involvement on mathematics performance, with t value of 1.39 the dominant and significant factor in influencing the students' mathematics performance in higher education in riau province during the covid-19 pandemic was learning motivation and self-regulation. keywords: covid-19 pandemic, higher education, mathematics performance this is an open access article under the cc by-sa license. corresponding author: nofriyandi, department of mathematics education, universitas islam riau jl. kaharuddin nasution no.113, pekanbaru city, riau 28284, indonesia. email: nofriyandi@edu.uir.ac.id how to cite: nofriyandi, n., & andrian, d. (2022). factors that affect students’ mathematics performance at higher education in riau province during the covid-19 pandemic. infinity, 11(2), 367-380. 1. introduction mathematics learning has a function as a means to develop critical, logical, creative, and collaborative thinking skills, where these abilities are indispensable in modern life (acharya, 2017). mathematics learning has an important contribution to the development of the abilities of each student to become a quality human resource (li & schoenfeld, 2019; mazana et al., 2019; risnawati et al., 2019). the need for the development of mathematics learning in students has the aim of building sensitivity to the surrounding environment and being able to solve problems that occur around them. https://doi.org/10.22460/infinity.v11i2.p367-380 https://creativecommons.org/licenses/by-sa/4.0/ nofriyandi & andrian, factors that affect students’ mathematics performance at higher … 368 the number of challenges for students (prospective teachers) in indonesia related to mathematics is the low achievement of indonesian students in international forums such as the international mathematics and science olympiads that need to be improved. referring to the results of the trends in international mathematics and science study (timss), indonesia's ranking in 2015 was ranked 44th out of 49 countries with a score of 397. from the program for international students assessment (pisa) in 2018, indonesia was ranked 75th out of 80 countries. indonesia received scores from three aspects of the assessment, namely reading at 371 out of an average of 453.1, mathematics at 379 from an average of 458.3, and science at 396 from an average of 457.6. the low ranking of indonesian students at the international level is the responsibility of lecturers as a component of the education system in indonesia. the role of the lecturer is to provide, demonstrate, guide, and motivate students to interact with various learning resources. lecturers are required not only as presenters or conveyers of knowledge to students but also must be able to help students to develop their skills. a lecturer must be able to create active, innovative, creative, effective, and fun learning so that the learning process generates and attracts student interest and motivation. interest in learning is a very important factor in supporting the achievement of the effectiveness of the learning process, which will directly affect learning outcomes. according to slameto (2015), interest is a sense of interest in a thing or activity without anyone telling. interest is basically the acceptance of a relationship between oneself and something outside oneself. the stronger or closer the relationship, the greater the interest. motivation is one factor that determines the learning process's effectiveness. motivation is also an internal and external encouragement for someone who is learning with several indicators or supporting elements. these indicators, namely; the desire to succeed, encouragement and needs in learning, hopes and aspirations for the future, appreciation of learning, and a conducive learning environment (uno, 2021). this is in line with the opinion of sardiman (2014) that motivation is the overall driving force in students that creates, ensures continuity, and provides direction for learning activities so that it is hoped that the goals can be achieved. students must have interest and motivation in every lesson because motivation will make it easier for students to achieve learning goals. susanto (2012) states that interest is a significant contribution to the success of students. sardiman (2014) states that a person will succeed in learning if there is a desire to learn. this desire is called motivation. the success of a person's learning is not solely determined by his abilities but also by his interest and learning motivation. it is often found that students who have high abilities but fail in learning are caused by a lack of interest and motivation to learn. interest and motivation are essentially an effort to produce maximum learning achievement. self-regulated learning and self-efficacy can affect student learning success. when a person's independence in learning and self-confidence is high, it is expected that learning achievement is also high. if students have good interest, motivation, self-regulated learning, and self-efficacy, certainly, the student's performance is also good. in the field of mathematics, involvement, interest, motivation, independence, and self-confidence of students will result in achievement or mathematical performance that can achieve what students expect from learning. however, learning motivation, interest in learning, selfregulation, self-efficacy, and learning involvement of students in university mathematics education study programs in riau province is still low. students do not submit assignments on time, make assignments not optimal, attend lectures lazily, do not want to ask questions when they don't understand, make assignments by cheating friends, and are often late for online lectures. these problems require changes in behavior and interrelated actions. volume 11, no 2, september 2022, pp. 367-380 369 therefore, the factors that affect the mathematics performance of students majoring in mathematics education at universities in riau province need to be investigated. slameto (2015) states interest is a persistent tendency to pay attention and remember some activities. interest in learning is a student's interest in participating in learning without coercion from anyone (pedro et al., 2018). interest in learning is a source of high student motivation in participating in learning and improving student achievement (fryer et al., 2019). high and low student achievement is caused by high interest in learning, so that which increases motivation to succeed in learning (azmidar et al., 2017). interest in learning is defined as a tendency towards something because it is profitable. when someone sees that something is profitable, then that person has a sense of interest because it can bring satisfaction (renninger & hidi, 2019; vitasari et al., 2010). based on the opinions of the experts above, it can be concluded that interest is a sense of interest, attention, and desire that a person has to do something. uno (2021) explains that the term motivation comes from the word motive and is defined as a force within a person that causes the individual to act. santrock (2003) states motivation is why individuals behave, think, and feel, with an emphasis on the activation and direction of behavior. motivation is encouragement and desire so that a person does an activity by giving the best of himself, both time and energy (lin & chen, 2017; tokan & imakulata, 2019). dimyati and mudjiono (2013) states that motivation is seen as a mental impulse that moves and directs human behavior, including learning behavior. based on some of the motivation definitions, it can be concluded that motivation is an impulse that a person has to do something. nofriyandi (2016) explains that self-regulated is an individual's attitude in facing various situations so that they can think and act on their own to overcome various situations. schunk and zimmerman (1998) stated that self-regulated learning is the ability to monitor understanding, decide when he is ready to be tested, to choose a good information processing strategy. self-regulation describes students' ability to control themselves to learn actively (panadero et al., 2017; wong et al., 2019). self-regulation is a picture of student's ability to manage themselves to succeed in learning (carter jr et al., 2020; panadero, 2017). there are three main stages of the learning independence cycle: planning one's study, monitoring progress when implementing the plan, and evaluating the results of the completed plan. self-efficacy is a person's self-evaluation of his ability or competence to do something or solve an obstacle (byrne & baron, 1977; hatlevik et al., 2018). according to joët et al. (2011) and schunk and dibenedetto (2020), self-efficacy is basically the result of a cognitive process in the form of a decision, belief, or expectation about the extent to which the individual estimates his or her ability to complete a task. self-efficacy will affect several aspects of one's cognition and behavior. self-efficacy can affect student performance in college because students' confidence in their ability to learn will increase their efforts to succeed (joët et al., 2011). students who have high self-efficacy will try hard to learn because the goal of learning is learning achievement (bartimote-aufflick et al., 2016; van dinther et al., 2011). self-efficacy can strengthen students in achieving their academic achievements (yeşilyurt et al., 2016). therefore, increasing student confidence in learning needs to be done so that student learning achievement in higher education also increases. learning involvement is a student's effort with various strategies, facilities, and all sources that can support success in learning (staley et al., 2017; zuber‐skerritt, 2002). involvement in learning mathematics describes a student's efforts to achieve success in the learning process (takeuchi, 2018). learning engagement is a way for students to create tools that involve complex ways to succeed in learning (turner et al., 1998). parental involvement is part of the way students are involved in learning with parental motivation or support (panaoura, 2021). parental motivation can improve students' efforts in learning so that nofriyandi & andrian, factors that affect students’ mathematics performance at higher … 370 students use time effectively to engage in learning (gómez-garcía et al., 2020). involvement is a definition in which a student is actively involved in learning to get maximum learning outcomes. student performance describes student achievement in the teaching and learning process in the classroom (yeşilyurt et al., 2016). student performance or achievement is the impact of learning activities that have been carried out by teachers and students (chukwuyenum, 2013; maliki et al., 2009). performance describes the results of interactions between educators and students in the form of numbers measured through tests or learning outcomes instruments (gómez-garcía et al., 2020; imms & byers, 2017). student performance explains the extent to which students have worked optimally in the learning process so that students get maximum learning outcomes. student performance in learning is described by learning outcomes both cognitively, affectively, and psychomotor (awofala, 2017). student performance is the impact of learning that can be measured through test instruments or non-test instruments. this research objective was to find the variables that affected the student's performance during the covid-19 pandemic. the pandemic made education activities in the classroom unable to run maximally (darras et al., 2021). online learning becomes the best strategy and does not contribute to transferring knowledge because of less interaction between students and teachers (alzahrani, 2022). learning activities do not significantly contribute to education development because students are less interested and motivated to follow the class activities (singh et al., 2021). the students can not control and manage time to solve homework and are always late in collecting them (mesghina et al., 2021). students are not involved maximally in learning because the students have difficulties finding the learning source and only rely on the internet (diaz et al., 2021). from the above problem, the six variables were considered to solve the problem so that the affected variables can be given the best strategy to improve learning activities in higher education. 2. method the design of this research is quantitative with a correlational approach. the research population was students in higher education majoring in mathematics education which consisted of riau islamic university (uir), sultan syarif qasim state islamic university (uin), pasir pengarayan university (upp), tambusai university, and riau university (ur) with total students of 1238. the samples of this research were some students in higher education majoring in mathematics education which were taken randomly with a proportional random sampling approach. the calculation of the sample number uses the slovin formula with 303 samples. the instrument in this research was a questionnaire. the six variables in this research were the students' interest (feeling happy, student interest, attention students), motivation (persevere in the face of the task, tenacious in the face of adversity, interest in various issues, prefer to work independently, and can defend his opinion), student self-regulated (study initiative, diagnose learning needs, set learning goals, planning and controlling learning, happy challenge, the wise use of time and learning resources), student self-efficacy (magnitude, strength, generality), student involvement (agentic engagement, behavioral engagement, emotional engagement), and student mathematics performance in college (quantity, quality, effectiveness, independence). the instrument validation has been done in two-way content and construct. in content, the instrument is validated by three experts in measurement, evaluation, and education development. the content validation result showed that seven items are needed to improve. in construct, validation is done by spreading the instrument to 120 mathematics students and analyzed with cfa (confirmatory factor analysis). the construct validity result showed volume 11, no 2, september 2022, pp. 367-380 371 that the 25 indicators of six variables were valid with a loading factor greater than 0.3. data analysis in this study used path analysis. path analysis will show the effect of exogenous to endogenous variables and endogenous to other endogenous variables. path analysis shows a complete analysis of the results of each variable that had an influence. 3. result and discussion 3.1. result path analysis was conducted to see the effect of the independent variable on the moderating variable and the moderating variable on the dependent variable. before the path analysis is carried out, checking the normality of the data and multicollinearity is also necessary. the assumption of normality can be seen in table 1. table 1. data normality from skewness and kurtosis variables zskewness decision zkurtosis decision conclusion self-regulation 2.665094 moderate 0.739373 normal normal self-efficacy -0.13274 normal 1.303854 normal normal motivation 1.936273 normal 1.502399 normal normal interest 0.524394 normal 1.788018 normal normal involvement 1.760995 normal -1.32269 normal normal performance 2.414667 moderate 4.070203 normal normal table 1 shows that variables self-regulation, self-efficacy, motivation, interest, involvement, and performance were normally distributed. this assumption explained that six variables could be analyzed using regression, path analysis, or structural equation modeling. the next assumption that will be checked is multicollinearity (see table 2). table 2. multicolinieritas from product moment correlation variables self-reg self-eff mot interest involve performance self-reg 1 .183* .612** .659** .609** .547** self-eff .183* 1 .401** .355** .300** .324** mot .612** .401** 1 .696** .536** .580** interest .659** .355** .696** 1 .719** .578** involve .609** .300** .536** .719** 1 .520** performance .547** .324** .580** .578** .520** 1 nofriyandi & andrian, factors that affect students’ mathematics performance at higher … 372 table 2 shows the acquired correlation coefficient of self-regulation on self-efficacy, learning motivation, learning interest, learning involvement, and students' performance: 0.183, 0.612, 0.659, 0.609, and 0.547. the correlation coefficient of self-efficacy with learning motivation, learning interest, learning involvement, and students' performance, respectively, was 0.401, 0.355, 0.300, and 0.324. the coefficient correlation of learning motivation, learning interest, learning involvement, and students' performance, respectively, was 0.696, 0.536, and 0.580. the correlation coefficient of learning interest on learning involvement and students' performance was 0.719 and 0.578. the correlation coefficient of learning involvement on students' performance was 0.520. from correlation analysis, it can be concluded that the highest coefficient was 0.719 (correlation between learning interest on learning involvement) with a high category. the highest correlation from the analysis showed that all variables did not have the perfect coefficient correlation, so the multicollinearity assumption was met. path analysis showed the effect of exogenous variables on endogenous variables, the endogenous variables to other endogenous variables. the path analysis result can be seen in figures 1 and 2 and table 3. figure 1. path analysis with standardized volume 11, no 2, september 2022, pp. 367-380 373 figure 2. path analysis with t-value table 3 shows that some of the results of the path analysis were (1) there was an effect of learning motivation on learning interest, with a t-value of 11.49, (2) there was no effect of learning motivation on learning involvement with t values of -0.31, (3) there was an effect of learning motivation on students' self-regulated, with a t-value of 3.64, (4) there was an effect of learning motivation on students' self-efficacy, with t-value of 2.79, (5) there was an effect of learning motivation on mathematics performance, with t-value of 2.59, (6) there was an effect of learning interest on self-regulated students, with a t-value of 5.55, (7) there was no effect of learning interest on students' self-efficacy, with t-value of 1.37, (8) there was an effect of learning interest on learning involvement, with t-value of 6.26, (9) there was no effect of learning interest on mathematics performance, with tvalue of 1.28, (10) there was no effect of students’ self-efficacy on self-regulated, with t values of -1.75, (11) there was no effect of self-efficacy on learning involvement, with t values of 1.12, (12) there was no effect of self -efficacy on mathematics performance, with t values of 1.38, (13) there was an effect of self-regulated on learning envolvement, with t values of 3.19, (14) there was an effect of self-regulated on mathematics performance, with t values of 2.23, and (15) there was no effect of learning involvement on mathematics performance, with t value of 1.39. table 3. summary of path analysis variables t-values r conclusion motivation*interest 11.49 0.70 significant motivation*involvement -0.31 -0.03 no significant motivation*self-regulated 3.64 0.33 significant motivation*self efficacy 2.79 0.30 significant motivation* performance 2.59 0.25 significant interest*self-regulated 5.55 0.47 significant interest*self efficacy 1.37 0.15 no significant interest*involvement 6.26 0.55 significant nofriyandi & andrian, factors that affect students’ mathematics performance at higher … 374 variables t-values r conclusion interest*performance 1.28 0.14 no significant self-efficacy*self-regulated -1.75 -0.12 no significant self-efficacy*involvement 1.12 0.07 no significant self-efficacy*performance 1.38 0.10 no significant self-regulated*involvement 3.19 0.25 significant self-regulated*performance 2.23 0.20 significant involvement*performance 1.39 0.13 no significant 3.2. discussion the analysis results show that motivation and self-regulation directly affected students' performance in the covid-19 pandemic. this result explains that motivation and self-regulated the best variables need to improve students' performance in the covid-19 pandemic. motivation always is needed to improve and generate the best outcome for the education process in the classroom (o’shea et al., 2017; orehek et al., 2017). self-regulation and motivation are highly correlated in improving students' performance because these variables will solve every problem in the classroom learning activities (wilby, 2022). high motivation affects self-regulation, and self-regulation maximally will increase the students’ performance (köpetz et al., 2013). motivation and self-regulation become the best concept to improve the learning quality, so the interaction between teachers and students become meaningful (faílde-garrido et al., 2022). motivation and self-regulation will be the best variables to support the learning process in the classroom so the learning process will run maximally and contribute to the education progress. the results of the analysis show a direct influence between the motivation on the learning interests and mathematics performance. the motivation of students gives a good contribution to the learning interest and mathematics performance of students. bye et al. (2007) said that motivation and interest had a positive effect on university students. lavasani et al. (2011) explained that academic motivation is a motivation that initiates and guides students to have an interest in learning. in addition to student learning interest, motivation also directly affected student self-regulation. academic motivation had a significant positive relationship with learning independence (self-regulated) (akbay & akbay, 2016; mahmoodi et al., 2014). the analysis results also show that there was a significant effect between student motivation and student self-efficacy. köseoglu (2015) explained that selfefficacy believes in the abilities and strengths possessed in working, studying, and seeking success in obtaining academic achievement. good motivation possessed by students will be able to obtain good academic results, too, because students who have motivation can act according to their abilities or competencies to carry out tasks and achieve and resolve obstacles to achieve goals. student motivation didn't affect learning involvement in learning. this happens because students feel more directly involved when learning in class than just being charged with individual tasks. students can discuss with friends or with lecturers during class learning compared to just doing assignments. interest in learning has a significant effect on self-regulated and learning involved in learning. students with a good interest in learning can influence their self-regulation and involvement in learning because the interest that grows in students is a source of encouragement to do something they want (bernacki & walkington, 2018; tsai et al., 2018). the learning that students wanted was direct learning in class, not just giving individual assignments. learning that is carried out directly makes volume 11, no 2, september 2022, pp. 367-380 375 students feel involved in learning, producing positive results and independence for students. velayutham et al. (2011) states that students' confidence in their inner self and self-regulated can be used as an instrument to influence student involvement in the learning process. student interest did not affect self-efficacy and student mathematics performance. this happens because many students have a sense of love or interest in learning mathematics but are not sure about their abilities, so low student confidence has an impact on students' mathematics performance. based on the analysis results, it can be seen that student selfefficacy did not affect self-regulation, learning engagement, or student mathematics performance. this result occurs because students did not consistently maintain motivation and enthusiasm for learning. students who have the motivation to learn will maintain selfefficacy and improve self-regulated strategies in their lives. cobb (2003) said that selfregulated learning is influenced by many factors, including self-efficacy, motivation, and goals. according to alafgani and purwandari (2019), self-efficacy, motivation, and learning goals possessed by students are positively related to self-regulated learning. self-efficacy that consistently exists in students can have a positive effect on self-regulation (bradley et al., 2017). schunk and zimmerman (1998) states that self-regulated was the concept of how a student becomes a regulator for his learning. there was a significant effect of self-regulated on student involvement and mathematics performance. this shows that mathematics education students in riau province can manage themselves in learning, be active during the learning process, and can regulate their learning performance. self-regulation can significantly affect student involvement in learning so that they achieve achievement in learning (uzun & kilis, 2019). students' self-regulated causes students to be persistent in achieving learning goals so that the effectiveness of the learning process can run optimally (yerdelen & sungur, 2019). student involvement in learning did not affect students' mathematics performance. there was no effect of student involvement on student performance due to the lack of parental involvement in motivating students to learn (shute et al., 2011; yawman et al., 2019). another factor that caused students' involvement did not affect mathematics performance was the low involvement of students in learning. the low involvement of students because learning during the pandemic focuses on online learning, potentially increasing or decreasing students' mathematics performance. the low involvement of students in learning is due to the lack of student participation in learning activities which indirectly affects student performance (maolida & savitri, 2016). 4. conclusion variables that affected students' mathematical performance were motivation and self-regulated. students' self-regulated affected learning involvement and mathematics performance. the motivation variable did not affect the learning engagement variable, but it affected learning interest, self-regulated self-efficacy, and mathematics performance of mathematics education students of the riau province university. the learning interest variable affected students' mathematical performance and self-efficacy but did not affect self-regulation and learning engagement. the self-efficacy variable did not affect the selfregulated variable, learning engagement, and student mathematics performance. motivation and self-regulated were significant variables that affected students' mathematics performance in higher education in riau province during the covid-19 pandemic. nofriyandi & andrian, factors that affect students’ mathematics performance at higher … 376 acknowledgements the authors would like to thank the universitas islam riau, who provided financial assistance to complete this research. hopefully, this research can provide great benefits for universitas islam riau and the development of education in indonesia. references acharya, b. r. 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(2002). the concept of action learning. the learning organization, 9(3), 114-124. https://doi.org/10.1108/09696470210428831 https://doi.org/10.1037/0022-0663.90.4.730 https://doi.org/10.1016/j.chb.2018.08.045 https://doi.org/10.1016/j.edurev.2010.10.003 https://doi.org/10.1080/09500693.2010.541529 https://doi.org/10.1016/j.sbspro.2010.12.067 https://doi.org/10.1177/1362168820917323 https://doi.org/10.1080/10447318.2018.1543084 https://doi.org/10.18535/ijsrm/v7i3.el04 https://doi.org/10.1007/s10763-018-9921-z https://doi.org/10.1016/j.chb.2016.07.038 https://doi.org/10.1108/09696470210428831 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p35-44 35 the correlation between cognitive style and students’ learning achievement on geometry subject udiyono 1 , muhammad ridlo yuwono 2 1,2 universitas widya dharma, jl. ki hajar dewantara, klaten 57438, central java, indonesia 1 udiyono@unwidha.ac.id, 2 ridloyuwono90@gmail.com received: september 25, 2017 ; accepted: january 27, 2018 abstract the objective of the research is to identify whether there is a positive correlation between cognitive style and students’ learning achievement on geometry subject. the research is classified as correlational quantitative research. the population of the research is all students of mathematics education program at widya dharma university in the academic year of 2015/2016. the sample is the students in semester iv b. it was taken by cluster random sampling. the instrument is a psychiatric test, geft and learning achievement test. the technique of data analysis is simple linear regression analysis. the result of the research is there is a positive correlation between cognitive style and students’ learning achievement on geometry subject. the coefficient determination is r2 = 0.6209. it means the increase and decrease of students’ learning result on geometry subject 62.09% can be explained by cognitive style with linear correlation equation ŷ= -2.9650 + 4.6513x. meanwhile, 37.91% is influenced by another factor. 13 out of 17 samples are categorized as students fd and 4 students are classified as fi. the mean score of students fd is 16 while students fi is 59.5385. it means students fi has better learning achievement than students fd on geometry subject. keywords: cognitive style, geometry, learning achievement. abstrak penelitian ini bertujuan untuk mengetahui apakah terdapat pengaruh yang positif gaya kognitif terhadap hasil belajar mahasiswa pada mata kuliah geometri. metode yang digunakan adalah metode kuantitatif dengan pendekatan korelasional. populasi pada penelitian ini adalah seluruh mahasiswa program studi pendidikan matematika universitas widya dharma tahun akademik 2015/2016. sampel penelitian ini adalah mahasiswa program studi pendidikan matematika universitas widya dharma semester iv b dengan menggunakan teknik sampling cluster random sampling. instrumen yang digunakan berupa tes psikiatrik geft dan tes hasil belajar. analisis data yang digunakan pada penelitian ini adalah analisis regresi linear sederhana. hasil penelitian ini adalah terdapat pengaruh positif gaya kognitif terhadap hasil belajar mata kuliah geometri mahasiswa program studi pendidikan matematika universitas widya dharma klaten dengan koefisien determinasi diperoleh . ini berarti bahwa, meningkat atau menurunnya hasil belajar 62,09% dapat dijelaskan oleh gaya kognitif mahasiswa melalui hubungan linear dengan persamaan ̂ . sedangkan 37,91% dipengaruhi oleh faktor lain. dari 17 sampel yang diambil, mahasiswa yang termasuk kategori fd sebanyak 4 orang dan termasuk kategori fi sebanyak 13 orang. nilai rata-rata thb kategori fd 16 dan kategori fi 59,5385; yang berarti bahwa mahasiswa fi mempunyai hasil belajar yang lebih baik daripada mahasiswa fd. kata kunci: gaya kognitif, geometri, hasil belajar. how to cite: udiyono & yuwono, m. r. (2018). the correlation between cognitive style and students’ learning achievement on geometry subject. infinity, 7 (1), 35-44. doi:10.22460/infinity.v7i1.p35-44 udiyono & yuwono, the correlation between cognitive style … 36 introduction mathematics is one of basic knowledge that should be mastered. this basic knowledge is used to master other knowledge. since it is categorized as abstract, learning this knowledge needs deep thinking and skill. a teacher should consider students’ ability when he/she wants to teach mathematics. the teacher should know students’ cognitive development and how to teach learning material based on their cognitive level. according to piaget, a human at more than 11 years of age is in formal operational level with some main cognitive development characteristics: hypothesis, abstract, deductive and inductive, logic and probability as well (suprijono, 2009). students are still having trouble doing the analytic geometry. the results of noto, hartono, & sundawan show that the ability of representation and mathematical connections of students in the course of analytic geometry (noto, hartono, & sundawan, 2016). although an individual experiences the same process of cognitive development, he/she will not have the same cognitive style. according to borich and tombari, cognitive style is the way how an individual process and think about what they are learning. there are two kinds of cognitive style: field dependent (fd) and field independent (fi) (razali, jantan, & hashim, 2003). generally, students with cognitive style fd are different from those with fi. students fd tend to depend on their environment perceptions; have difficulty to focus on, find the main idea, and use prominent instruction; hard to give ambiguous information structure; have difficulty in arranging new information and relating it with the previous one; and have difficulty in retrieving information from long-term memory (altun & cakan, 2006). meanwhile, students fi are able looking at the part of shadow separated from its form; separating relevant thing from irrelevant form; providing separated information structure from the provided one; rearranging information from previous knowledge context; and tending to be more precise in taking part of the memory. based on the characteristics, students fd has different way from students fi in learning mathematics. a teacher should consider students’ cognitive style while he/she is teaching geometry because they have different characteristics that ask different treatment too during the learning process. the characteristics can be the guidance in arranging lesson plan for appropriate teaching geometry. this should be done because it is indicated that there is a correlation between cognitive style and students’ learning achievement on geometry. appropriate treatment will result maximum progress. this research is in line with apriliantis’ study about cognitive style. the conclusion of her research is there is a significant positive correlation between cognitive style with students’ learning achievement on physics subject in students class x sma kotabaru jambi (aprilianti, 2014). moreover, suryanti’s study releases that there is a correlation between cognitive style toward students’ learning achievement on accounting subject (suryanti, 2014). in addition, ulya in her research, states that there is high level positive correlation between cognitive style and students’ ability in solving a problem (ulya, 2015). based on the background of the study, the problem statement of the research is to find out whether there is a correlation between cognitive style and students’ learning achievement on geometry subject. the objective of the research is to identify whether there is a correlation between cognitive style toward students’ learning achievement on geometry subject. while, volume 7, no. 1, february 2018 pp 35-44. 37 the benefit of the research are (1) through the research, it would be identified a description about the correlation between cognitive style and students’ learning achievement on geometry subject; (2) the teacher would be helped to arrange a lesson plan on geometry subject that take cognitive style into account. method the method that is used in this research is quantitative method with correlation approach. independent variable (x) of this research is cognitive style while the dependent variable (y) is students’ learning achievement on geometry. the population of the research is all students of mathematics program at widya dharma university in the academic year of 2015/2016. the sample of the research is the students of mathematics program in semester iv b widya dharma university. it was taken by using cluster random sampling. the instrument that is used to determine students’ cognitive style is psychiatry. the test was developed by witkin, moore, goodenough, & cox (1977). the name of the test is geft. the test had been measured its reliability by the previous researcher. its reliability score based on alpha cornbach is 0.84. it means the reliability of geft is very high (khodadady & tafaghodi, 2013). geft is valid because it has been used in the previous researches. the example of geft test is shown in figure 1. figure 1. the example of geft test instrument that is used as students’ achievement data is taken from students’ test. the test instrument had been tried out first before it was used in the research. tryout was done to know the validity, reliability, discrimination power, and level of difficulty of the question. the contents of the learning achievement test are shown in figure 2. udiyono & yuwono, the correlation between cognitive style … 38 figure 2. the contents of the learning achievement test data analysis that was used in the research is simple linear regression. it includes (1) normality test, (2) regression equation, (3) linearity test, (4) significant regression test, (5) significant coefficient regression test, (6) significant coefficient correlation test, (7) coefficient determination. results and discussion results the result of normality test that used lilliefors method for the data of cognitive style and students’ achievement test on geometrycan be seen in table 1 as follows. table 1. the result of normality test no variable lobs l0.05;17 decision 1 cognitive style 0.1251 0.2060 h0is accepted 2 learning achievement 0.1721 0.2060 h0is accepted volume 7, no. 1, february 2018 pp 35-44. 39 based on the data in table 1, the conclusion that can be drawn is cognitive style and students’ achievement data come from normal distribution population. meanwhile, simple linear regression counting generates regression equation ŷ = -2.9650 + 4.6513x. the regression shows that there is an increase for students’ learning achievement 4.6513 for every one unit cognitive style increase. the result of linearity test is shown in table 2. table 2. linearity test result sources of variance sum of squares degree of freedom mean square fobs fɑ decision regression 4869. 1248 1 lack of fit 2117.9046 8 264.7381 2.1687 3.7257 h0 is accepted pure error 854.5000 7 122.0714 total 7841.5294 16 based on the data in table, the conclusion is the correlation between cognitive style and students’ learning achievement on geometry is linear. the result of regression significance test is presented in table 3. table 3.the result of regression significance test sources of variance sum of squares degree of freedom mean square fobs fɑ decision regression 4869.1248 1 4869.1248 24.5716 4.5431 h0 is rejected error 2972.4046 15 198.1603 total 7841.5294 16 based on the data in table 3, the conclusion is the correlation between cognitive style and students’ learning achievement on geometry is significant. the result of regression coefficient significance test is tobs = 4. 9570. critical area t0. 025; 15 = 2.1314 is dk = {t | t> 1.7531 or t > 2.1314}. becausetobsϵ dk, so h0 is rejected. it can be concluded that the regression coefficient is significant. based on the correlation coefficient calculation, the value of rxy = 0.7880. meanwhile, correlation coefficient significant test is t obs = 4.9570. critical area t0.05;15 = 1.7531 is dk ={t| t>1.7531}. becausetobsϵ dk, h0 is rejected. therefore, it can be summarized that there is a positive correlation between cognitive style and students’ learning achievement. based on the determination coefficient calculation, the value r 2 = 0.6209. it means the increase and decrease of students’ learning achievement 62.09% can be explained by cognitive style through linear correlation equation ŷ = -2.9650 + 4.6513x. while 37.91% is influenced by other factor. four students out of 17 samples are categorized as students fd and 13 students are classified as students fi. the mean score of students fd is 16.0000 while students fi is 59.5385. it means students fi have better learning achievement than students fd. udiyono & yuwono, the correlation between cognitive style … 40 discussion based on the research data analysis, it can be summarized that there is a positive correlation cognitive style toward students’ learning achievement on geometry subject. based on altun & calkan (2006), the following is the tendency of students fd and fi in general. an individual with fd type tendsto depend on their environment perceptions; have difficulty to focus on, find the main idea, and use prominent instruction; hard to give ambiguous information structure; have difficulty in arranging new information and relating it with the previous one; and have difficulty in retrieving information from long-term memory. individual with fi type tends to be able looking at the part of shadow separated from its form; separating relevant thing from irrelevant form; providing separated information structure from the provided one; rearranging information from previous knowledge context; and tending to be more precise in taking part of the memory. the correctness of hypothesis is supported by some related research. aprilianti’s research shows that there is significant positive correlation between cognitive style with students’ learning achievement on physics subject (aprilianti, 2014). moreover, the result of suryanti’s research is there is correlation between cognitive style toward students’ learning achievement on accounting i subject (suryanti, 2014). in addition, ulya in her research concluded that there is high level correlation between cognitive style and students’ ability to solve problem (ulya, 2015). moreover, this research is accordance with the sulaiman’s research (2013). the research uses naturalistic qualitative approach and takes one student fi and one student fd as the research subject. the method uses worksheet with six activities to gain information about the geometry thought profile of subject research. the activities are (1) drawing rectangles; (2) showing and defining rectangles; (3) selecting rectangles; (4) guessing mysterious figure; (5) identifying the equvalency of two parallelogram definitions; (6) and applying rectangles. the result of the sulaiman’s research (2013) was drawn by analyzing the geometry thought of students that can be identified through six activities. the geometry thought of students fi was identified as follows: they were able to draw 6 different rectangular figures and gave appropriate argument in differenciating them; in showing and defining rectangular figure, they were able to appoint some kinds of rectangular figures like square, rectangle, rhombus, parallelogram, trapezoidal, and kite; they were able to mention the characteristics of the figures and define them; in selecting rectangular figure, the students could guess mysterious figures although they needed some repetitions; in equvalencing two parallelogramdefinitions, the students had not understood about the definition of two equivalent parallelogram so the figures that they appointed were rectangle and parallelogram; in applying rectangular figures, the students were able to find two rectangular figures only and they had not understood the association between them. meanwhile, the geometry thought of students fd through 6 activities were identified as follows: they were only able to draw 5 different rectangular figures; their argument in differenciating the figures was not appropriate; in showing and defining rectangular figures, they were not able to mention the characteristics and the definition of rectangular figure correctly; in selecting rectangular figures, students did inappropriate selection; students were able to guess mysterious figure but they needed many repetitions; in defining parallelogram, the students did not understand about the definition of parallelogramand they appointed incorrect figure; in applying the rectangular figure, the students just found one rectangular thing and they hadn’t found the association between them. volume 7, no. 1, february 2018 pp 35-44. 41 analysis of geometry answers from fi and fd students is shown below. figure 3. the answer of number 1 of student fi based on figure 3, students fi can name two pairs of base and height. the answer is not perfect, because the parallelogram has eight pairs of base and high. figure 4. the answer of number 1 of student fd based on figure 4, the students fd actually make the parallelogram to be like a rectangle with its four right angles. student fd makes two isosceles to determine the base and height of the parallelogram. what this fd student has done is not quite right, because the one asked for about the angle of parallelogram is not right-angled. udiyono & yuwono, the correlation between cognitive style … 42 figure 5. the answer of number 2 of student fi based on figure 5, students fi have drawn the model appropriately. however, the student has not yet made the process of determining the size of the rest of the carton by using the concept of two similar constructions. figure 6. the answer of number 2 of student fd based on figure 6, students fd are wrong in drawing the model. this causes the result of the calculation to be wrong. figure 7. the answer of number 3 of student fi volume 7, no. 1, february 2018 pp 35-44. 43 based on figure 7, students fi prove the formula of the area of the triangle by using a parallelogram image that forms two conjugent triangles. the student immediately stated that the area of the triangle is equal to half of the parallelogram area. however, the student's answers were less systematic. the student does not show the base and height of the image. figure 8. the answer of number 3 of student fd based on figure 8, students fd prove the formula of the area of triangle area by using the resulted image of jejang gejang with rectangle. the results of these intersions form a right triangle. in fact, the triangle that the question maker asks is an arbitrary triangle, not just elbow. thus, the proof is less precise, because it only applies to right-angled triangles only. the result of the research supports dimention theory of field dependent and independent by witkin, moore, goodenough, & cox (1977) that stated cognitive style approach dimention fi and fd is useful to be implemented in education such as giving description about how students learn, how teachers teach, how students and teachers interact each other, and how students make a decision in choosing job. students field independent (fi) and students field dependent (fd) is very different. students fd focus on social environment and depend on external social standard while students fi less focus on social environment and tend to use internal standard use. in the other words, students fd have more interpersonal skill than students fi that tend to be comfortable when they work alone without any interaction with other people. the characteristics of both cognitive syle dimension can be taken into account when teachers determine learning strategy. when the students are taught using appropriate learning strategy they learn better and the result of learning will increase including in learning geometry subject. udiyono & yuwono, the correlation between cognitive style … 44 conclusion the conclusion that can be drawn is there is positive correlation between cognitive style toward students’ learning achievement on geometry subject for mathematics students at mathematics education program widya dharma university. the determination coefficient is r 2 = 0.6209. it means the increase and decrease of students’ learning achievement on geometry 62.09% can be explained by cognitive style through linear correlation formula equation ŷ = -2. 9650 + 4.6513x. the regression obtained showed that the average change in learning outcomes increased by 4.6513 for each increase in cognitive style of students by one unit. while 37.91% is influenced by other factor. teacher especially geometry teacher is suggested to notice students’ cognitive style and then take it into accountin arranging effective learning. by considering students’ cognitive style, learning achievement can be improved because students experience appropriate learning process based on their characteristics. references altun, a., & cakan, m. (2006). undergraduate students' academic achievement, field dependent/independent cognitive styles and attitude toward computers. journal of educational technologyand society, 9(1), 289. aprilianti, e. (2014). hubungan gaya kognitif dengan hasil belajar fisika siswa kelas x sma negeri se-kecamatan kota baru jambi. artikel ilmiah, 1-10. khodadady, e., & tafaghodi, a. (2013). cognitive styles and fluid intelligence: are they related?. journal of studies in social sciences, 3(2), 138-150. noto, m. s., hartono, w., & sundawan, d. (2016). analysis of students mathematical representation and connection on analytical geometry subject. infinity journal, 5(2), 99-108. razali, m., jantan, r., & hashim, s. (2003). psikologi pendidikan. pts professional. sulaiman, s. (2013). profil berpikir geometri siswa smp ditinjau dari perbedaan gaya kognitif. jurnal ilmiah mitsu, 1(2). suprijono, a. (2009). cooperative learning: teori & aplikasi paikem. pustaka pelajar. suryanti, n. (2014). pengaruh gaya kognitif terhadap hasil belajar akuntansi keuangan menengah 1. jinah (jurnal ilmiah akuntansi dan humanika), 4(1), 1393-1406. ulya, h. (2015). hubungan gaya kognitif dengan kemampuan pemecahan masalah matematika siswa. jurnal konseling gusjigang, 1(2). witkin, h. a., moore, c. a., goodenough, d. r., & cox, p. w. (1977). field-dependent and field-independent cognitive styles and their educational implications. review of educational research, 47(1), 1-64. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p193-206 193 utilizing lesson study in teaching synthetic division for procedural fluency in a postpandemic classroom jessa christine dedal agsalon, joy meribeles anore, hanna arnedo salinas, princess pera dipasupil, minie rose caramoan lapinid* de la salle university manila, philippines article info abstract article history: received mar 9, 2023 revised apr 10, 2023 accepted may 2, 2023 published online jun 14, 2023 rigid planning, implementation, and evaluation of the learning activities has proved crucial in reflective teaching practice, especially in collaboration through a lesson study. this study was conducted in the philippine postpandemic context with the aim of using lesson study to improve lesson delivery in a hyflex classroom setup in teaching synthetic division for procedural fluency. participants included four collaborating full-time teachers and an intact class of twenty-two online and seventeen in-person learners. research instruments were a self-assessment tool, classroom observation, and a focus group discussion. most students could perform synthetic division but some failed to achieve procedural fluency due to poor prior knowledge in performing operations on real numbers and arranging terms in descending order of degree, and inadequate understanding of the concepts behind the algorithm. these findings underscore the importance of striking a balance between procedural fluency and conceptual understanding in a lesson. the challenges in conducting lesson study were difficulty in scheduling and conducting online meetings. the challenges in implementing the research lesson were intermittent and weak internet connection, hyflex learning classroom management, and getting students to express their mathematical ideas. on the basis of these findings, the research lesson is then revised and improved for future implementation. keywords: hyflex learning, lesson study, post-pandemic classroom, procedural fluency, synthetic division this is an open access article under the cc by-sa license. corresponding author: minie rose caramoan lapinid, faculty of the science education department, de la salle university manila 2401 taft ave, malate, manila, 1004 metro manila, philippines. email: minie.lapinid@dlsu.edu.ph how to cite: agsalon, j. c. d., anore, j. m., salinas, h. a., dipasupil, p. p., lapinid, m. r. c. (2023). utilizing lesson study in teaching synthetic division for procedural fluency in a post-pandemic classroom. infinity, 12(2), 193206. 1. introduction the covid-19 pandemic has drastically impacted the education sector (keshavarz, 2020) and the sudden transition to online learning has paved the way for online synchronous learning to become mainstream (chellathurai, 2020). it has forced schools to a new method https://doi.org/10.22460/infinity.v12i2.p193-206 https://creativecommons.org/licenses/by-sa/4.0/ agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 194 of teaching and learning, which is foreign to teachers and learners (miller et al., 2021) as there has been a shift from the pre-pandemic face-to-face to online or modular distance learning during the pandemic and subsequently to post-pandemic hybrid learning (unesco, 2021). schools in the philippines post-pandemic education have been considering different modalities of instruction deliveries for the gradual transition to faceto-face modality (estrellado, 2021). a post-pandemic classroom (chellathurai, 2020) would combine online synchronous learning through zoom, google meet, and other online meeting platforms and in-person campus learning. the participation of the philippines in large-scale international assessments prepandemic allows us to understand how our students perform with global benchmarks. for example, the philippines scored 249 in timss (trends in international mathematics and science study, 2020) for grade eight mathematics, while pisa (organization for economic co-operation and development, 2018) results show that average 15-year old students in the philippines scored 353, the second lowest score in mathematics compared to an average of 489 points in organization for economic co-operation and development (oecd) countries. the covid-19 pandemic has worsened the learning crisis in the philippines (cho et al., 2021). in effect, learning losses caused by the closure of schools may lead to significant adverse long-term impacts on learning-adjusted years of schooling, learning outcomes, income in the long run, and the basic proficiency level. it was suggested that post-covid19 schools should adapt to the learning needs of every child and should continue to blur the walls to allow children to still learn both in school and at home. these changes have forced the education system to adopt a modality that is new to the stakeholders – the hybrid learning modality (nelson et al., 2022). hyflex (hybrid-flexible) learning is a modality that allows learners both in school and those who cannot come to school to learn skills and concepts equally with equivalent tasks, the same learning objects, and similar, if the same authentic assessments (beatty, 2019). some of the advantages of hyflex learning include flexibility for both learners and teachers (beatty, 2019), while some of the disadvantages include lack of student focus, the teacher’s tendency to overwhelm the students with tasks and not provide similar activities that would be engaging for both the online learners and face-to-face learners (college of dupage, 2016), and the lack of gadgets and poor internet connectivity (unesco, 2021). the difficulty of student feedback is also one of the challenges in a hyflex learning environment, which is rooted in communication (kohnke & moorhouse, 2021). lesson study (ls) is a professional development approach from japan that follows a four-step process: (1) investigation; (2) planning; (3) research lesson; and (4) reflection (lewis, 2015). it aims to improve the lesson delivery (nelson et al., 2022), which could also help students understand the lesson and enhance their procedural fluency. ls is a platform for teachers to collaborate and improve their teaching practice such as to become more technologically literate (mullis et al., 2020) through hybrid learning classes. moreover, ls helps teachers to be well-rounded mathematically (lewis, 2015), be able to utilize school and classroom contexts as venues of inquiry (elipane, 2012), create challenging and appropriate tasks and activities that promote student engagement (ferrer & lapinid, 2017), and make effective changes to the lesson that show noticeable improvements in student learning (appova, 2018). one of the strands of mathematical proficiency is procedural fluency. professional fluency (pf) is the skill of an individual to carry out procedures flexibly, accurately, efficiently, appropriately, and correctly (kilpatrick et al., 2001). high pf tends to provide learners to gain confidence in task performance (williams et al., 2020). moreover, procedural fluency is one of the five strands of mathematical proficiency. it does not only refer to knowing how to apply a particular formula but knowing how to apply the procedure volume 12, no 2, september 2023, pp. 193-206 195 in problem-solving (kilpatrick et al., 2001). knowing how to use a procedure efficiently is also part of a learner’s procedural fluency (laswadi et al., 2016). student’s procedural fluency may refer to students’ knowledge about procedures, knowledge of when to use and how to use each procedure correctly, and the ability to use these procedures effectively and accurately (laswadi et al., 2016). this study was conducted in the philippine post-pandemic context the overarching aim of which is to use lesson study to improve lesson delivery in a hyflex classroom setup by looking at students’ procedural fluency in synthetic division. this aim prompted the researchers to answer the following questions: (1) what discussion episodes target procedural fluency in synthetic division? (2) what is the students' procedural fluency in synthetic division? (3) how do students assess their procedural fluency in synthetic division? (4) what challenges are encountered in conducting the lesson study? (5) what challenges are encountered in implementing the research lesson in the hyflex modality? 2. method the lesson study process followed a three-stage procedure: the pre-lesson implementation stage, the lesson implementation stage, and the post-lesson implementation stage. the pre-lesson implementation stage constitutes planning the research lesson by the ls group, securing the school’s permission to conduct the research lesson implementation in the class, and securing students’ and guardians’ informed consent. the post-lesson implementation stage consists of the focus group discussion (fgd) to discuss areas the research lesson needs to improve based on class observation field notes. the ls process underwent only one cycle due to the limitated availability of the classes. the lesson study consisted of four professional teachers from different schools in the philippine national capital region who were likewise graduate students of the teaching of mathematics course. their professor randomly chose the four professional teachers in their graduate class to form the ls group. they met four times, two to four hours each time, to decide the lesson topic, the schedule of implementation, the teacher implementor, and what to focus on in the lesson study, to plan and develop the research lesson, and to prepare the materials for lesson implementation. one of the ls group members was selected to teach the lesson while the other ls group members observed the class. all ls group members participated in the post-lesson fgd for one hour. the research lesson was implemented to an intact class of twenty-two (22) students who participated virtually and seventeen (17) students who participated physically, following beatty’s model of hyflex learning (beatty, 2019) that allows a mix of both faceto-face and online students be in the class at the same time (detyna et al., 2023). the class belongs to a private school in a city in the national capital region of the philippines. the school is following an odd/even scheme of assigning students’ mode of attendance wherein students with odd and even id numbers attend the classes alternately in person or online. during this research study, students with odd id numbers attended the class online, while the rest attended the class in person. the choice for the class was based on the availability of the ls group members and the schedule assigned to the class by the school to avoid disruption. all these student participants were allowed to use their gadgets in classroom activities per school regulations. after the hyflex class instruction, the teacher administered a self-assessment tool to students and assigned them an error analysis asynchronous task as homework. learners’ procedural fluency self-assessment responses were analyzed using descriptive statistics, while learners’ responses to the open-ended questions were analyzed using thematic analysis (braun & clarke, 2012). the study adopted the use of the self agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 196 assessment form by the school with a 4-point likert scale where responses 1 and 4 correspond to ‘strongly disagree’ and ‘strongly agree’, respectively. cronbach’s alpha was determined (α = 0.995), which indicates the instrument is deemed reliable. the open-ended questions are: (1) how did you find/understand your lessons? (2) which part of the topic/lesson was easy? explain why. (3) which part of the topic/lesson was difficult? explain why. the research lesson class was video recorded. other pertinent data constitute students’ written works and observation field notes of the ls implementers. 3. result and discussion 3.1. discussion episodes targeting procedural fluency in synthetic division three notable episodes took place during the implementation of the lesson. in an attempt to connect the current with the previous meeting’s lesson on long division of polynomials, the first episode was the comparison and contrast of synthetic and long divisions. episode 1: comparison and contrast of synthetic and long division teacher : what are the similarities and differences between synthetic and long division? teacher : let's talk about similarities first. student a : both are methods of division. the lesson was the first time the students encountered synthetic division as a topic to study in mathematics 10. the teacher needed to repeat the question several times and wait for the responses until student a raised his hand and answered the question. episode 2: definition of numerical coefficients teacher : so, when we say coefficient, what does it mean? for example, 3x2. what is the coefficient of that term? student b : 3 teacher : so, what is the definition of a coefficient? in your own words. yes, student c? student c : same [the class laughed] teacher : what do you mean the same? anyone who can help with the class? student d : value of the variable. teacher : it's the numerical part of the expression. thus, the coefficient in 3x2 is 3. so, don't be confused. when i say “coefficient”, i refer to the numerical coefficient. the teacher asked the students to define coefficients based on their understanding by giving the mathematical expression 3x2. the teacher posed another question to verify if they understood numerical coefficients. the students could hardly express in words the concept of numerical coefficients. however, they can quickly identify it when given a mathematical expression. the teacher gave an idea of numerical coefficients. however, a better terminology or description could have been used, such as the numerical factor of a term in an algebraic expression. nonetheless, the teacher underscored the numerical coefficient concept’s importance as a prerequisite to synthetic division. episode 3: the implication of a nonzero remainder student e : miss, does the remainder always have to be 0? volume 12, no 2, september 2023, pp. 193-206 197 teacher : [it] doesn't have to be 0. when you divide numbers, is the remainder always 0? is it always like that? class : no teacher : no, right? it is similar to polynomial expressions. student e : what if the remainder isn't 0? how will we interpret kung hindi siya [if it is not] zero? teacher : the teacher introduced an example. in episode 3, the student was very inquisitive and used critical thinking, as all the given examples had zero as the remainder. although the primary goal of the examples was to check how students understood the procedures of synthetic division, without this student raising this question, students might misconstrue synthetic division always yields a zero remainder. this incident happened just about the time the teacher was to give an example with a nonzero remainder. the teacher should have directly answered the question. instead, she gave the next example. the situation allowed students to use critical thinking, collaborate, and interpret observations based on the examples. 3.2. students' procedural fluency in synthetic division the instruction given in the homework was for the students to identify the error in a given solution and correct it. please see figure 1. students who identified the mistake correctly got one point, provided the correct solution, received two points, and answered correctly, received 1 point, for a total of four points. figure 1. error analysis homework table 1 shows the mean and standard deviation of students’ scores in the asynchronous task. the average score of thirty (30) students who submitted their works on the 4-item asynchronous task is 3.47 (sd = 1.14) out of 4 points signifying a good turnout of results. this result means students generally could discern which aspects of the solution went wrong, and provided correct solutions and answers. consequently, implying most students have acquired procedural fluency. table 1. descriptive statistics of students’ scores in homework n mean standard deviation 30 3.467 1.137 agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 198 table 2. distribution of students’ scores in homework scores counts % of total cumulative % 0 2 6.7 % 6.7 % 1 0 0% 6.7% 2 3 10.0 % 16.7 % 3 2 6.7 % 23.3 % 4 23 76.7 % 100.0 % table 2 shows the distribution of students’ scores. specifically, 23 (76.7%) students received a perfect score, while 5 (16.7%) students received a score of 2 or below. only two students scored 0 points. these results show that most learners know how to find the error in the given solution. afterward, they provided the correct solution by uploading a photo or screenshot of their work. the following were examples of students’ responses to the given questions. item #1. what error did joshua make in his solution? student c : he did not arrange the dividend in descending order. student d : the error joshua made in the solution was the arrangement of the polynomial, which was not in descending order. student e : joshua made an error in the solution where he should first rearrange the polynomial's terms in descending form. the proper arrangement should be 2, 1, -7, and 14 for the coefficients. student f : the coefficients 14 and – 26 were not combined correctly. it should be 14-(-26) = 40. both students c and d identified the error, albeit they did not specify the terms in the dividend needed to be arranged in descending order of their degree. student e identified and determined that the terms must be arranged and, at the same time, listed down the correct arrangement of numerical coefficients of the dividend, albeit the degree of the term as the basis of the descending order arrangement was left out. student f should have recognized the error and misconstrued the correct operation (addition) as erroneous since the additive inverse has been taken care of in the divisor. item #2. divide using the synthetic division: −7𝑥 − 𝑥2 + 14 + 2𝑥3 𝑥+2 . show your complete solution and final answer. volume 12, no 2, september 2023, pp. 193-206 199 figure 2. solution of student h figure 2 and figure 3 show the correct solution by students h and i for the given item. on the other hand, in figure 4, student j failed to recognize the error in the solution and performed the operations incorrectly. student j did not arrange the terms in descending order, but the student could determine the correct divisor. student j was also able to follow the algorithm of synthetic division since the latter steps of his solution were correct. in figure 5, student k identified the error but used an incorrect value for the divisor. it can be observed that both students j and k were able to follow the algorithm of procedural fluency but tend to forget the initial and most crucial steps. these mistakes necessitate students to clearly understand the concepts behind each procedure in the synthetic division: why it is necessary to convert the binomial divisor into the form (x c) and use the numerical value c as the divisor in the synthetic division. figure 3. solution of student i agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 200 figure 4. solution of student j figure 5. solution of student k 3.3. students’ self-assessment on procedural fluency in synthetic division only twenty-five (25) of the thirty-nine (39) enrolled learners answered a selfassessment questionnaire through google forms. table 3 shows students' perceived procedural fluency in synthetic division. the mean for statements 1 and 2 is 3.72 (sd = 0.54), the highest mean score. the skills in identifying the dividend and the divisor and rewriting expressions in descending order are essential before performing the process of synthetic division. statement 4 has a mean of 3.64 (sd = 0.57), showing that students can determine and differentiate quotient from remainder. statements 3 and 5 have a mean of 3.60 (sd = 0.65 and 0.58, respectively). performing operations on real numbers got the lowest mean among the prerequisite skills. overall, students think they can perform division of polynomials using synthetic division. table 3. descriptive statistics of students’ self-assessment of their procedural fluency in synthetic division n mean standard deviation 1. i can identify the dividend and divisor in a given expression 25 3.72 0.54 2. i can rewrite expressions in descending order. 25 3.72 0.54 3. i can perform operations on real numbers. 25 3.60 0.65 4. i can determine the quotient and remainder when dividing polynomials by binomials using synthetic division. 25 3.64 0.57 5. i can perform division of polynomials using synthetic division. 25 3.60 0.58 volume 12, no 2, september 2023, pp. 193-206 201 students’ responses to the open-ended questions: “which part of the topic/lesson was easy?” and “which part of the topic/lesson was difficult?” were analyzed using thematic analysis. one student responded that, “they [synthetic division and long division] were equally okay.” some students (n=5) found the solving part of synthetic division easy. three students stated that following the steps was easy. most of the students (n=9) classified identifying the degree of the polynomial and the coefficients to be an easy task. three students responded that everything in the lesson was easy, and two students answered that none was easy. there was one student who did not answer any of the two questions. on the other hand, some students (n=2) responded that they have a hard time solving when the divisor is a fraction. two students had difficulty in the actual computation, and two students found it challenging to identify the remainder and the quotient since there were terms that they needed help understanding. a total of 18 students wrote “none” as their answers meaning they did not find any part of the lesson difficult to do. the students did not include their learning modality in answering the forms, hence we are not able to compare which set of learners had difficulty following the lesson instruction. but, majority (n=18) of the students claimed no part of the lesson was difficult to do. 3.4. challenges encountered in conducting the lesson study due to different schedules and workplaces, the members of the lesson study group struggled to find a common time during the planning stage. the group separated the tasks into those that could be done asynchronously and those that needed to be done synchronously as a way to go about this. the approval of the study’s research ethics delayed the research lesson class implementation. consequently, we changed the lesson topic to avoid disrupting the classes. likewise, there was an unforeseen class suspension before the lesson implementation, so the researchers had to change the topic from long division of polynomials to synthetic division and make necessary adjustments in the research lesson, instructional materials, and assessment tools. in the lesson planning during the lesson study, the teachers had a difficult time thinking of strategies to make the lesson more engaging and with a deeper conceptual understanding. thus, the original lesson plan did not show the connectivity between long and synthetic division. furthermore, the ls group conducted collaborative research lesson planning through an online platform meeting that was new for them. in the next cycle, we highly recommend that the teachers schedule a weekly appointment to develop the lesson plan to maximize available resources and exhaust the best strategies to use to maximize learning results. 3.5. challenges encountered in implementing the research lesson in hyflex modality since the study was conducted in a hyflex learning environment, intermittent and weak internet connection became a challenge during implementation. additionally, there was an unforeseen electrical problem at the beginning of the lesson, so the class had to transfer to another room with a more stable connection, which led to a reduced time for discussion. the hyflex setup itself was a challenge. during the class, the teacher was confronted with so many things that needed her attention delivering the lesson, interacting with the students in the physical and virtual classroom, or achieving interaction between the two groups of students. in addition, in addition, the teacher barely used the physical whiteboard. agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 202 instead, she used the annotation feature in powerpoint so that both groups could see the solutions. moreover, it was difficult for the teacher to write on the actual board because the teacher needed to be close to the virtual classroom equipment in order to get a stable audio reception. roaming around the classroom to have more active interaction also became a struggle because the audio quality for the virtual group would be compromised. on the other hand, the planned error analysis for more in-depth procedural fluency was not executed in class as originally planned due to time constraints. instead, this activity was assigned as an asynchronous homework to the students. lastly, the students practically had no idea about the lesson. had the ls group thought of an activity integrated into the research lesson plan and implementation to link concepts in long division to the synthetic division algorithm, it could have been less challenging for the teacher to interact with students who have become passive. 3.6. discussion most of the students could easily follow the given procedure in synthetic division. however, this raises questions about whether they understand the concepts behind each step of the process. after the lesson implementation in the class, the lesson study group reflected and realized that we could improve the research lesson by interweaving the concepts behind the algorithm and the algorithm per se. since the students already took up long division of numbers and long division of polynomials in previous lessons, the teacher may direct them to observe the similarities and differences between the long division method and the synthetic division method. consequently, students could see that the synthetic division is a shorter version derived from the long division algorithm. the teacher may prompt students to think about why synthetic division works and why it is a shorthand of long division method. the teacher may also ask why a crucial synthetic division step is to equate the divisor by zero and solving for x. another way to deliberately show a more vital link between the two methods of division is through a brief recall on long division, such as watching a video available on the internet with focus questions to answer, preferably before the class. such activities may be included for the students to discover by themselves and make sense of why synthetic division is a shorthand method of long division, that the two approaches are not different except we have foregone changing of signs in the partial subtrahends with a one-time change of sign in the divisor for the synthetic division. the researchers also realized that foreseeing or anticipating learners’ possible responses and thinking processes could be helpful in the development of any lesson plan, as it would create a structure for the effective delivery of the lesson. the members of the lesson study group realized that to implement the lesson better, the teacher needs to anticipate the possible challenges of recalling previous skills and concepts to prepare appropriate courses of action in case there are no responses from the students. some students could not completely follow the algorithm as they made mistakes in the first few steps in rearranging the terms in descending order of their degree, identifying the correct numerical divisor, and getting confused about whether to add or subtract the coefficient and the partial product in each term. these imply some learners still need to achieve procedural fluency, partly because they lack conceptual understanding. this tendency to be confused and make mistakes supports hiebert and lefevre’s (1986) argument that students failing to attain procedural fluency may be due to their lack of conceptual understanding. as this study was conducted in a hyflex setup, internet connectivity due to the sudden power outage hindered the continuous flow or delivery of the lesson. this challenge is consistent with teachers’ experiences in the study of rodriguez (2022). online volume 12, no 2, september 2023, pp. 193-206 203 synchronous learners were not able to maximize mathematical discourse due to limited access to the internet. the whiteboard was also not used in the lesson proper since no cameras were pointing to the board, nor were there drawing tablets/ipad for the teacher to use in the annotation. nonetheless, there currently needs to be more studies on the integration of ls in hyflex learning. thus, we recommend teachers conduct similar studies utilizing a hyflex learning environment since it makes learning more accessible to students as they have a choice to be physically or virtually present in the class. hyflex learning is a way to cater to students’ different needs (rodriguez, 2022) as the locus of control lies on the student (beatty, 2019) because it allows students who cannot attend the class physically to participate online real-time (romero-hall & ripine, 2021), thus, making education more accessible and equitable. lohmann et al. (2021) suggest that teachers utilize their best practices in face-to-face classroom management as hyflex learning poses unique challenges in terms of expectations and managing classroom behaviors. teachers must lay down expectations explicitly, model desired behavior, and provide timely precise feedback to support students’ learning. the lesson study group members realized that hyflex learning demands more for teachers to be at their best when it comes to classroom management than in either face-to-face or online learning environments alone. similar to the experience of cheng and yee (2013), one of the constraints in doing ls is time. teaching full-time in different schools with conflicting schedules and taking up graduate studies in the evenings became a challenge for the lesson study group members to regularly meet as ls requires considerable time and commitment (rock & wilson, 2005). nonetheless, this did not hinder them from drafting and preparing a research lesson plan. the lesson in its initial cycle definitely needed much improvement to maximize learning. areas for improvement were identified among collaborators of the ls through collegial and friendly feedback, peer learning, and reflections on actual classroom practice (lee, 2008). 4. conclusion a lesson study in a hyflex learning environment was conducted to improve lesson delivery by looking at students’ procedural fluency in synthetic division. the episodes in the implementation of the lesson revealed some learners have forgotten the term ‘coefficient’. moreover, we noted that some students were very attentive and were thinking critically since somebody asked the question of the possibility of a nonzero remainder. the majority of the learners claimed they did not have a hard time following the lesson. most students got a perfect score, while only two got 0 in the error analysis asynchronous task. nonetheless, some students failed to achieve procedural fluency because of a lack of understanding of the concepts behind the procedure and prior knowledge/skill in performing operations on real numbers. their solutions indicate some computational errors, failure to arrange terms in descending order of degree, incorrect sign in the divisor, and subtracting partial product from the dividend term instead of adding them. the conduct of a lesson study has helped the researchers to reflect on how vital conceptual understanding is to procedural fluency for students’ meaningful learning and retention. lesson study allowed the proponents to realize aspects of their teaching practice that need improvement. through this lesson study, we found out that although it is beneficial for the learners to be algorithmically adept in manipulating expressions and following procedures, understanding the underlying concepts proved valuable so they could make sense of the solutions they create. the challenges encountered in planning the lesson study were unforeseen circumstances that contributed to the delays in the class implementation, difficulty in agsalon et al., utilizing lesson study in teaching synthetic division for procedural fluency … 204 scheduling, and conducting meetings on online platforms. these delays made the researchers realize the importance of time and commitment in conducting a lesson study. the challenges in implementing the research lesson were intermittent and weak internet connection, hyflex learning classroom management, and getting students to express their mathematical ideas. better infrastructures and some contingency measures, establishing classroom protocols in hyflex learning, and anticipating student difficulty in classroom interaction based on their prior knowledge in planning the research lesson, are suggested for the next cycle of the research lesson implementation. acknowledgements the authors would like to thank the institution, students, and fellow educators who gave their precious time and effort to make this study possible. references appova, a. k. 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(2022). the rise and fall of the hyflex approach in mexico. techtrends, 66(6), 911-913. https://doi.org/10.1007/s11528-022-00780-3 romero-hall, e., & ripine, c. (2021). hybrid flexible instruction: exploring faculty preparedness. online learning, 25(3), 289-312. https://doi.org/10.24059/olj.v25i3.2426 trends in international mathematics and science study. (2020). timss 2019 international reports – timss & pirls international study center at boston college. retrieved from https://timss2019.org/reports unesco. (2021). acting for recovery, resilience, and reimagining education: the global education coalition in action. unesco. retrieved from https://unesdoc.unesco.org/ark:/48223/pf0000379797 williams, e. f., duke, k. e., & dunning, d. (2020). consistency just feels right: procedural fluency increases confidence in performance. journal of experimental psychology: general, 149(12), 2395-2405. https://doi.org/10.1037/xge0000779 https://gpseducation.oecd.org/countryprofile?primarycountry=phl&treshold=5&topic=pi https://gpseducation.oecd.org/countryprofile?primarycountry=phl&treshold=5&topic=pi https://doi.org/10.1007/s11528-022-00780-3 https://doi.org/10.24059/olj.v25i3.2426 https://timss2019.org/reports https://unesdoc.unesco.org/ark:/48223/pf0000379797 https://doi.org/10.1037/xge0000779 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p151-164 151 is communicating mathematics part of the ease of online learning factor? eka fitria ningsih1*, sugiman1, c. asri budiningsih1, dita surwanti2 1universitas negeri yogyakarta, indonesia 2utrecht university, netherlands article info abstract article history: received jan 15, 2023 revised feb 10, 2023 accepted feb 27, 2023 in this study, we focus on self-disclosure, communicating mathematics, and infrastructure support in determining the ease of online learning for students of mathematics education study programs. the sample in this study were students (n=465) who were asked voluntarily to fill out an online questionnaire. participants consisted of 335 female students (72%) and 130 male students (28%) from various universities in indonesia. the data were analyzed quantitatively using structural equation modeling. the results of the path analysis show that the ease of online learning is influenced by selfdisclosure, communicating mathematics, and infrastructure support. the r square results show that these factors influence 47%. each path analysis shows that self-disclosure (r = 0.556 p= 0.000) and infrastructure support (r = 0.243 p = 0.000) have a significant positive relationship with the ease of online learning. meanwhile, communicating mathematics (r = -0.025 p =0.507) has an insignificant negative relationship to the ease of online learning. further research is needed to see how the impact of mathematical application support communicating mathematics in online learning. keywords: higher education, infrastructure support, mathematical communication, online learning this is an open access article under the cc by-sa license. corresponding author: eka fitria ningsih, department of educational science, universitas negeri yogyakarta jl. colombo yogyakarta no.1, sleman, daerah istimewa yogyakarta 55281, indonesia. email: ekafitria@umala.ac.id how to cite: ningsih, e. f., sugiman, s., budiningsih, c. a., & surwanti, d. (2023). is communicating mathematics part of the ease of online learning factor? infinity, 12(1), 151-164. 1. introduction as we all know, the development of the internet is characterized by its use in all aspects of life. the internet of things provides convenience and breadth of face-to-face communication. this can be seen from the behavior of the people who are able to solve various social challenges and problems by utilizing various innovations that were born in the current era. as in the e-learning system is a distance learning process by combining the principles of the learning process with technology. this system utilizes internet infrastructure in the teaching and learning process, so that students can access and understand a topic or material. then, students can access learning materials anywhere and anytime. https://doi.org/10.22460/infinity.v12i1.p151-164 https://creativecommons.org/licenses/by-sa/4.0/ ningsih, sugiman, budiningsih, & surwanti, is communicating mathematics part of the ease … 152 communication carried out online by teachers and students greatly influences the implementation of education. at least with internet facilities, online learning can provide flexibility (bussey, 2021; pham et al., 2019). although for some areas which are considered blank spots, online learning experiences many problems, but it must be understood that expanding access to education through online learning provides opportunities (szopiński & bachnik, 2022) for people who are geographically distant from the center of the education provider to participate in education. online learning has differences with offline learning. offline learning can be interpreted as direct learning such as students face to face directly with teachers without intermediaries, for example conventional learning. online learning is a teaching and learning process that utilizes the internet and digital media in delivering material. for this reason, online learning requires teachers to be creative in preparing teaching tools, because the teaching tools that can be used in online learning are digital teaching devices. conditions in the field, online learning is essentially the same as scheduled learning, it's just that it is held online so that face-to-face teacher-student interactions are carried out in a virtual space. the covid-19 epidemic has prompted significant educational changes, particularly in adopting e-learning platforms like zoom and google meet (irfan et al., 2020; szopiński & bachnik, 2022). the abrupt shift in online learning (mseleku, 2020) has increased learners’ interest in engaging (assunção flores & gago, 2020; szopiński & bachnik, 2022). obstacles to establishing online learning include equipment restrictions (assunção flores & gago, 2020; gay, 2016) and teachers’ inability to use technology (rasheed et al., 2020). anxiety resulting from the abrupt shift in modes from offline to online must be addressed (bao, 2020). the implementation of online learning in indonesia has only recently been crowded due to the covid-19 outbreak. the indonesian government provides a policy to work from home, including implementing learning. this is aimed at reducing the spread of covid-19. online learning is undoubtedly the best alternative during the pandemic because learning can continue. however, online learning, which is still new, requires adjustment because, in indonesia, almost all learning is face-to-face. education providers usually provide feedback as part of the evaluation. the implementation of learning that is still new certainly raises the perception of students—moreover, indonesia's vast geographical conditions and the characteristics of each existing island. perception is an essential part of the psychological aspects of students that impact the achievement of learning outcomes. this project focuses on developing an instrument to measure students' perceptions of online learning. various developments of instruments regarding perceptions and attitudes in online learning have been carried out. however, the development still needs to include infrastructure support in the instrument. at the same time, the implementation of infrastructure learning impacts the sustainability of knowledge, which also becomes part of the determining factor of how perceptions are formed during online learning. students' readiness is an essential factor that educators must address. thorndike's (law of readiness) law states that a person's learning success is determined by readiness (steiner, 1997). watson differentiates learner preparation into psychological and pedagogical (watson, 1998). the pedagogical framework is concerned with learners' preparedness to participate in instruction. readiness to learn can aid in effective learning. readiness to learn relates to a student's physical, mental, and emotional readiness (dangol & shrestha, 2019). learners' readiness to learn is also associated with their ability to be independent and control their behavior to achieve learning goals (chorrojprasert, 2020). for decades, people have been experimenting with online learning (volery & lord, 2000). the adoption of online education has been widespread in industrialized countries (joosten & cusatis, 2020). online learning is based on the ease and expansion of access. in addition, due to the increased interest in continuing education, the required capacity is being volume 12, no 1, february 2023, pp. 151-164 153 raised. growth in the number of students necessitates the construction of new buildings and accompanying infrastructure. evidence shows that public awareness of the necessity of education has grown. not only are young people between the ages of 18 and 24 encouraged to pursue higher education, but people of all ages are encouraged to do so. this is relevant to the implementation of online learning, which is flexible and hence easy to follow. organizing online learning enables various accomplishments for individuals, institutions, and countries (akaslan & law, 2011). students participating in online learning situations show significant differences (muilenburg & berge, 2005). this learning experience also influences the impression of learning, which affects learning outcomes. as a result, support becomes a key component in achieving learning objectives (muilenburg & berge, 2005). problems can occur in any learning; thus, while online learning provides convenient access (volery & lord, 2000), technical help is required (lee et al., 2011) to minimize problems. online learning during the pandemic is a new approach for indonesia. the covid19 pandemic has resulted in considerable educational improvements, particularly in using elearning platforms such as zoom and google meet (szopiński & bachnik, 2022). the rapid transition in online learning (mseleku, 2020) has piqued learners' enthusiasm to participate (assunção flores & gago, 2020; szopiński & bachnik, 2022). obstacles to establishing online learning include equipment restrictions (assunção flores & gago, 2020; gay, 2016) and teachers’ inability to use technology (rasheed et al., 2020). anxiety resulting from the abrupt shift in modes from offline to online must be addressed (bao, 2020). students must be mentally and physically prepared to attain learning experiences through online learning. as a result, assessing learners' readiness to engage in online learning in various circumstances is critical—the ability and willingness to use technology influence pupils' preparedness to use technology. online learning preparedness is influenced by technical issues, content, human resources, and finances (rohayani et al., 2015). research on developing online learning instruments has been widely conducted (hashim & tasir, 2014; hung et al., 2010; smith et al., 2003). ledbetter created an online learning readiness test with five dimensions: self-efficacy, motivation, self-directed learning, learner control, and online communication efficacy (hung et al., 2010). hung had developed a more complex instrument than smith had. however, hung's instrument continues to focus on aspects of self-efficacy, while infrastructural supports significantly influence online learning readiness. because they depend on individual psychological effects, efficacy, and attitude are inextricably linked. instruments for measuring attitudes toward online learning have been developed (bernhold & rice, 2020; ledbetter, 2014; mazer & ledbetter, 2012). in learning mathematics, mathematical communication refers to the involvement of students not only in solving mathematical problems but also the involvement of students in speaking and listening activities to share ideas, opinions, and solutions (kosko & gao, 2017; kosko & wilkins, 2010). mathematical communication skills include expressing real problem situations into mathematical models, explaining and evaluating ideas orally and writing using symbols and mathematical language (hidayat & sumarmo, 2013; lomibao et al., 2016; silver & cai, 1996). mathematical representations are also an essential part of mathematical communication (tong et al., 2021). online learning systems during the covid-19 pandemic and the closeness of the younger generation to online media have encouraged higher education institutions to implement online learning systems (elmunsyah et al., 2020). however, it is necessary to explore communicating mathematics in online learning. in this study, we focus on self-disclosure, communicating mathematics, and infrastructure support in determining the ease of online learning for students of mathematics education study programs. communicating mathematics is an essential part of learning ningsih, sugiman, budiningsih, & surwanti, is communicating mathematics part of the ease … 154 mathematics. mathematics has the characteristics of an abstract science different from other social sciences, so it is necessary to explore how the response to online mathematics learning choices, especially in mathematics lessons. 2. method this study aims to investigate factors that significantly impact the ease of online learning for mathematics education students in indonesia. in addition, this research wants to see how these factors affect the ease of online learning. researchers identified selfdisclosure, infrastructure support, communicating mathematics, and ease of online learning variables in the study. based on a literature review, self-disclosure, infrastructure support, and communication impact the ease of online learning. this research more specifically explores communication in mathematics learning. communication in learning mathematics in this study leads to the convenience of students to communicate when learning mathematics is held online. as for the instruments for self-disclosure, infrastructure support, and ease of online learning, the researchers adapted various instruments that researchers had previously developed. the researcher made 18 question items (see table 1). the questionnaire is presented in indonesian. the question items were designed using the 5-point likert-type scale ranging from 1 (strongly disagree) to 5 (strongly agree) for the level of student agreement with each of these items. the questionnaire that has been developed is presented in google form. table 1. items in the questionnaire aspect items indicator self-disclosure items 1 when i'm learning online, i feel comfortable discussing. items 2 i'm more open to communicating in online. items 3 when i communicate online, i feel more confident. items 4 i am comfortable sharing personal information online. communicating mathematics items 5 when i learn online, i can think clearly items 6 when discussing online, i’m able to express mathematical ideas items 7 when i'm learning online, i can understand symbols, pictures and mathematical formulas items 8 when learning online, i can understand the explanation from the teacher and students items 9 when learning online, i can explain mathematical ideas systematically ease in online learning items 10 i like online learning because i don't have to come to campus items 11 i like online learning because i can give feedback anytime items 12 online learning is the best way during the covid-19 pandemic items 13 i enjoy communicating online infrastructure support items 14 since the location where i live has an excellent internet connection, it facilitates online learning. items 15 i have no objections to funding internet packages to support online learning. items 16 when i study online, i don't have to deal with inclement weather. items 17 when i'm learning online, i don't have any issues with power outages. items 18 i have a suitable device for online learning (eg laptop, mobile phone) researchers distributed questionnaires through social media networks. researchers contacted lecturers from various universities in lampung. in the questionnaire, participants were asked to tick the answer that best represented them with the statement. questionnaires are distributed at the end of the odd semester of the 2021/2022 academic year. the volume 12, no 1, february 2023, pp. 151-164 155 questionnaire was broadcast for one week. answering all of the participant's questionnaire items takes about 10 minutes. after answering all the questions, participants submitted the answers and recorded them on the researcher's google drive. each questionnaire question is requested to be filled out so that all questionnaires must be filled out before they can be submitted. table 2. demographic information of participants demograpics n percentage gender male 130 28% female 335 72% academic year first 157 34% third 116 25% fifth 102 22% seventh 90 19% the sample in this study were students (n=465) who were asked voluntarily to fill out an online questionnaire. participants consisted of 335 female students (72%) and 130 male students (28%) from various universities in lampung province, indonesia. in this study, most participants came from the first semester, with as many as 157 students (34%). the following highest number came from semester 3, with as many as 116 students (25%), followed by 102 students (22%) in semester five, and the remaining students in semester seven as many as 90 students (19%) (see table 2). this research is quantitative research with multivariate analysis. multivariate analysis is used to look for the effect of various variables on one object simultaneously. these variables are interrelated, with at least one dependent variable and more than one independent variable. sem – pls (structural equation modeling – partial least square) analysis measures the impact of self-disclosure, infrastructure support, and communication in mathematics learning on the ease of online learning. sem is a multivariate analysis technique that combines factor analysis and regression (correlation) analysis to examine the relationship between variables in a model, both between indicators and their constructs and relationships between constructs. pls (partial least square) is a variant-based sem structural equation model. pls is an alternative approach that shifts from a covariance-based sem approach to a variance-based one. the data is then recapitulated in ms excel and smart pls 3 for analysis. the use of pls-sem in this study was divided into two analysis steps. the first step, the analysis, focuses on the measurement model, determining validity and reliability. the validity and reliability criteria refer to cr, cronbach's alpha value, loading factor, and ave. the validity of each indicator is determined using standardized factor loading (sfl). if the sfl is higher than 0.4, the item is said to be valid (wawan, 2020). according to mehrens and lehmann, determining a reliability coefficient above 0.85 is a good criterion (retnawati, 2016). in the second step, evaluate the structural model to test the hypothesis. in this study, the analysis used the bootstrap method with 5000 subsamples in smart pls to calculate path coefficients, t-values, and p-values. in sem, model fit is determined by four criteria r square, srmr, rms thea, and nfi (hair et al., 2019). ningsih, sugiman, budiningsih, & surwanti, is communicating mathematics part of the ease … 156 3. result and discussion the analysis results using confirmatory factor analysis (cfa). the cfa test results show that this instrument's development can be accepted as a fit model (see table 3). table 3. validity and reliability test based cfa variabel laten manifest variabel elf dlta note cr ave decision self-disclosure items 1 0.834 0.305 valid 0.870 0.721 reliable items 2 0.877 0.292 valid items 3 0.890 0.298 valid items 4 0.793 0.283 valid communicating mathematics items 5 0.715 0.165 valid 0.848 0.617 reliable items 6 0.729 0.155 valid items 7 0.850 0.321 valid items 8 0.864 0.328 valid items 9 0.758 0.280 valid ease of online learning items 10 0.785 0.265 valid 0.885 0.746 reliable items 11 0.870 0.290 valid items 12 0.907 0.299 valid items 13 0.887 0.302 valid infrastructure support items 14 0.826 0.261 valid 0.866 0.651 reliable items 15 0.708 0.207 valid items 16 0.842 0.226 valid items 17 0.795 0.232 valid items 18 0.857 0.308 valid the analysis results show that all indicators contribute significantly to measuring the latent variable (see figure 1). the standardized factor loading (sfl) value for each hand determines validity. if the efl is higher than 0.7, the item is said to be valid (wawan, 2020). the minor loading factor is 0.715 (item 5), while the largest is 0.907 (item 17). all of the 18 items have a loading factor higher than 0.7 (see table 3). figure 1. t value and outer loading factor volume 12, no 1, february 2023, pp. 151-164 157 the results of the path analysis show that the ease of online learning is influenced by self-disclosure, communicating mathematics, and infrastructure support. the r square results show that these factors influence 47%. each path analysis shows that self-disclosure (r = 0.556 and p= 0.000) and infrastructure support (r = 0.243 and p = 0.000) has a positive and significant relationship to the ease of online learning. meanwhile, mathematical communication (r = -0.025 and p =0.507) has an insignificant negative connection to the ease of online learning (see figure 2). figure 2. path coeficient and p value the srmr value or standardized root mean square model fit criteria must be smaller than 0.08 (hair et al., 2019). rms theta or root mean square theta value 0.102; and nfi value > 0.90 are the model fit requirements. based on the model criteria, the nfi 0.865 < 0.90 does not match the fit model criteria. however, based on the srmr or standardized root mean square value, the value is 0.056 < 0.08 and rms theta or root mean square theta 0.158 > 0.102, the model is fit based on these two criteria. based on the model criteria, the value of rms theta or root mean square theta 0.158 > 0.102 matches the fit model criteria. 3.1. the relationship between self-disclosure and ease of online learning accessibility, which includes the facilities that are available and their adaptability. students are welcome to come in at any time of day or night. interaction (chorrojprasert, 2020) in online learning occurs in virtual classrooms, eliminating the need for teachers and students to sit together in the school to communicate. virtual classrooms necessarily require a teacher's ability to facilitate active discussion interaction between students. the disadvantage is that there is no direct interaction, as there is in face-to-face learning. however, this allows students to enjoy themselves more because it removes the rigidity of implementing learning. strong self-confidence (mcsporran & young, 1969; ramírezcorrea et al., 2015) in learning mathematics leads to high achievement in online learning ningsih, sugiman, budiningsih, & surwanti, is communicating mathematics part of the ease … 158 (lee et al., 2021; ramírez-correa et al., 2015). the comfort and confidence in speaking, debating, and sharing information are recognized as indicators of self-disclosure. ease of access is part of the advantages of online learning. learning that is held in a virtual space allows every student to take part in learning from anywhere without having to come to campus (lópez meneses et al., 2020; okoye et al., 2022). the study results a show that implementing online learning can attract students from various countries to meet in one virtual space (ma & lee, 2019). online learning makes it easy for students to express their ideas (joo et al., 2011). learning that finds students online provides confidence for students to engage in online communication. learners can be more open when communicating online. students not used to appearing directly in front of the class get the opportunity to express ideas through virtual space. self-disclosure is an act of sharing information with others. the online learning environment provides a space that allows students to connect with one another. students can share and utilize learning resources together. for today's young generation, online communication has become a common thing to do. they are barely separated from the gadgets that connect them to online social networks. this makes students feel comfortable carrying out discussions online. learners do not hesitate to disclose personal information in the online environment. online social media allows one person to share personal information with others. the survey results show that 66% of adults use social networks. in addition, schools also use social networking sites to support educational goals. self-disclosure in online social networks helps convenience when participating in online learning (chen & sharma, 2015). when participating in online learning, students consciously choose profile information to display to teachers and friends. the convenience of sharing information online affects students' interest in participating in online learning. students enjoy online learning because of its flexibility which does not require them to be present in class. the online discussion provides convenience for students to comment and provide feedback. in contrast to face-to-face learning, sometimes students withdraw to provide responses and wait for the teacher to choose them. students who are not used to appearing directly in front of the class get the opportunity to express ideas through virtual space. 3.2. the relationship between infrastructure support and ease of online learning infrastructure support has been identified as a factor in online learning readiness. one of the indicators is fund readiness. this is consistent with the belief that financial assistance plays a role in online learning (muilenburg & berge, 2005). technical issues appear to be part of the online learning infrastructure support, including the availability of laptop equipment, cell phones, internet networks, and electricity networks (taskin & erzurumlu, 2021). when there is a power outage, some internet service providers have problems. power grid outages are sometimes caused by bad weather. this confirms that the weather is a part of the infrastructure support in online learning. the need to organize online learning during the pandemic encourages all parties to become involved with technology. of course, technological mastery is required for online learning readiness (olayemi et al., 2021). in contrast to face-to-face learning, laptops, and internet access, although needed, have yet to become a vital part of organizing education. infrastructure support makes a substantial contribution to the implementation of online learning. however, explicitly learning mathematics requires the support of facilities that are no less important, for example, the application of mathematics (loch et al., 2011) and video support that links mathematics to life (engelbrecht et al., 2020). learners need help that represents the mathematical language presented online. even with the backing for explaining, a teacher volume 12, no 1, february 2023, pp. 151-164 159 needs additional equipment to write virtually. research shows that mathematics learning media were developed online to support online learning tools (borba, 2012; hariyono et al., 2021; hwang et al., 2006). the limitations of this research infrastructure support online learning still focuses on primary facilities, namely laptops, internet and electricity networks, and funding. furthermore, especially in learning mathematics requires the support of other means. learning mathematics about formulas, theorems, and proofs involves mathematical symbols and terms. mathematical material cannot be discussed abstractly but requires visual presentation. facilities such as digital whiteboards, digital pens, and math applications are parts that must be prepared during online learning. further research is needed on how this infrastructure influences the ease of online learning. 3.3. the relationship between communicating mathematics and ease of online learning interaction in a virtual space using online communication. of course, providing comfort in touch, including discussions with teachers and other students, is necessary (hung et al., 2010). according to ledbetter, one aspect of online communication attitude is the selfdisclosure and online social connection (ledbetter, 2009). individuals feel fear, anxiety, nervousness, or despair when participating in online communication (bernhold & rice, 2020). indicators of confirmed self-disclosure include discomfort with expressing opinions and pain after online learning. in the previous results, it was found that online learning provides easy access and communication for students. however, on a more specific aspect of mathematical communication, the ease of online education is relatively minor. the relationship shown tends to be negative, although not significant. this indicates that the online learning system still needs to support students' mathematical communication in learning mathematics. the study results a show that teachers must put extra effort into teaching geometry material online (diana et al., 2021). learning mathematics requires clear thinking and direct contact with the teacher/face-to-face. mathematical anxiety also influences students to express opinions and convey ideas systematically. education providers need to provide support to facilitate virtual mathematical communication. direct learning provides space for students to express their ideas. writing mathematical symbols and making representations in pictures can be done directly. this convenience cannot be felt now during online learning. however, research shows that learning through google meet can improve mathematical communication (hutajulu, 2022). during mathematics learning, students are involved in activities to share ideas. on online learning, students express their views by speaking directly online. in addition, opinions are usually conveyed through voice recordings and writing in the comment’s column. problems that can be faced when student express ideas through voice messages, symbols and mathematical terms become more abstract. in addition, writing and drawing shapes related to mathematical concepts is limited when students have online learning tools in the form of gadgets and laptops only. the ease of expressing mathematical ideas is yet to be part of the convenience obtained from online learning. these results align with research findings trenholm and peschke (2020) which show the problems encountered during online learning, namely limitations in communicating mathematical concepts that involve notation and diagrams. in face-to-face learning, the teacher usually writes a mathematical note and mathematical symbols on a large blackboard. in addition, verbal cues, gestures, and facial expressions help cognitive mastery during learning. these resources are less felt during this online learning process, causing difficulties in developing students' conceptual understanding. johnson recommends using online ningsih, sugiman, budiningsih, & surwanti, is communicating mathematics part of the ease … 160 whiteboards to communicate mathematical ideas or justify students' reasoning (johnson & green, 2007). support online platforms to use virtual whiteboards that enable interactive learning. it's just that the obstacle for students is not being able to make pictures flexibly and write symbols, especially since there is no additional support for using virtual whiteboards and pens. mathematics which has abstract characteristics and is full of characters, certainly involves the language of mathematics in providing mathematical explanations and arguments. 4. conclusion the results of this study highlight three aspects related to self-disclosure, infrastructure support, and communicating mathematics as determining factors for ease of online learning. self-disclosure and infrastructure support strongly influence the ease of online learning. meanwhile, communicating mathematics does not affect online learning. teachers need to provide more specific means of support for math content. virtual mathematical communication requires additional media and applications that support mathematical content. infrastructure support in this research instrument is limited to facilities in general, while for communicating mathematics, it is not specific to mathematical communication. this is a limitation of this research. further research is needed to see how the impact of mathematical application support on mathematical communication skills in online learning. references akaslan, d., & law, e. l.-c. 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(2018). students’ geometric thinking based on van hiele’s theory. infinity, 7 (1), 55-60. doi:10.22460/infinity.v7i1.p5560. mailto:harina.fitriyani@pmat.uad.ac.id mailto:sriadi@ustjogja.ac.id fitriyani, widodo, & hendroanto, students’ geometric thinking based on … 56 introduction geometry is part of mathematics that has been taught to students since elementary school. learning geometry can train students’ logic, systematic, and creative thinking skills. such skills are indispensable for studying other branches of mathematics as well as for solving problems in everyday life. therefore, it is necessary to have a good geometric thinking skill. nevertheless, students still have difficulty in geometric thinking (hardianti, priatna & priatna, 2017; abidin, 2010) which is indicated by the level of geometric concept mastery is still not maximized at the elementary school (yudianto, 2011), junior high school (lestariyani, 2013; apriyanti & fitriyani, 2017; amimah & fitriyani, 2017), high school (sunardi, 2016) and higher education (jupri, 2005; utomo & wardhani, 2015; darta, 2013; rafianti, 2016; noto, 2015). therefore, as students of mathematics education department are prepared to become teachers in schools after graduation, it is very important to understand their geometric thinking skills so that their geometric thinking can be maximized. based on researchers’ experiences during the course of geometry in the mathematics education program uad, it was found that students’ understanding of the concepts of geometry is very lacking. this can be seen from the number of graduated students who take geometry courses is still less than 65%. this is allegedly due to the students' geometric thinking ability is low so they tend to avoid the courses. to help the students develop their geometric thinking then a reasearch is needed to investigate students’ level so that counter measurers can be prepared. in relation to geometric thinking, van hiele (van de walle, 1998; crowley 1987) proposed a theory of geometric thinking which includes 5 levels of level 0 (visualization), level 1 (analysis), level 2 (informal deduction ), level 3 (deduction), and level 4 rigor (accuracy). at level 0 (visualization), according to van de walle (1998) "the objects of thought at level 0 are shapes and what they look like". the characteristic of students at this level is that they begin to learn to understand the shapes of geometrical objects in general, but not yet know their properties. in addition, van de walle (1998) also stressed that "the products of thought at level 0 are classes or groupings of shapes that seem to be alike." it means that at level 0 students will group abjects with similar shapes. level 1 (analysis) according to van de walle (1998) is "the objects of thought at level 1 are classes of shapes rather than individual shapes". this means that students have begun to learn the properties of the geometrical objects. in addition, students have been able to mention the regularity contained in these objects. but, at this stage students have not yet been able to know the relation among these geometrical objects. level 2 (informal deduction), according to van de walle (1998), is that "the objects of thought at level 2 are the properties of shapes". it means what is thought at level 2 is the objects’ properties. at this level, students have begun to carry out conclusions called deductive thinking. but this ability is not fully developed yet. in addition, students at this stage have begun to sort, determine the relationship between one object and another objects. in other words, "the products of thought at level 2 are relationships among properties of geometric objects" as presented by van de walle (1998). as for level 3 (deduction), van de walle (1998) states that "the objects of thought at level 3 are relationships among properties of geometric objects". at this level, students are able to deductively draw conclusions from general into more specific. in addition, students have volume 7, no. 1, february 2018 pp 55-60. 57 understood the importance of the role of undefined elements in addition to defined elements. at this stage, students also have begun to use the axioms or postulates to prove many things. but, the students still do not understand why it is a postulate or a theorem. specifically, van walle (1998) adds that "the products of thought at level 3 are deductive axiomatic systems for geometry". at level 4 (rigor) according to van de walle (1998), the target of thinking is "the objects of thought at level 4 are deductive axiomatic systems for geometry". at this level students have begun to realize how important the precision of the basic principles in a proof. for example, he knows the importance of axioms or postulates from euclid's geometry. accuracy stage is a high stage of thinking, complicated and complex. therefore, it is not surprising that students, even if they are already in high school or even college students, still have not reached this stage of thinking. level 4, according to van de walle (1998), is "the products of thought at level 4 are comparisons and contrasts between different axiomatic systems of geometry". before students start teaching geometry, it is better to identify their level of geometric thinking based on van hiele's theory. students’ level of geometric thinking need to be studied to determine the extent to which their geometric thinking so that we can help them to develop more. the result of the study of rafianti (2016) stated that geometric thinking level of elementary school teachers candidate according to van hiele’s theory is mostly only reached phase 1 or introduction stage that is 50%. from this study, it is necessary to identify also the level of students’ geometric thinking in mathematics education program to prepare better mathematics teachers in the future. method this study is a descriptive research with qualitative approach. the subjects of the study were students of mathematics education program class of 2014, amounting to 129 students. data collection techniques in this study consist of two methods of test and interview. the instrument used to collect data on the level of the students’ development of geometry concepts is a test developed by usiskin (1982) and it has been translated into bahasa by yudianto (2011). this test is designed to measure and identify the developmental levels of students’ geometric thinking based on van hiele's theory and constructed to classify students into five levels. the test consists of 25 items where each 5 items will indicate van hiele geometry thinking from level 0 4. criteria for determining the levels of student geometric thinking, according to yudianto (2011), is stipulated by the following rules: 1. students are classified at the n th level if: at least 3 out of 5 items at the n th level are answered correctly and also in previous levels too. if the student does not meet the criteria, then the student is classified into the pravisualization level. 2. students are classified transition level between the n th and (n + 1) th level if: a. at least 3 out of 5 items are answered correctly at the n th level and every previous level, and b. 2 out of 5 items are answered correctly at the (n + 1) th level 3. students are difficult to classify if: a. at least 3 out of 5 items are answered correctly at the n th level and every previous level, b. a maximum of 2 out of 5 items are answered correctly on the (n + 1) th level, and fitriyani, widodo, & hendroanto, students’ geometric thinking based on … 58 c. at least 3 out of 5 items are answered correctly at the n th level (n + 2) th or any subsequent level. 4. students can not be classified, if less than or equal to 1 of 5 items are answered correctly at the n th level and consistent for the next level. data analysis used to reveal the level of students' geometric thinking based on van hiele theory refers to miles and huberman (2014) model which are data reduction, data presentation, and drawing conclusion. results and discussion the results showed that the geometric thinking level of mathematics education students is spread over level 0 (visualization), level 1 (analysis), level 2 (informal deduction) and level 3 (formal deduction). on the other hand, no single student has reached level 4 (rigor). the number of student that is difficult and can not be classified or identified is significant enough to be at the level of pravisualization (before visualization). table 1 shows the result of data analysis of the development levels of students’ geometric thinking. tabel 1. level of development of students’ geometric thinking level % pravisualization 30,65% level 0 21,51% level 1 29,03% level 2 16,67% level 3 2,15% level 4 0,00% total 100,00% based on table 1, most students of mathematics education program are at level 1 (analysis). only few students (2.15%) have fulfilled level 3 (deduction) while no students meet level 4 (rigor) (0%), and 30.67% was not at the level of the development of geometry van hiele. this data shows that most of the development of students’ geometric thinking is at the level of analysis that is understanding the concept of geometry done by informal analysis of parts of the geometrical objects. the ability of students in doing deductive thinking is still weak and similar to the result of jupri (2005) and darta (2013). for sixth semester students who have taken all geometry courses consisting of geometry, space geometry, analytical geometry of the field, space analytic geometry and transformation geometry are supposed to be at higher level than this result. especially, if the student has taken the optional course of geometry systems. ideally, their development level of geometric thinking shall be already at the top level. but that is not the case. this shows that learning activities in geometry courses need to be improved so that the development of students' geometric thinking can be boosted. tabel 2. level of transition between developmental levels of student geometry transition level f % pra 1 32 17,20% pra 2 25 13,44% volume 7, no. 1, february 2018 pp 55-60. 59 transition level f % pra 3 12 6,45% pra 4 2 1,08% total 71 38,17% of the 129 respondents who have beend identified, 71 respondents (38.17%) are at the transition level between the levels of geometric thinking. the results of the transition level analysis are presented in table 2. the highest percentage of transition level at pre-1 level (pre-analysis) is 17.2% and the lowest percentage at pre-rigor level is 1.08%. students who have reached the transition level can improve their geometric thinking level through learning that supports the improvement of geometric thinking. in addition to students who are at transition level, there are also students who are difficult to classify their level of geometric thinking that is as many as 75 students (40.32%). this number is quite significant considering almost half the total number of students are difficult to identify their level of thinking. these findings support the findings of sunardi (2002) and yudianto (2011). this is possible because the respondents are less serious in doing the test, especially if the test is not in accordance with their development of thinking. there were also 8 students (4.3%) who could not be classified in any category from the development of geometric thinking. this is because the respondent did not seriously take the test given so that they answered randomly or cheated during the test, or it could be because their development of geometric thinking have not met any level on van hiele’s theory. conclusion based on the discussion, it can be concluded that the development of geometric thinking of mathematics education students still have not yet reached rigor level based on van hiele’s level of geometric thinking. most students are still at the analytical level. in addition, there was found students at transition level between the level of development of geometric thinking in pre-analysis, informal pre-deduction, pre-deduction and pre-rigor which are 17,20%; 13.44%; 6.45%; 1.08% respectively. another finding is that 40.32% of students are difficult to classify and 4.3% of students can not be classified or identified. based on the results of this study, the researchers suggested that lecturers consider the development of students’ geometric thinking in preparing and planning activities in geometry courses. in addition, it is also suggested that lecturers apply learning strategies that can stimulate and assist students to develop their geometric thinking. researchers can examine further about the development of student geometric thinking, especially new enrolled students which are still in transition from high school to college life. references abidin, z. z., & abu, m. s. (2011). alleviating geometry levels of thinking among indonesian students using van hiele based interactive visual. edupress. https://core.ac.uk/download/pdf/11790040.pdf. amimah, h. s., & fitriyani, h. (2017). level berpikir siswa smp bergaya kognitif refleksif dan impulsive menurut teori van hiele pada materi segitiga. prosiding seminar nasional unimus. fitriyani, widodo, & hendroanto, students’ geometric thinking based on … 60 apriyanti, s & fitriyani, h. (2017). teori van hiele: tingkat berpikir siswa smp bergaya kognitif refleksif dan impulsif pada materi segiempat. prosiding seminar nasional unimus. crowley, m. l. (1987). the van hiele model of the development of geometric thought. learning and teaching geometry, k-12, 1-16. darta, m. (2013). kemampuan deduksi matematika mahasiswa tingkat pertama prodi pendidikan matematika unpas (studi kasus untuk tahap berpikir deduksi geometri dari van hiele). jurnal pengajaran mipa, 18(1), 16-21. hardianti, d., priatna, n., & priatna, b. a. (2017). analysis of geometric thinking students’ and process-guided inquiry learning model. in journal of physics: conference series (vol. 895, no. 1, p. 012088). iop publishing. jupri, a. l. (2005). banyak cara, satu jawaban: analisis terhadap strategi pemecahan masalah geometri. aljupri. staf. upi. edu, 697-703. lestariyani, s. (2013). identifikasi tahap berpikir geometri siswa smp negeri 2 ambarawa berdasarkan teori van hiele(doctoral dissertation, program studi pendidikan matematika fkip-uksw). miles, m. b., & huberman, a. m. (2007). analisis data kualitatif: buku sumber tentang metode-metode baru. jakarta : ui press. noto, m. s. (2015). efektivitas pendekatan metakognisi terhadap penalaran matematis pada mata kuliah geometri transformasi. infinity journal, 4(1), 22-31. rafianti, i. (2016). identifikasi tahap berpikir geometri calon guru sekolah dasar ditinjau dari tahap berpikir van hiele. jurnal penelitian dan pembelajaran matematika, 9(2). sunardi, s. (2016). hubungan antara tingkat penalaran formal dan tingkat perkembangan konsep geometri siswa. jurnal ilmu pendidikan, 9(1). usiskin, z. (1982). van hiele levels and achievement in secondary school geometry. cdassg project. utomo, f. h., wardhani, i. s., & asrori, m. a. r. (2015). komunikasi matematika berdasarkan teori van hiele pada mata kuliah geometri ditinjau dari gaya belajar mahasiswa program studi pendidikan matematika. cendekia: journal of education and teaching, 9(2), 159-170. van de walle, j. a. (1998). elementary and middle school mathematics: teaching developmentally. addison-wesley longman, inc., 1 jacob way, reading, ma 01867; toll-free. yudianto, e. (2011). perkembangan kognitif siswa sekolah dasar di jember kota berdasarkan teori van hiele. prosiding seminar nasional matematika dan pendidikan matematika program studi pendidikan matematika unej. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p69-84 69 developing realistic mathematics educationbased worksheets for improving students’ critical thinking skills reza lestari1, rully charitas indra prahmana2*, maureen siew fang chong3, masitah shahrill4 1stkip muhammadiyah pagaralam, indonesia 2universitas ahmad dahlan, indonesia 3department of educators management, ministry of education, brunei darussalam 4sultan hassanal bolkiah institute of education, universiti brunei darussalam, brunei darussalam article info abstract article history: received july 15, 2022 revised dec 24, 2022 accepted feb 14, 2023 applying critical thinking is an essential skill in the 21st century. however, teaching materials that do not facilitate students to improve these skills impact the achievement of learning objectives. therefore, educators need appropriate teaching materials that encourage students to enhance their thinking skills. this study aims to develop teaching materials based on realistic mathematics education (rme) to improve students’ critical thinking skills. the development model used is addie consisting of analysis, design, development, implementation, and evaluation phases. the instruments used, consist of validated student worksheets based on material experts and media experts, pretest questions, posttest questions, and the practicality of student worksheets. the results showed that the student worksheets developed were feasible regarding validity, practicality, and effectiveness. the validity of the student worksheets is indicated by the average score of two material expert validators and two media expert validators, each of which is in the good and excellent categories. the practicality of the student worksheets is denoted by the average value of student assessments included in the practical category. then its effectiveness is shown by increasing students’ critical thinking skills after being given intervention using the student worksheets. keywords: critical thinking skills, realistic mathematics education, research and development, student worksheets this is an open access article under the cc by-sa license. corresponding author: rully charitas indra prahmana, department of mathematics education, universitas ahmad dahlan jl. pramuka no. 42, pandeyan, umbulharjo, daerah istimewa yogyakarta 55161, indonesia. email: rully.indra@mpmat.uad.ac.id how to cite: lestari, r., prahmana, r. c. i., chong, m. s. f., & shahrill, m. (2023). developing realistic mathematics education-based worksheets for improving students’ critical thinking skills. infinity, 12(1), 69-84. 1. introduction critical thinking skills are higher-order thinking skills that are part of 21st century skills (hujjatusnaini et al., 2022; zetriuslita et al., 2018). efforts to develop critical thinking skills have become the main goal in the mathematics education curriculum worldwide (weng et al., 2022; yildirim et al., 2011). critical thinking skills are needed in the https://doi.org/10.22460/infinity.v12i1.p69-84 https://creativecommons.org/licenses/by-sa/4.0/ lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 70 development of students’ thinking (hsu et al., 2022; mahanal et al., 2019) as it is one of the key skills needed in the 21st century. critical thinking is the central pillar to preparing students in the 21st century at the education level (koyunlu ünlü & dökme, 2022; sulaiman et al., 2008). thus, the ability to think critically is an essential basic ability possessed by students to obtain the truth from various problems. critical thinking skills are important for students to have and develop to deal with problems and everyday problems (wang et al., 2022). critical thinking skills make students careful in solving problems so that the resulting decisions become appropriate and reasonable solutions (berestova et al., 2022). critical thinking skills familiarize students with developing rational attitudes in determining the best alternative choices (plummer et al., 2022). in addition, critical thinking allows students to study problems systematically, face challenges in an organized manner, formulate innovative questions, and design original solutions (ramírez-montoya et al., 2022; sasson et al., 2022). thus, the ability to think critically must be owned by students because it provides many benefits. critical thinking skills are the basis of the thinking process to analyze arguments and stimulate ideas on any meaning or other forms of interpretation to develop logical thinking patterns (marzuki et al., 2021). it involves the ability to use prior knowledge to draw logical decisions from an issue, so that the truth that is considered the best solution can be done through scientific method steps (ennis, 2011). students’ critical thinking skills need to be trained using contextual problems in everyday life (shavelson et al., 2019; yuliati et al., 2018). one of the causes of the low level of critical thinking is the application of teachercentered learning, where students are passive recipients and do not have the opportunity to think (khalid et al., 2020). thus, critical thinking skills are fundamental for students to improve and develop in the mathematics learning process. realizing the importance of developing critical thinking skills, it is essential to have mathematics learning that involves more active students’ participation and engagement in the learning process (lugosi & uribe, 2022). critical thinking skills need to be stimulated and developed in the mathematics learning process (hafni et al., 2019; yumiati & kusumah, 2019; zetriuslita et al., 2018) because it is one of the skills needed in the 21st century. however, the facts in the field show that critical thinking-oriented learning is still lacking. this is evident by indonesia’s pisa results in 2018, which decreased compared to 2015 pisa results. for the mathematics category, indonesia is ranked 7th from the bottom, 73rd out of 79 countries, with the score of 379 compared to the international average score of 459 (oecd, 2019). in this case, indonesian students’ critical thinking ability is still relatively low. therefore, critical thinking skills are very important to be improved through learning models that involve learners playing an active role in the learning process. realistic mathematics education (henceforth, referred to as rme) based learning is one of the effective learning alternatives where mathematical concepts can be conveyed well (prahmana et al., 2020; risdiyanti & prahmana, 2021). rme is a domain-specific instruction theory that focuses on problems that are “real” or that have been experienced by learners, emphasizes the skills of the process of doing mathematics, discussing, and collaborating, brainstorming with classmates, finding their mathematical concepts, and using mathematics to solve problems (van den heuvel-panhuizen & drijvers, 2020). realistic learning is not only related to the real-world context but also related to the emphasis on imagination so that the problems given in learning can be imagined by learners (basuki & wijaya, 2019; zulkardi, 2002). the rme approach is suitable for middle school learners through the teaching materials provided, namely module, student worksheet, and learning trajectory (risdiyanti & prahmana, 2021). learners did not see difficulties in working on teaching materials with rme approach. therefore, with the help of teaching materials, the rme approach can be the best choice to facilitate learners in improving critical thinking skills. volume 12, no 1, february 2023, pp. 69-84 71 teaching materials are all forms of materials used to help teachers or instructors carry out teaching and learning activities in the classroom (de jong et al., 2019). one of the teaching materials used in the school is the student worksheets. the student worksheets guide learners in understanding process skills and material concepts that will be studied (dewi et al., 2023; nurfadhillah et al., 2018). rme-based student worksheets can improve students’ critical thinking skills (samura et al., 2022; susandi & widyawati, 2022) and suitable for use and fall into the category of both and improved critical thinking skills of learners who use rme-based student worksheets as high (sari & putri, 2021). therefore, we develop teaching materials, namely student worksheet, based on the rme approach that can improve students’ critical thinking skills. 2. method this research is development research using the addie model, which aims to develop valid, practical, and effective student worksheets. five stages are applied to developing the student worksheets using the addie model, including analysis, design, development, implementation, and evaluation (branch, 2009). the analysis phase includes material, mathematics curriculum, student characteristics, and work plan analyses. next is the design stage; the student worksheets are designed according to the results analysis that has been done, the required instruments, and the determination of the validator. at the same time, the preparation of student worksheets and validation are included in the development stage. the first author carried out several teaching and learning activities at the implementation stage, including a pretest, student worksheets trial, posttest, and assessment. finally, the evaluation stage focuses on assessing the results of the analyses of validity, practicality, and effectiveness and making improvements to the student worksheets. figure 1 illustrates the research procedure. figure 1. research procedure the population used in this study were seventh-grade students from one of private school in pagaralam, indonesia, which consisted of 10 classes. furthermore, the selected lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 72 research sample was 26 students from class vii 5, which would later be used as a largescale trial class. as for the small-scale trial sample using 22 students of class vii 2. 3. result and discussion 3.1. stage of analysis the analysis stage is the initial stage before designing teaching materials. at this stage, curriculum analysis, analysis of learners’ characteristics, material analysis, and work plan analysis so that the teaching materials can be developed based on the situation’s needs. 3.1.1. curriculum analysis before reviewing the mathematics curriculum applied at this school, teachers were interviewed regarding the curriculum's implementation in schools. the interview results show that the curriculum used in all classes is the 2013 curriculum. furthermore, assessing the set material in the 2013 curriculum and the extent to which the material is taught in class vii by the core competencies (cc) and basic competencies (bc) determined in the curriculum. teachers at this school use cc and bc knowledge and skills in the 2013 curriculum. still, it is not fully implemented because the competency achievement indicators (cai) compiled by teachers are still not included. based on the interview results, the cai is prepared by the bc, which is entirely contained in the curriculum. the cai indicates competence that students must achieve according to the curriculum sequence. the cai compiled in this study are presented in table 1. table 1. curriculum analysis core competencies (cc) basic competence (bc) competency achievement indicator (cai) a. understand knowledge (factual, conceptual, and procedural) based on his curiosity about science, technology, art, culturerelated phenomena, and visible events. b. try, process, and recite in the concrete realm (using, parsing, stringing, modifying, and creating) and the abstract realm (writing, reading, counting, drawing, and composing) according to those studied in the same school and other sources in the same point of view/ theory. a. bc knowledge: 3.4 describes and states sets, subsets, universe sets, empty sets, and set complements using contextual problems. b. bc knowledge: 3.5 describes and performs binary operations on sets using contextual problems. c. bc skills: 4.4 solves contextual problems relating to sets, subsets, universe sets, empty sets, and set complements. d. bc skills: 4.5 resolves contextual problems related to binary operations on the set a. states set and not set. b. determines subsets, universe sets, empty sets, set complements c. determines binary operations on a set d. solves problems related to sets, subsets, universe sets, empty sets, set complements e. resolves problems related to binary operations on the set volume 12, no 1, february 2023, pp. 69-84 73 3.1.2. material analysis the material analysis is based on the cc, bc, and cai that has been prepared at the stage of competency analysis that students must achieve, then analyze what material will be contained in the student worksheets through collecting references related to material that is by the scope of bc. subsequently, determine the content and main materials as well as subsections of the main material to be developed in the student worksheets as listed here: 1) understanding of the set; 2) stating members and not members of the set; 3) notation of the set; 4) determining the empty set, the set universe, subsets and venn diagrams; and 5) define set operations. 3.1.3. analysis of characteristics of learners and work plan analysis at this stage, interviews were conducted with mathematics teachers at this school regarding students’ initial knowledge, prerequisite materials and subject matter on sets, difficulties encountered in learning, and written tests to determine the critical thinking skills of students who will become student worksheet users. according to the teacher, the interview results showed that the student's critical thinking ability was still low. the teacher said that students had difficulties in understanding the questions, namely in the form of students being able to read all the words in the questions but not understanding or understanding the overall meaning of the words in the questions. students cannot write down what is known and what is being asked of the question. in stating the set and registering the members of the stage, students have difficulty writing down what is known and what the question is asking. students are still glued to the teacher’s explanation and have not been able to identify ideas and arguments, perform calculations and draw conclusions. whereas, work plan analysis consists of seven phases: set development goals, design student worksheets, develop student worksheets validation instruments, perform validation, test the validity of student worksheets, implement, test the practicality and effectiveness of student worksheets, and evaluation. 3.2. design stage in the design stage, the student worksheets are designed according to the analysis results that have been done. the developed student worksheets contain set material for class vii and rme characteristics. the first and second authors design the presentation of materials that have elements of rme which include the introduction of concepts through problems to learners, organizing learners to research, group investigation assistance, development and presentation of interpretation and problem solving of learners, as well as the analysis and assessment of the process and results of solving learners’ problems. the characteristic content design of rme in student worksheets is designed as a strategy to align materials and improve learners’ critical thinking skills. the design stage is done by selecting symbols as icons to distinguish between rme characteristics and indicators of critical thinking skills. in addition to the student worksheets design, at this stage, also prepared the design of critical thinking test instruments in the form of pretest and posttest questions, as well as various instrument designs needed in the development of student worksheets, including student worksheets validation sheets by material experts, student worksheets validation sheets by media experts, and student worksheets assessment sheets by learners. at this stage also began to determine the candidate’s validator. there are six validator candidates have lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 74 been determined in this study. the details of each prospective validator can be seen in table 2. table 2. list of validators validator information validator 1 validation of material expert instruments (student worksheets validation sheet by a material expert) validation of media expert instruments (student worksheets validation sheet by media expert) validation of participant response instruments (student worksheets assessment sheet by learners) validator 2 validation of critical thinking test instruments (pretest problem) validation of critical thinking test instruments (posttest problem) validator 3 validation of student worksheets as a media expert validator 4 validation of student worksheets as a media expert validator 5 validation of student worksheets as a material expert validator 6 validation of student worksheets as a material expert 3.3. development stage the development stage involves the activity of translating design specifications at the design stage into physical form. this activity produces a prototype development product in the form of student worksheets based on rme to improve the ability to think critically of set materials; in addition to developing products, material expert validity instruments are also developed, including instruments for the expert media validity, learners’ response questionnaire, pretest and posttest critical thinking skills. after each instrument is declared fit for use, the next step is to prove its validity. the validity of student worksheet products is measured using validated instruments, namely questionnaire. evaluation of the validity of student worksheets by material experts is contained in table 3. table 3. recapitulation of student worksheets validity by materials expert validator total score category validator 5 60 good validator 6 61 good the total score of the two validators 121 average 60.5 validity category valid evaluation of the validity of student worksheets products by media experts is contained in table 4. volume 12, no 1, february 2023, pp. 69-84 75 table 4. recapitulation of student worksheets validity by media experts validator total score category validator 3 122 good validator 4 134 excellent the total score of the two validators 256 average 128 validity category highly valid table 3 shows that the average score of both validator expert materials indicates that student worksheet products fall into the valid category. table 4 shows that the average value of both media expert validators indicates that student worksheet products fall into the category of highly valid, so student worksheet products have been valid and worthy of the aspects of material experts and media experts. in contrast, the results of media expert validation in giving comments or suggestions can be seen in table 5. table 5. comments or suggestions from media experts validator comments/suggestions validator 3 add the back cover of the student worksheets. in the instructions for using student worksheets, it would be nice to explain every symbol. for example, in the use of context, what does it mean? what should learners do in the use of this context? in the concept map section, it is seen that the set material consists of knowing the set and the operation of the set. for the empty set and so on, there is no concept map from where and where? the builder needs to create a concept map that is easier to understand. validator 4 the back cover does not exist. spacing between text raised at least 1.5 spaces. on page 7 of the 2021 calendar image should be brought up. after validating the product, improvements were made based on material and media experts’ comments and suggestions. some of the improvements can be seen in the following image. the student worksheet back cover was added according to comments and suggestions from experts, as shown in figure 2. figure 2. before revision (left image) and after revision (right image) lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 76 in the instructions for using the student worksheets, each symbol is explained according to the validator’s input. furthermore, improvements were made to the concept map section to make it easier for learners to understand. 3.4. implementation stage the student worksheets validated and declared eligible for use by the validator can be tested in large-scale or experimental classes. however, before being tested in a largescale class, the student worksheets were first tested in a small-scale class, namely on 22 randomly selected students. after carrying out teaching and learning activities using student worksheets, they filled out student response questionnaires to assess the practicality of student worksheets, which material experts and media experts had previously validated. after implementing a small-scale class trial and then applying it to a large-scale class with 26 students, all activities carried out by students during learning are centered on student worksheets. the following are the details of the implementation activities in large-scale classes. 3.4.1. first meeting activities the first meeting was held on thursday, july 29, 2021. at the first meeting, students were given pretest questions. pretest questions were used to determine the material for the initial set of students before learning using the student worksheets. the pretest questions consist of three description questions with a time allocation of 80 minutes. 3.4.2. second meeting activities the second meeting was held on monday, august 2, 2021. the student worksheets were introduced to the students, formed several groups, explained instructions for using the student worksheets, and then students studied the preliminary material. the activities of the second meeting can be seen in figure 3. figure 3. activities of the second meeting volume 12, no 1, february 2023, pp. 69-84 77 3.4.3. third meeting activities the third meeting was held on thursday, august 5, 2021. all class vii 5 students started learning using the student worksheets at this meeting. the first and second authors guide and supervise students in this learning process. we engage students through student worksheets and students study activity sheet 1, which contains set material that is given realistic problems about several menus in the school canteen. 3.4.4. fourth meeting activities the fourth meeting was held on monday, august 9, 2021. at this meeting, students continued learning using student worksheets, namely studying activity sheet 2, which contains set material given realistic problems. students pay attention to a calendar included in student worksheets, then follow the steps consisting of the use of context, use of models, use of learner construction, interactivity, and linkage. they said that learning sets using student worksheets equipped with steps to solve each problem made it easier to understand the set material. student activities at this meeting can be seen in figure 4. figure 4. activities of the fourth meeting 3.4.5. fifth meeting activities the fifth meeting was held on thursday, august 12, 2021. at this meeting, students resumed learning using student worksheets, namely studying activity sheet 3, which contains set material given realistic problems about various kinds of empek-empek (a type of indonesian snack), with the aim that students can solve contextual problems related to set operations. this student worksheet is equipped with steps from the characteristics of rme, and the objective is to train students to develop their thinking skills. 3.4.6. sixth meeting activities the sixth meeting was held on monday, august 16, 2021. at this meeting, students were given posttest questions. posttest questions are used to see student learning outcomes after using student worksheets. the posttest questions consist of three description questions lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 78 with a time allocation of 80 minutes. student activities working on posttest questions can be seen in figure 5. figure 5. students working on posttest questions 3.5. evaluation stage in the evaluation stage, it is done by comparing the value of the results of critical thinking ability tests in the form of pretest that have been validated in the experimental class with the value of the critical thinking ability test results in the form of posttest that have also been validated in the experimental class. the first author gave the product to the experimental class before giving a posttest problem to the experimental class. the comparison between the pretest and posttest values is used to see the effectiveness of the use of the product in improving the ability to think critically in the set material. 3.6. discussion this research was conducted using the addie model in developing student worksheets based on realistic mathematics education, which is used to improve students' critical thinking skills in learning a set. risdiyanti and prahmana (2021) designed a learning trajectory set using rme with indonesian shadow puppets and mahabharata stories as a learning context. it means that rme can be used as a learning approach in teaching the learning of a set. the developed student worksheets have undergone several phases to obtain valid, practical, and effective criteria. the addie model is divided into five stages: analysis, design, development, implementation, and evaluation (peterson, 2003). in the initial stage, namely analysis, the researcher analyzed the school's needs as a reference for developing products. needs analysis includes an analysis of the competencies students must achieve, material analysis, and student characteristics that consistently show that students need worksheet teaching materials that can improve the ability to think critically about a set. it is known that when students are given math problems and the results of their work when examined from indicators of critical thinking skills, most students have yet to be able to determine what is known or do analysis, evaluation, and conclusions. because of these conditions, the researchers start to think about finding solutions to these problems. one is by developing teaching materials in the form of student worksheets based on realistic mathematics education with the hope that students would find it easier to learn set material and be more active in the learning process (prahmana et al., 2020). before researchers develop students' volume 12, no 1, february 2023, pp. 69-84 79 worksheets based on realistic mathematics education, researchers determine the critical thinking skills students possess through pre-test and post-test questions. expert validators have validated the pre-test and post-test questions, then tested on students who have received set material to test the validity and reliability. furthermore, the researcher carried out the product design stage. at this stage, the researcher designed the student worksheet according to the needs analysis (spatioti et al., 2022). this worksheet is designed based on facts in the field that students need, which makes the problem as a starting point and can improve students' critical thinking skills. at this stage, the researcher made a concept map of the student worksheet presentation, determined the format, and made learning stages in the student worksheet based on realistic mathematics education to improve students' critical thinking skills. in the development stage, the first activities were the content and media expert validation tests (davis, 2013). the content expert validation test results obtained an average rating of two validators 60.5 in the good category and an average rating of two media expert validators, 128 in the very good category. the acquisition of this value indicates that the developed student worksheet is content and media valid. a student worksheet is said to be valid if the average score of the validity assessment meets the minimum standards of good (mulbar & zaki, 2018). at the implementation stage, the activities carried out were the implementation of student worksheets in small classes, and then a small class practicality test was carried out. the small class practicality test results obtained an average rating of 77.8 with the good rating criterion, so the student worksheets prepared are practically used in small classes. then the student worksheets are applied to the large or experimental class, and a significant class practicality test is carried out. the large class practicality test results obtained an average rating of 80.07 with the good rating criterion, so the student worksheets prepared are practically used in large classes. the student worksheets are practical if the average practicality assessment score meets the minimum standards of good (hasibuan et al., 2019). the final stage is the evaluation stage. this stage determines the effectiveness of student worksheets based on realistic mathematics education on students' critical thinking skills. the researchers gave the post-test to the students and then analyzed it to obtain data on the effectiveness of the student worksheets. student worksheets' effectiveness data were obtained from the results of the experimental class post-test scores. the technique used by researchers to test effectiveness is comparing students' critical thinking skills test results in the form of pre-test questions. students did it before being given student worksheets based on realistic mathematics education developed with the results of critical thinking ability tests in the form of post-test questions after being given a realistic mathematics education-based student worksheets developed. at this stage, to compare the results of the critical thinking ability test in the form of pre-test questions and the results of the critical thinking ability test in the form of posttest questions, the researcher conducted a paired sample t-test. the paired sample t-test shows that the value of sig. (2-tailed) is 0.000 < 0.05, so a significant difference exists between the results of learning mathematics in the pre-test and post-test data. the average pre-test score obtained by students is 37.31, while the average post-test score obtained by students is 67.50. the n-gain value is 0.48, so the increase is included in the moderate category when viewed from the ability to think critically. therefore, based on the data obtained, student worksheet is effective in terms of students' critical thinking skills. in other words, there was an increase between the results of the critical thinking ability test in the form of pre-test questions before being given learning materials in the form of a realistic mathematics education-based student worksheet and the results of the critical thinking ability test in the form of post-test questions after being given lestari, prahmana, chong, & shahrill, developing realistic mathematics education … 80 learning materials in the form of realistic mathematics education-based student worksheet. it aligns with the results of erita et al. (2022) research that learning using a realistic mathematics education-based student worksheet can improve students' critical thinking skills. it means there is an increase in the critical thinking skills in the experimental class before teaching materials in the form of student worksheets based on realistic mathematics education and after being given teaching materials in the form of student worksheets based on realistic mathematics education (hasibuan et al., 2019). they can also use the developed student worksheet with these results because it meets the product eligibility requirements: valid, practical, and effective. it shows that student worksheets based on realistic mathematics education can improve students' critical thinking skills. the results of this study indicate that the characteristics of realistic mathematics education contained in the realistic mathematics education approach theoretically have elements that can grow indicators included in critical thinking skills. this happens because realistic mathematics education brings students to real-world experiences daily so that mathematics lessons are not separated from students' daily lives (prahmana et al., 2020; samura et al., 2022; susandi & widyawati, 2022; zetriuslita et al., 2018). this research aligns with development research conducted by wewe and juliawan (2019) that realistic mathematics education-based worksheets to improve critical thinking skills meet good criteria and are suitable for use as alternative mathematics teaching materials (basuki & wijaya, 2019). therefore the results of this study add to empirical evidence, which states that the realistic mathematics education approach implemented in student worksheets can improve students' critical thinking skills. 4. conclusion students can use these worksheets based on rme to improve their critical thinking skills because of their validity, practicality, and effective criteria. the rme-based student worksheets met the valid criteria. it is measured by assessing material experts, who obtained an average score of 60.5 with a good category, and assessments from media experts with a score of 128 with excellent types. the student worksheets met the practical criteria based on the learner’s response questionnaire assessment, with an assessment score of 80.07 and a maximum score of 100. the student worksheets developed based on rme on quality critical thinking skills judged by effectiveness. there was a significant difference between the pretest and post-test scores for critical thinking skills. the average post-test score is greater than the average pre-test score, which shows an increase in the moderate category between the pre-test and post-test scores. therefore, the student worksheets meet the effective criteria. the findings of this study also emphasize the effect of developing teaching materials based on rme to improve students’ critical thinking skills with positive outcomes. acknowledgements the authors thank chika rahayu, m.pd., and meryansumayeka, s.pd, m.sc., as experts in validating our e-module focusing on the content. furthermore, we thank dr. sri adi widodo, m.pd., and dr. bambang riyanto, m.pd., as the expert to validate our e-module focusing on the media. we also would like to thank anggit prabowo, m.pd., and indah widyaningrum, m.pd. as the expert to validate the research instrument for this research. lastly, we thank universitas ahmad dahlan and smp muhammadiyah pagaralam for providing facilities and opportunities to develop and carry out this research. volume 12, no 1, february 2023, pp. 69-84 81 references basuki, w. a., & wijaya, a. 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(2018). association among mathematical critical thinking skill, communication, and curiosity attitude as the impact of problem-based learning and cognitive conflict strategy (pblccs) in number theory course. infinity journal, 7(1), 15-24. https://doi.org/10.22460/infinity.v7i1.p15-24 zulkardi, z. (2002). developing a learning environment on realistic mathematics education for indonesian student teachers. doctoral dissertation. enschede: university of twente. retrieved from https://repository.unsri.ac.id/871 https://doi.org/10.22460/infinity.v7i1.p15-24 https://repository.unsri.ac.id/871 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p275-290 275 grade 10 namibian learners' problem-solving skills in algebraic word problems hesekiel k. iilonga*, ugorji i. ogbonnaya university of pretoria, south africa article info abstract article history: received apr 4, 2023 revised jun 1, 2023 accepted jun 12, 2023 published online jul 20, 2023 this study investigated grade 10 namibian learners' problem-solving skills in algebraic word problems. a sample of 351 grade 10 learners from ten secondary schools in the ohangwena region in namibia participated in this study. the study followed a qualitative approach and adopted polya's problemsolving model as the framework. the data were collected using the algebra word problem-solving test and interview. the findings showed that the learners needed better problem-solving skills in algebraic word problems. only 6% and 7% of the learners showed an ability to understand the problems and devise a plan, respectively, while only 5% could carry out the plans. based on the findings, it is recommended that teachers introduce learners to polya's steps of problem-solving and incorporate word problems into mathematics teaching. keywords: algebraic word problem, polya’s problem-solving steps, problem-solving, problem-solving skills this is an open access article under the cc by-sa license. corresponding author: hesekiel k. iilonga, department of science, mathematics and technology education, university of pretoria groenkloof campus, corner of george storrar drive and leyds street, groenkloof, pretoria, south africa. email: iihesekiel@gmail.com how to cite: iilonga, h. k., & ogbonnaya, u. i. (2023). grade 10 namibian learners' problem-solving skills in algebraic word problems. infinity, 12(2), 275-290. 1. introduction problem-solving skills refer to the ability to use specific approaches and strategies to arrive at a meaningful solution(s) to a situation or problem (i̇ncebacak & ersoy, 2016). this requires the ability to identify the nature of a problem, deconstruct it (break it down), and design an effective set of activities to handle the obstacles associated with the problem (abazov, 2016). problem-solving skills are important in both school and real-life situations. it is posited that a person with problem-solving skills develops into a self-confident, creative, and autonomous thinker (yöyen et al., 2017). great problem solvers strive to discover and understand the underlying causes of a difficult situation; the essence of a specific problem that can be recognised, addressed, and finally, resolved (al-mutawah et al., 2019). when the educational policy for independent namibia was formulated in 1996, the ministry of education instucted that "the namibia national curriculum guideline should provide opportunities for developing essential problem-solving skills in grade 10–11 https://doi.org/10.22460/infinity.v12i2.p275-290 https://creativecommons.org/licenses/by-sa/4.0/ iilonga & ogbonnaya, grade 10 namibian learners' problem-solving skills … 276 mathematics curriculum" (national institute for educational development, 2016, p. 46). however, problem-solving skills cannot be developed in isolation; instead, they must be developed across the mathematics curriculum for namibian learners to improve learners’ skills in this regard. the use of word problem-solving activities in mathematics could be one way of developing problem-solving skills. mathematical word problems are special types of mathematical problems; they are "verbal descriptions of problem situations; they refer to an existing or imaginable meaningful context; and they can be closed or open, algorithmic or non-algorithmic" (van dooren et al., 2019, p. 99). palm (2006) elaborates that a word problem is a set of sentences that describes a ‘real-life’ scenario in which a problem must be solved using mathematical calculations. due to the fact that word problems sometimes incorporate a form of narrative, they are sometimes referred to as story problems, and can vary in terms of the amount of language employed. the use of natural language in mathematics instruction is typically justified in two ways. first, by bringing the actual world into classrooms, word problems aid learners in developing links between the real world and mathematics (krawitz et al., 2018). second, some scholars suggest that word problems may provide tangible meaning to abstract mathematical things (greer et al., 2002). several studies show that for many high school learners, solving word problems is one of the biggest challenges in algebra (bush & karp, 2013; capraro & joffrion, 2006; jupri & drijvers, 2016; van amerom, 2003). this has been corroborated in namibia, as some studies have shown that namibian learners have difficulties when it comes to solving algebra word problems (albin & von watzdorf, 2019; sikukumwa, 2017). as stated in the examiner’s reports from 2014 to 2018, algebra word problems were found to be one of the most challenging topics in the junior secondary certificate mathematics examination in namibia (directorate of national examinations and assessment, 2014, 2015, 2016, 2017, 2018). the reports note that algebra word problems were always poorly addressed. however, the examiner’s reports did not specify the learners’ skills in solving algebra word problems. it is against this backdrop that this study explored grade 10 learners’ problem-solving skills in algebraic word problems. specifically, it explored the learners’ abilities to understand algebraic word problems, devise a plan to solve the problem, and carry out the plan. some scholars have investigated learners’ problem-solving skills at various levels of schooling. for example, lupahla (2014) used polya’s problem-solving model to investigate the algebraic problem-solving skills of grade 12 learners in the oshana region in namibia. the results show that only 34.6% and 29.1% of the learners, respectively, were successful in the ‘understanding the problem’ and ‘devising a plan’ steps of polya’s model. alternatively, 26.1% and 23.8% learners, respectively, successfully executed the devised plan, and were able to look back at the steps taken and reevaluate if necessary. in a similar study, lubis et al. (2017) explored junior high school students’ problem-solving skills in linear equations and inequalities in one variable using polya’s problem-solving steps. their study showed that the performance of the students decreased from the first step (understanding the problem) to the third step (carrying out the plan) of the problem-solving process. however, they performed better in the last step (looking back) than in the second and third steps. sipayung and anzelina (2019) investigated the mathematics problem-solving skills of junior high school learners in nusantara lubuk pakam, indonesia. the study was based on using a realistic mathematics approach to three aspects of integers (the recognition and comparison of integers, the addition and subtraction of integers, and the multiplication and division of integers). the study showed that the percentages representing the average score of the learners’ problem-solving abilities in the four steps were 81.6%, 73.5%, 78.6%, and volume 12, no 2, september 2023, pp. 275-290 277 76.1% respectively. this finding suggests that using a realistic approach to mathematics teaching augments learners’ skills in mathematical problem solving in terms of integers. riyadi et al. (2021) conducted a qualitative study to examine learners’ problemsolving skills in algebra word problems at third, fourth, and fifth-grade elementary school level in indonesia using polya's four-steps model. the study revealed that the learners' problem-solving skills across the grades decreased from the first step (understanding the problem) to the fourth step (looking back). hendriana et al. (2018) investigated the mathematics problem-solving ability of grade 12 senior high school learners in cimahi, indonesia. the data were collected using a test. the investigation found that the learners’ mathematics problem-solving ability was extremely poor, and that the typical learner could not comprehend the problems or express them in mathematical form. in their study of elementary school teachers’ mathematics problem-solving ability in indonesia, based on polya's steps, yayuk and husamah (2020) found that only 5.3% of the participants were successful at understanding the problem, devising a plan, and carrying out the plan. only 8% of the teachers were successful in re-checking answers. in general, the study showed that the pre-service teachers had poor mathematics problem-solving abilities, based on polya’s problem-solving steps. a similar study was conducted by akyüz (2020) to determine pre-service elementary school mathematics teachers’ mathematical problemsolving performance in a university using polya’s problem-solving steps. the study showed that the pre-service teachers’ performance decreased from the first step to the fourth step of polya’s steps, and that the overall problem-solving performance level was low. in general, most of the studies highlighted here, and many more (e.g., aljaberi, 2015; hijada jr & cruz, 2022; pentang et al., 2021) suggest that most students, at different school levels, have poor mathematical problem-solving abilities. this study was framed using polya’s (1945) problem-solving model. the model consists of four steps: understanding the problem, devising a plan, carrying out the plan, and looking back. this study focused on the first three steps. understanding the problem refers to people’s ability to figure out what is being asked, what is known, what is not known, and what type of answer is required (polya, 1957). individuals are able to understand a problem when they are familiar with the relevant vocabulary, when they know what the question is asking for (required or unknown), and know what information is in the problem (given or known) to help them solve it. this first step concerns comprehending the problem's given circumstances and limitations, expanding on the objective and the unknown, and making the required assumptions (schoenfeld, 2014). understanding the problem is a key part of finding the solution to the problem. in other words, before a problem can be solved, the problem solver must understand the problem (berlinghoff & gouvêa, 2021; niss & højgaard, 2019; polya, 1981). lee (2016) highlights that in the ‘understanding the problem’ step, polya proposes that teachers should first ask learners questions, for example: do you understand all the terminologies used to denote the problem? could you repeat the problem in your own words? can you think of an image or a diagram that would help you understand the problem? what is it that you are required to do or find? what unknowns are there? and, what information is missing or required, if any? according to daulay and ruhaimah (2019), understanding the problem is split into two phases: getting to know the problem, and searching for a greater understanding. the phase of getting to know the problem is where a problem solver can restate the problem in their own words, while the phase of searching for a greater understanding is where the problem solver can identify what is required (what is unknown or what you are required to find) and known information (the given information that can help you to solve the problem). iilonga & ogbonnaya, grade 10 namibian learners' problem-solving skills … 278 devising a plan refers to coming up with a way (translating the problem) to solve the problem by setting up an equation, drawing a diagram, and making a chart (polya, 1957; wickramasinghe & valles, 2015). ersoy (2016) points out that at the stage of devising a plan, learners are supposed to choose which actions, such as computation, sketching, and so on, to perform to attain the desired result. nurkaeti (2018) states that, in order for a problem solver to solve a problem successfully, the person should formulate a plan, which could be in the form of a drawing or a math-based solution to the problem; and figure out the concepts of the question, formula, or mathematical ideas that will be used to solve the problem. polya mentions that there could be many ways to solve a rational problem. the best way to develop the ability to choose an effective plan is for learners to solve as many problems as possible (maulyda et al., 2019). learners will find it increasingly convenient to choose a proper plan to solve a problem once they are exposed to different problems. a problem solver is said to have a plan when he/she knows what calculations, computations, or constructions they must perform to get the unknown, or at least know the outline. understanding a problem and coming up with a solution might be a difficult and drawn-out process. however, the main key to solving a problem is to devise a plan. this idea may emerge slowly, after unsuccessful trials and a time of delay, or it may suddenly appear in a flash as a "bright idea" (polya, 1973, p. 8). a learner may start using one plan and then realise that the plan does not fit the information given or lead to the desired solution. in this case, the learner has to choose a different plan. in some instances, gray (2018) mentions that a variety of techniques might need to be used in conjunction with the developed plan, such as guessing and checking, looking for patterns, making an orderly list, drawing a picture, eliminating possibilities, using a model, working backward, using a formula, or being ingenious. this will allow learners to solve the given problem and improve their problem-solving skills. carrying out the plan is putting the strategy into action by doing any necessary tasks or calculations, carrying out each step of the plan as one goes, and keeping an accurate record of one’s work (polya, 1957). polya (1973) emphasises that, normally, this step is simpler than devising the plan. all that is needed, in general, is care and diligence as the problem solver should have the expertise needed. therefore, the problem solver must keep the selected plan in mind at all times and use it. if it does not continue to work, they must discard it and choose another one. in'am (2014) clarifies that ‘carrying out the plan’ is how a learner or problem solver puts the good plan they have chosen into action. in'am also explains that it does not help to understand a problem and devise a good plan to solve it if the plan is not implemented. instead, an effort should be made to put the plan into action. according to lupahla (2014), once a problem has been carefully scrutinised and a plan has been developed, assuming the plan is appropriate for the problem, carrying out and implementing the plan will be a fairly simple process. putting the plan into action can be viewed as a mathematical operation that must be performed to obtain the result of completion (nurkaeti, 2018). polya (1945) referred to ‘looking back’ as revisiting the completed answer, and evaluating and re-examining the result, as well as the road that led to it. chang (2019) defines looking back as a reflection process in problem solving. it is the process of looking over the results, verifying the solution against the problem, comparing the problem with the offered solution, comparing the mathematical terms in the problem with the answer, and being confident in the response provided (nurkaeti, 2018). volume 12, no 2, september 2023, pp. 275-290 279 2. method this research employed a qualitative approach and descriptive research design. the participants comprised 351 learners from ten sampled secondary schools in the ohangwena region, namibia. all 351 learners wrote an algebra word problem-solving test, while 20 randomly selected learners (two from each school) were interviewed after the test. the test was not time constrained; the learners were allowed to submit their scripts when they felt that they had attempted the questions to the best of their abilities. this was done to enable the learners to demonstrate all of their knowledge and skills in solving the problems. the test and interview questions were developed by the first author with the guidance of the namibia senior secondary certificate grade 10 – 11 mathematics syllabus, and polya’s problem-solving steps. the test comprised six algebraic word problems (see figure 1). figure 1. algebra word problem solving test questions the interviews were semi-structured in nature, and were used to further explore the learners’ understanding of the problem, devising a plan, and carrying out the plan in solving the test questions. some of the questions asked at the interviews were: “restate the problem (question) in your own words”, “how did you approach the problem?”, “what did you do to solve the problem?”, and “are there any steps that you know or used to solve the problem?” the learners’ solutions to the test questions were deductively analysed and categorised using a modified rubric adapted from charles et al.’s (1987) and sumaryanta’s (2015) problem-solving analysis rubrics. for the ‘understanding the problem’ category, the sub-categories were: ‘identified all knowns and unknowns’, ‘identified some knowns and unknowns’, ‘identified no known or unknown,’ and ‘did not attempt the question’. for the ‘devising a plan’ category, the sub-categories were: ‘the statement is translated into a correct algebraic form’, ‘the statement is translated into a partially correct algebraic form’, ‘the iilonga & ogbonnaya, grade 10 namibian learners' problem-solving skills … 280 statement is translated into an incorrect algebraic form,’ and ‘no translation of the statement’. for the ‘carrying out the plan’ category, the sub-categories were: ‘correct procedures and correct answer’, ‘correct procedures and incorrect answer’, ‘incorrect procedures and correct answers,’ and ‘wrong procedures with the wrong answers or problems not attempted’. an interpretive data analysis process was used to make sense of the learners' responses to the interview questions. 3. result and discussion 3.1. result the findings are presented according to the first three steps of the polya problemsolving model explored in this study. understanding the problem table 1 shows the learners' performance in the ‘understanding the problem’ step in all six test questions. the table shows that most of the learners an average of 260, or 74% were not able to identify the known or unknown conditions in the problems. table 1. descriptive statistics of the learners’ performance in understanding the question identified all knowns and unknowns identified some knowns and unknowns identified no known or unknown did not attempt the question q1 36 89 222 4 q2 6 63 282 0 q3 31 72 245 3 q4 21 94 235 1 q5 3 39 308 1 q6 21 61 266 3 mean 20 70 260 2 an average of 70 learners (20%) partially identified the known or unknown, while only an average of 5.6% (20 learners) showed an understanding of the problems by identifying all known and unknown terms in the questions. some learners’ understanding of the problem presented in question 2 is shown in figure 2. volume 12, no 2, september 2023, pp. 275-290 281 (a) all known and unknown identified (b) known or unknown partially identified (c) no known or unknown identified figure 2. examples of learners’ solutions showing their understanding of question 2 the learner in figure 2(a) identified the present ages of the father, son, and daughter, as well as their ages in the past three years. in figure 2(b), the learner only identified the present age of the son. the learner used different variables to identify the different ages of the father, son, and daughter, which made it difficult to solve the problem. the learner in figure 2(c) could not identify the unknown and the known in the question. most of the learners had similar solutions to the problems, showing a lack of understanding of the questions. furthermore, the interview revealed that most of the learners lacked an understanding of the questions. one of the learners interviewed said, “…i seriously don’t understand this question. once i realised the question was tricky, i just tried my best to do some calculations. i'm just not sure if it’s correct or not.” even when the learners were able to restate the problem in their own words during the interview, most of them could not identify the given information from the problems. devising a plan the descriptive statistics of the learners’ performance in devising a plan are presented in table 2. the table shows that most of the learners could not devise a plan to solve the problem as, on average, 260 (74%) of them translated the problems into incorrect algebraic forms; and, on average, only 24 (approximately 7%) of them successfully devised plans to solve the problems by translating them into the correct algebraic equations. iilonga & ogbonnaya, grade 10 namibian learners' problem-solving skills … 282 table 2. learners’ performance in devising a plan (n =351) statement is translated into a correct algebraic form statement is translated into a partially correct algebraic form statement is translated into an incorrect algebraic form no translation of the statement at all q 1 33 97 217 4 q 2 14 64 273 0 q3 36 51 261 3 q 4 30 93 227 1 q 5 3 44 303 1 q 6 30 37 281 3 mean 24 64 260 2 figure 3 shows how some learners devised a plan to solve question 1. the learner in figure 3(a) translated the problem statement into the correct algebraic representations; x + y = 14 and (2 + y) + y = 14. (a) algebraic statement translated into correct algebraic form (b) algebraic statement translated into partially correct algebraic form (c) algebraic statement translated into an incorrect algebraic form figure 3. examples of learners’ devised plans to solve question 1. the learner in figure 3(b) translated the problem statement into a partially correct x + y = 14, continuing to write another expression, y + 2 = x 2, which leads to y x = -2, which is wrong. alternatively, the learner in figure 3(c) translated the statement into an incorrect x + 14 (y + 14), following this up with x + 14y = 28, which does not make sense. during the interview, only five learners out of the 20 interviewed said that they had deduced the formulae or equations to answer the problems presented. the remaining 15 learners explained that they simply solved the problems by guessing the answers. this volume 12, no 2, september 2023, pp. 275-290 283 further demonstrates that most of the learners could not devise a plan for solving algebraic word problems. carrying out the plan table 3 shows the learners' overall performance in the step of carrying out the plan across all six test questions. most of the learners, 283 (81%), on average, followed incorrect procedures to arrive at incorrect answers in their attempt to solve the problems. a few leaners (18, accounting for 5%) used incorrect procedures, but got the final answeres correct. only 17 learners (approximately 5%) used the correct procedure to get the correct answers to the questions. in all questions, the overwhelming majority of the learners' (between 70% ‒ 95%) did not use a correct procedure to attempt any of the questions. table 3. learners’ performance in carrying out the plan (n =351) correct procedures and correct answers correct procedures and incorrect answers incorrect procedures and correct answers wrong procedures with the wrong answers or problems not attempted q 1 34 72 13 232 q 2 5 24 13 309 q3 27 14 12 298 q 4 24 36 40 251 q 5 1 18 14 318 q 6 12 34 13 292 mean 17 33 18 283 some examples of how the learners carried out their plan to answer question 6 are shown in figure 4. the learner in figure 4(a) showed an ability to carry out the devised plan to solve the problem. (a) correct procedures and correct answer (b) correct procedures and incorrect answer iilonga & ogbonnaya, grade 10 namibian learners' problem-solving skills … 284 (c) incorrect procedure and answer figure 4. examples of learners’ work on carrying out the plan for question 6 the learner in figure 4(b) used the correct procedure, but made an incorrect calculation that led to the incorrect answer. figure 4(c) shows a learner who used incorrect procedures and got an incorrect answer. most of the learners did not seem to have followed a well-defined plan for solving the problem, as confirmed in the interview. one of the learners replied, “i just did my calculation,” when asked if he followed any specific steps in solving the problem. another learner stated, “i just guessed by putting the numbers on one side and the letter (variables) on the other side of the equal sign”. when further asked if she used any equation or formula to solve the problem, the student asked, “equation from where? no, there were no equations given in my paper”. in general, the step of carrying out a devised plan was not successfully performed as most of the learners solved the problems without following any derived plan, resulting in incorrect steps being followed to solve the problems. 3.2. discussion the study revealed that the learners had poor mathematical problem-solving skills concerning understanding a problem. most of the learners could not identify the known(s) and unknown(s) of the algebraic word problems. this finding corroborates those of hendriana et al. (2018), lupahla (2014), and yayuk and husamah (2020). furthermore, it was found from the interview that although some learners could state the problem in their own words, only a few could identify the known and what was asked. this is in alignment with peterson et al.’s (2017) finding that most learners find it difficult to understand mathematical word problems and communicate their views in writing. similarly, in their study, kusumadewi and retnawati (2020) concluded that learners' problem-solving ability was low due to their difficulty locating keywords from the problems. this resulted in learners guessing the solution without completing the process. some learners failed to restate the problem in their own words since they did not understand the question. learners' reading skills are a crucial element of problem solving (steinbrink, 2009); it may be the case that some of the learners were not able to read and understand the questions due to literacy problems. this concurs with meutia et al.’s (2020) observation that the learners in their study could not express their understanding of the problems by restating the questions. in addition, it was found that most of the learners in the current study were not able to devise a plan to solve the algebraic word problems. this finding aligns with those of previous studies, for example, those of hendriana et al. (2018), lupahla (2014), and yayuk volume 12, no 2, september 2023, pp. 275-290 285 and husamah (2020). this is not surprising because most of the learners did not show any understanding of the problems, which is the foundation for devising and caring out a plan. brijlall (2015) notes that learners' challenges in understanding mathematical problems prevent them from taking the essential measures to solve these problems. if a learner does not know what the question is asking, it will be difficult for them to come up with a plan that can be used to solve the problem. similarly, nikmah et al. (2019) have noted that learners have difficulties identifying the known and unknown(s) in the first step, making it difficult for them to come up with an effective plan that will allow them to solve the problem. furthermore, the study found that the learners’ overall performance in carrying out a plan was dismal. this might be because most of the learners could not establish a plan to solve the problems. this is in agreement with that of franestian et al. (2020), who find that students' performance in carrying out a plan was poorer than their performance in the first two stages. a similar finding was also made by khabibah et al. (2018), who showed that the students in their study could not implement a plan to solve the given problems. in general, the learners’ problem-solving performance decreased from the first step of the polya problem-solving steps to the third step, as was found in other studies (akyüz, 2020; lubis et al., 2017; lupahla, 2014; riyadi et al., 2021). this buttresses the hierarchical nature of polya’s problem-solving steps in that success in subsequent steps depends much on success in the preceding steps of the problem-solving process. as noted by tambychik and meerah (2010), if the first two steps of the problem-solving process are challenging for learners, the third step, and consequently the fourth step, will also be challenging. 4. conclusion one of the main goals of 21st century education is the development of learners’ problem-solving skills (gunawan et al., 2020; kivunja, 2015; rahman, 2019). in namibia, promoting learners’ mathematical problem-solving skills is a key issue in the school curriculum. however, research on namibian learners’ mathematical problem-solving skills is sparse. therefore, this study investigated grade 10 namibian learners' problem-solving skills in algebraic word problems. it was found that the learners' problem-solving skills in algebraic word problems, specifically concerning understanding the problem, devising a plan, and carrying out the plan, were extremely poor. in addition, it was found that the learners’ problem-solving performance decreased from the first step of polya’s problemsolving steps to the third step. one way of improving learners’ computational skills at primary and secondary school level is by using didactical games (hooshyar et al., 2021; yeh et al., 2019). we recommend that mathematics teachers in namibia incorporate educational games into their mathematics instruction to enhance learners’ computational skills. to enhance learners’ understanding of mathematical word problems, which was found to be a major problem for the learners in this study, we recommend that when teaching mathematical word problems, teachers should include the translation of mathematical word problems from english (the language of teaching and learning) to the native language of the learners, and vice versa, before translating the problems into algebraic form. in addition, we recommend that teachers model polya’s problem-solving steps when teaching algebra word problems in particular, and mathematics in general. references abazov, r. 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(2019). enhancing achievement and interest in mathematics learning through math-island. research and practice in technology enhanced learning, 14(1), 1-19. https://doi.org/10.1186/s41039-019-0100-9 yöyen, e. g., azakli, a., üney, r., demirci, o. o., & merdan, e. (2017). ergenlerin kişilik özelliklerinin problem çözme becerisi üzerine etkisi. doğu anadolu sosyal bilimlerde eğilimler dergisi, 1(1), 75-93. https://doi.org/10.17478/jegys.665833 https://doi.org/10.1186/s41039-019-0100-9 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p261-274 261 mathematics teacher educators’ noticing of pedagogical content knowledge on hierarchical classification of quadrilateral rooselyna ekawati1, ahmad wachidul kohar1*, tatag yuli eko siswono1, agung lukito1, kai-lin yang2, khoirun nisa1 1universitas negeri surabaya, indonesia 2national taiwan normal university, taiwan article info abstract article history: received apr 2, 2023 revised jun 1, 2023 accepted jun 12, 2023 published online jul 18, 2023 this study aims to investigate mathematics teacher educators’ (mte) knowledge in noticing preservice teachers’ pedagogical content knowledge (pck) on the hierarchical classification of the quadrilateral. a multiple case study was conducted to analyze the responses of ten mtes in an online moderated-forum group discussion (m-fgd) from their written work on the mte-pck test completed prior to the m-fgd. the pck test consisted of two tasks: the task that examines mtes’ knowledge to predict pre-service teachers’ reason in representing the hierarchical classification of quadrilateral in venn diagrams, and the task that examines mtes’ knowledge in making a flowchart as a recommendation to mathematics teacher to analyze the validity of quadrilateral classification. results show that the mtes indicate two considerations of noticing pre-service teachers’ pck on the quadrilateral classification: by definition and properties of quadrilaterals and by the visual appearance of quadrilaterals. despite this, 20% of them were indicated to perform a lack of understanding of the hierarchical classification of quadrilaterals, as indicated by invalid flowcharts of validating the hierarchical classification of the quadrilateral. keywords: mathematics teacher educator, pedagogical content knowledge, quadrilateral classification this is an open access article under the cc by-sa license. corresponding author: ahmad wachidul kohar, mathematics education study program, faculty of mathematics and natural science, universitas negeri surabaya jl. ketintang wiyata no.36, gayungan, surabaya, east java 60231, indonesia. email: ahmadkohar@unesa.ac.id how to cite: ekawati, r., kohar, a. w., siswono, t. y. e., lukito, a., yang, k.-l., & nisa, k. (2023). mathematics teacher educators’ noticing of pedagogical content knowledge on hierarchical classification of quadrilateral. infinity, 12(2), 261-274. 1. introduction pedagogical content knowledge (pck) has become one of the focuses of educational research in the last decade. initially developed by shulman (1986) with seven categories of knowledge as the basic knowledge for teachers to teach, several researchers https://doi.org/10.22460/infinity.v12i2.p261-274 https://creativecommons.org/licenses/by-sa/4.0/ ekawati et al., mathematics teacher educators’ noticing of pedagogical content knowledge … 262 have then developed various frameworks to understand the knowledge teachers need to teach (e.g., chick et al., 2006; rowland, 2013). chick et al. (2006) developed a pck framework that identifies various fields of knowledge in three different categories: clearly pck, content knowledge used in a pedagogical context, and pedagogical knowledge used in a content context. meanwhile, rowland (2013) uses a framework called "the knowledge quartet" which consists of four aspects: foundation, transformation, connection, and contingency, to identify teachers' pck. the framework has been used to interpret teacher pck through quantitative and qualitative exploration and observing teaching practice (maher et al., 2022; mishra, 2020). it has been proven to be effective as a means of identifying the pck of mathematics teachers. despite being a relatively new area of study, the knowledge needed by mathematics teacher educators to teach pck to their prospective teachers has seen substantial advancement in recent years. several researchers have also proposed various analytical frameworks, such as chick and beswick (2018) whose analytical framework describes an analogous relationship between pck for mathematics teachers’ pck for mte (mtepck). this framework was adapted from the pck framework for mathematics teachers to the mtepck framework. some researchers use this framework to explore mtepeck, such as that done by amador et al. (2021) to characterize the noticing of mathematics teacher educators (mte) and their ability to interpret students' thinking, pascual et al. (2021) to analyze mte pedagogical knowledge specifically on the topic of symmetry, and spangenberg (2021) who investigated the mte-pck of trigonometry. these three studies are found important as they serve as the basis for mathematics teacher educators' practice, including classroom-based examples and strategies, and give potential routes for mathematics teacher educators to undertake their own research on the nature and effectiveness of their course activities that promote and enhance prospective teachers' pedagogical topic knowledge. while research on mte-pck has been emerging on the aspects of students’ mathematical thinking and specific content in school mathematics, this type of knowledge is still scarce to be investigated in terms of geographical area. in the context of china, for example, the development of chinese teachers’ mte-pck has been found as one of the successes of the strong collaborative work of the mathematics education research team, which are also noted as mathematics teacher educators (mte) (paine et al., 2015), and mathematics teachers (wu & cai, 2021) to design and implement mathematics lesson although these findings need to be further studied in some untouched region in china (wu & cai, 2016). however, in indonesia, the studies reported on mathematics teacher knowledge reported are still around the pck of teachers, instead of the mtepck. numerous research has been investigated to assess teacher knowledge and how it affects teachers’ teaching practices, one of which indicates the indonesian teachers’ unsuccessfulness in integrating their pck into the teaching practices due to the lack of pck (e.g., ekawati et al., 2015; yunianto et al., 2021), or inconsistency between the pck and their actual teaching (e.g., muhtarom et al., 2019; siswono et al., 2017). thus, a deeper understanding of the underlying reasons for these findings needs to be investigated primarily on the teachers’ engagement and experiences in teacher education, where mathematics teacher educators (mtes) play a significant role in shaping the teachers’ pck (oates et al., 2021). in this regard, the framework of pck is then further developed to identify the pck of teacher educators, including those in indonesia, which in this case, is known as mathematics teacher educators (mte)’ pck. zaslavsky and leikin (2004) describe the mtes’ knowledge as knowledge possessed by mathematics teachers, namely knowledge of school mathematics. meanwhile, the pck of mte is the pck needed by mte to introduce pck to prospective mathematics teachers. this is in accordance with the term pck for volume 12, no 2, september 2023, pp. 261-274 263 mathematics teacher educators or mathematics teacher educator pedagogical content knowledge (mtepck) by chick and beswick (2018). as teacher educators, mtes’ knowledge should be more than the knowledge teachers need to help students learn mathematics. this means, just as teachers need certain knowledge in teaching mathematics, mte also requires certain knowledge to teach school mathematics to prospective teachers. this is in line with zopf (2010) who states that teachers' knowledge is part of educators' knowledge. however, beswick and goos (2018) believe that the conceptualization of mte knowledge is influenced by the school mathematics teacher's model of knowledge. in relation to pck, mte knowledge can be investigated on a specific topic related to two-dimensional figures as one of the most important topics in school mathematics. by understanding the pck of two-dimensional figures, it is arguable that the instrument measuring mtes’ knowledge of this pck can be better constructed. for example, the pck states that teachers need to recognize that a definition of a quadrilateral includes only necessary and sufficient conditions for this quadrilateral, identify several equivalents and non-equivalent definitions for the same quadrilateral, and understand the definitions of quadrilaterals that are more appropriate for younger students. those pieces of knowledge can be transformed into mte knowledge by thinking of what are the same and the differences between mtes’ noticing and prospective teachers’ noticing skills. in this sense, beswick and goos (2018) noted that what teachers should notice (i.e., students' thinking) and what mtes must eventually notice (i.e., teachers' thinking) focus on the thinking of specific groups with which they worked. while mtes’ noticing is to assist teachers in noticing students' mathematical thinking, they must be able to observe students' mathematical thinking in addition to teachers' thinking, specifically through interpretation and evidence. as interest in teacher noticing research has grown, so have conceptualizations of noticing. jacobs and spangler (2017) summarized the three versions of noticing component conceptualizations, beginning with attending only (e.g., star & strickland, 2008). they demonstrated the nested and integrated components of the two-component conceptualization, attending and interpreting (e.g., goldsmith & seago, 2011) and the threecomponent perspective, attending, interpreting, and deciding to respond (e.g., goldsmith & seago, 2011; jacobs & spangler, 2017). in the case of quadrilateral, more particularly, rectangle remains a special case of a parallelogram in pre-service teachers’ figural concepts leading them not to adopt the hierarchical relationship. the findings suggested that learners were likely to recognize quadrilaterals by a special case of them and prototypical figures, even though they knew the formal definition in general. this led learners to have difficulty understanding the inclusion relations of quadrilaterals in primary education, students are taught about the types of quadrilaterals by recognizing their shapes and studying their properties. in contrast, in advanced learning, a quadrilateral is a figure that has several special cases with regard to its properties. in order to teach quadrilateral, teachers should understand the relationship between quadrilateral properties and quadrilateral classification. in the context of quadrilateral classification, de villiers (1994) states that two types of classifications can be made regarding the relationships between quadrilaterals: hierarchical classification and partial classification. with hierarchical classification, quadrilaterals are associated with one another within the framework of their properties as subsets (de villiers, 1994). in other words, a quadrilateral can be said to be a special case as its properties are a subset of another quadrilateral’s properties. school mathematics classifies quadrilateral in a simple way using visual summarization. however, the above-mentioned classifications motivate further analyses in addition to simple and visual summarization of the information (craine & rubenstein, 1993) and require the establishment of appropriate relationships between concepts and images. it ekawati et al., mathematics teacher educators’ noticing of pedagogical content knowledge … 264 considers shapes as subsets of other shapes, so squares are seen as special cases of rectangles, and rhombi are included in the set of kites (forsythe, 2015). in addition to the common approach of hierarchical classification, partial classification is used as an alternative to classifying the figures (de villiers, 1994). in partial classification, quadrilaterals are independent of each other and, classified according to their properties as separate sets (erez & yerushalmy, 2006). partial classification and definition are not acceptable in mathematical terms. they are simply partial, sometimes necessary and beneficial for a clear distinction between concepts (de villiers, 1994). however, the partitional view can be held very strongly since it has been developed from "an early age", so students often find it difficult to accept the inclusion of some classes of shapes within others (okazaki, 2009). the hierarchical classification involves comprehending the relationships between quadrilaterals, which is a rather difficult activity for many learners (erez & yerushalmy, 2006; fujita, 2012). however, hierarchical classification allows mte to deeply understand quadrilaterals and the relationships between their properties. hence, pedagogical content knowledge is said to be important for mte to manage the learning activity such that it is suited to the pre-service teachers' needs. furthermore, the mtes’ skill in attending, interpreting, and responding to prospective teachers’ pck (simpson & haltiwanger, 2017), or known as mtes’ noticing skill is arguably important for mtes’ professional knowledge. this skill is known as essential for teachers and mtes since every aspect of teaching relies on notice (mason, 2002), including seeing what students or psts are doing, how they are responding, comparing what is being responded to the standards of mathematics teaching and expectations, and predicting about what might be reacted next. despite this, little focus has been given to those who teach teachers to notice, or in other words, research on how mtes notice their prospective teachers’ pck is still scant. however, pck can also depend on content knowledge possessed by an mte. additionally, understanding how mtes notice their psts’ pck would give benefits in providing insights on expanding discussions either practically or theoretically on the enlargement of teacher-knowledge-related issues in teacher education. hence, this study aims to explore mtes’ knowledge in noticing pre-service teacher knowledge on hierarchical quadrilateral classification. 2. method 2.1. research design this study used a multiple case study to observe mathematics teacher educators' (mte) knowledge of noticing pre-service teachers’ pck of the hierarchical classification of quadrilaterals. a multiple case study is a type of case study that includes two or more cases to investigate the same phenomena (lewis-beck et al., 2003; yin, 2017). the mtes’ knowledge of the hierarchical classification of quadrilateral was analysed using qualitative content analysis of the mtes’ response to the given question and repeated observations of the videotapes from online moderated forum group discussion (m-fgd) via an online meeting platform. the moderation process was led by the moderator, which is the first author. the moderation was held by optimizing the moderator's role in the discussion to maintain the rhythm of the discussion and reveal all aspects planned to be discussed in detail. 2.2. sample a total of ten mtes with various backgrounds in terms of teaching experience in university, sexes, and background knowledge participated in this study. all the participants volume 12, no 2, september 2023, pp. 261-274 265 were second-year students in doctoral mathematics education program at a public university, in surabaya, east java, indonesia. the participants were enrolled in a course, called knowledge and praxis of in-service and pre-service teachers. the m-fgd was held at the beginning of the course where they did not learn about the content of the course in an explicit course related to teacher knowledge. 2.3. research instrument and procedure this study began with developing several item problems related to the hierarchical classification of a quadrilateral. a paper-based test was given to the research samples. two tasks as shown in figure 1 were given to explore mtes’ noticing of pedagogical content knowledge on the hierarchical classification of quadrilaterals. the first problem was asking mte to predict (noticing practice) pre-service teachers’ reason for drawing a venn diagram about quadrilateral classification. meanwhile, the second problem was asking mte to draw a flowchart as recommendation guidance to validate pre-service teachers’ venn diagram. translation: three prospective mathematics teachers are asked to draw venn diagrams showing the classification of quadrilaterals. note: parallelogram (j), kite (l), rectangle (p), and trapezoid (t). a) write down the probable reasons that might underly each student: students a, b, and c create the diagram. b) given a parallelogram, a kite, a rectangle, a trapezoid, and a rhombus. create a flowchart you may recommend to your students (the prospective teachers) that can be used as a guideline to assess the validity of the classification of those five quadrilaterals which is in the form of venn diagrams. complete the flowchart with any relevant explanation. figure 1. hierarchical classification of quadrilateral tasks for mte to confirm mtes’ responses to the given problem and to explore more about mtes’ noticing pedagogical content knowledge on hierarchical classification of quadrilateral, mfgd was held. the m-fgd was held for about an hour through a recorded online meeting. ekawati et al., mathematics teacher educators’ noticing of pedagogical content knowledge … 266 a list of questions was created as a guideline for the moderators to organize the moderation process. the list of questions was made based on some mte responses to the problem given before the m-fgd. the mtes’ response to the problem was analyzed by categorizing it into some categories based on similar responses. then, the m-fgd was held to confirm the mtes’ response and explore the mtes’ knowledge of the hierarchical classification of quadrilaterals. the detailed list of questions is given in table 1. table 1. guidelines of the m-fgd problems questions predicting pre-service teachers’ reason for making a venn diagram about the hierarchical classification of quadrilateral 1. what is the reason for the pre-service teachers’ answer? 2. are there any quadrilateral properties that make pre-service teachers think that way? 3. which parts of the quadrilateral’s concept might not have been understood by students? make a flowchart as a recommendation to analyze the validity of quadrilateral classification 1. which part can be used as a guide to analyzing the validity of the classification of quadrilaterals in the venn diagram? 2. what is the meaning of the written flowchart? 3. how can the quadrilateral classification be validated using the written flowchart? 2.4. data analysis once the mtes’ responses were collected and the m-fgd was held, the samples’ responses on the lists of questions in table 1 were investigated based on leinhardt and greeno (1986) two fundamental systems of knowledge that must be mastered by teachers and teacher educators, which are subject matter and lesson structure knowledge. subject matter knowledge is content knowledge to be taught to pre-service teachers (muir et al., 2017). while lesson structure knowledge is knowledge to manage a lesson (turnuklu & yesildere, 2007). during learning activities, mte must be able to predict and notice preservice teachers' thinking processes, so that they can maintain learning activities according to the needs of participants. the research's aim is to observe mathematics teacher educators' (mte) knowledge of the hierarchical classification of quadrilaterals by responding to three venn diagrams about quadrilateral classification. the given venn diagrams were about quadrilateral classification based on their special cases. usiskin (2008) write the special types of quadrilaterals and their special cases as shown in table 2. table 2. special types of quadrilaterals and their special cases special types of quadrilaterals their special cases kite rhombus, square trapezoid parallelogram, rectangle, rhombus, square parallelogram rectangle, rhombus, square rectangle square rhombus square square not available volume 12, no 2, september 2023, pp. 261-274 267 in this study, data analysis was carried out concerning the mtepck framework by chick and beswick (2018) as shown in table 3. table 3. mtepck category and indicator category mtepck indicator clearly pck: students’ thinking discusses or addresses pst’s ways of thinking about an smtpck concept mte notice pre-service teachers’ reasons for making venn diagrams about the hierarchical classification of quadrilateral clearly pck: representation of concept describes or demonstrate ways to model or illustrate an smtpck concept mte create a flowchart as guidance to validate pre-service teachers' venn diagram about the hierarchical classification of quadrilateral while the framework of chick and beswick (2018) builds on existing research into pck and categorizes aspects of the work of teacher education, this research only focuses on two aspects of the mtepck, namely psts’ thinking and psts’ representation of a concept. 3. result and discussion in responding to the hierarchical quadrilateral classification problem, it is known that 80% of mtes have a piece of knowledge and notice pre-service teachers' reason for drawing a venn diagram related to the hierarchical classification of the quadrilateral. meanwhile, 20% of mte just re-describe the venn diagram drawn by pre-service teachers. then, mtes’ response was categorized into some categories based on the pre-service teachers' reason predicted by mte. table 4 shows the list of categories of mtes’ response. table 4. list of categories of mtes’ responses to the hierarchical classification of quadrilateral problem list of categories mtes’ response category 1 the pre-service teacher considers the definition and properties of a quadrilateral in a quadrilateral family category 2 the pre-service teacher considers the visual appearance of a quadrilateral category 3 pre-service teacher lack understanding of the meaning and the properties of quadrilaterals according to these results shown in figure 2, mte who said that the pre-service teacher considers the meaning and properties of quadrilaterals explained that parallelograms, rectangles, and kite are related to each other. as it is known that a rectangle is a parallelogram that has an angle of 90°, the pre-service teacher might understand that this only applies to some rectangles, so he drew a venn diagram that intersects each other. in this regard, the mte could notice that the pre-service teacher with this response can identify that trapezoid only has a pair of parallel sides. this shows that in predicting pre-service teachers' reasons for drawing such venn diagram, mte takes into account the pre-service teacher's knowledge of the special case of the quadrilateral which can be defined exclusively and inclusively (usiskin, 2008). an exclusive definition is said to be true for a specific quadrilateral, while an inclusive definition is valid for a family of quadrilaterals. for example, the definition of a trapezoid which states that a trapezoid has only one pair of parallel sides is called the ekawati et al., mathematics teacher educators’ noticing of pedagogical content knowledge … 268 exclusive definition. while the inclusive definition of a trapezoid states that a trapezoid is a quadrilateral with at least one pair of parallel sides. translation: student b could observe the relationship between rectangles, parallelograms, and kites. he could see that a parallelogram with a measure of 90 degrees is a rectangle. however, he saw the relationship between the sets of parallelograms and rectangles as intersecting sets, which means he probably did not see rectangles as a special form of parallelograms or that all rectangles are parallelograms which is more properly presented as rectangles are a subset of parallelograms on the venn diagram. figure 2. category 1 of mtes’ responses another reason mentioned by mte is that the pre-service teacher makes a venn diagram because the pre-service teacher only observes each shape of quadrilaterals that is different, such as the sides of the quadrilateral, the angles, or the diagonals. this means that the preservice teacher's knowledge of quadrilaterals is limited to exclusive definitions. in this case, category 2 is related to category 3 which states that the pre-service teacher experienced a misunderstanding of the definition and properties of a quadrilateral. however, several mtes in the fgd stated that they did not agree that the pre-service teacher did not understand the quadrilateral concept. pre-service teacher errors in making venn diagrams can occur because of the different perspectives of pre-service teachers in classifying quadrilaterals, where the pre-service teacher has not been able to accept or understand the quadrilateral classification hierarchically, but partially. as erez and yerushalmy (2006) said, the partial classification of quadrilaterals is carried out based on the properties of each quadrilateral in a separate set, not interrelated. hence, it can be said that the preservice teacher's knowledge of quadrilaterals is still limited to exclusive definitions (josefsson, 2013). the limitations of the preservice teacher's knowledge can occur because, during the learning period, the pre-service teacher started studying the special forms of quadrilaterals and the characteristics of each shape. it is different from the inclusive definition which states that quadrilateral properties can also apply to quadrilateral families. to assess the pre-service teacher's understanding of the quadrilateral classification hierarchically, mte was asked to make a flowchart as a guide to validating the pre-service teacher's answers. the results of this study indicate that 80% of mte know to create a flowchart as recommendation guidance for validating the hierarchical classification of the quadrilateral. while the rest 20% of mte do not know because they can’t create the flowchart. there are two categories of mte response to the second problem as shown in table 5. volume 12, no 2, september 2023, pp. 261-274 269 table 5. mtes’ response to the 2nd problem category 1 category 2 mtes’ response a quadrilateral flowchart that shows the special types of quadrilaterals and their special cases based on their properties and their explanation a quadrilateral flowchart that shows the special types of quadrilaterals and their special cases based on their properties without explanation referring to table 5 and the results of the m-fgd, some mtes have difficulty in making flowcharts. this shows that mtes’ pedagogy content knowledge in making guidance to assess pre-service teacher answers to the hierarchical quadrilateral classification problem has not been achieved. meanwhile, the other two categories provide a different flowchart description. category 1 describes flowcharts that are written based on the properties of the quadrilateral family and its special case that leads to an inclusive definition of a quadrilateral. the flowchart made by mte is shown in figure 3 and started with quadrilateral in general, where if one of the angles in the quadrilateral is greater than 180°, it is called a concave quadrilateral. meanwhile, if the interior angles in the quadrilateral are each less than 180°, it is called a convex quadrilateral. a convex quadrilateral that has at least one pair of parallel sides is called a trapezoid. if two pairs of parallel sides are the same length, it is called a parallelogram. furthermore, a parallelogram in which each angle is 90° is called a rectangle. while a kite is a quadrilateral that has two pairs of adjacent sides that are the same length. however, if all sides are the same length, then the kite is called a rhombus. mte also writes that a rhombus with an angle of 90° is a rectangle which according to its exclusive definition, is not quite right because the special case of a rhombus is a square (usiskin, 2008). however, inclusively, this could also be true because a square is also a special case of a rectangle. figure 3. category 2 of mtes’ response to the 2nd problem on the contrary, category 2 shown in figure 4 is a flowchart created based on the visual appearance of a quadrilateral, which is still related to the concept of quadrilateral, but ekawati et al., mathematics teacher educators’ noticing of pedagogical content knowledge … 270 without a detailed explanation. in this category, mte explains that parallelograms, kites, and rectangles are different types of quadrilaterals because of their different visual appearances, such as the length of the sides, pairs of parallel sides, and the angles they have. meanwhile, rhombi and trapezoids are special cases of rectangles. this shows that mte has not been able to determine a hierarchical classification of quadrilaterals and mtes’ knowledge of quadrilaterals’ definition is limited to an exclusive definition, so the guidance flowchart refers to the partial classification of quadrilaterals (de villiers, 1994). figure 4. category 3 of mtes’ responses to the 2nd problem based on the results of the m-fgd, several conclusions can be drawn as follows. mtes’ pedagogical content knowledge on the hierarchical classification of a quadrilateral is divided into some categories. those who understood the hierarchical classification could explain pre-service teachers’ reason related to the relation of quadrilateral properties, that some of them are subsets because of their special cases. however, in their understanding of the classification, although hierarchical, some were deficient in their concept images of parallelograms, as evidenced by the response in figure 3, where this finding also occurred in preservice teachers (birgin & özkan, 2022). while those who have difficulties in understanding the hierarchical classification, use partial classification and refer to the quadrilaterals’ different visual appearance. this difficulty can be explained by their incomplete informal deduction where there is a gap between their personal concept images and the personal concept definition of geometric figures (fujita & jones, 2007), which may lead them only depend on their visual perception of quadrilaterals, instead of properties of each fo quadrilateral (leton et al., 2020). pedagogically, to help students or pst achieve informal deduction, the mte should be encouraged to understand that investigating lists of properties of quadrilaterals, making comparisons, and reaching abstractions such as 'if the property... holds for quadrilaterals... then it holds true for quadrilaterals...', as suggested by fujita (2012), is necessary. mtes’ noticing of pre-service teachers’ hierarchical classification has an impact on mtes’ ability to draw a flowchart as a guideline for validating pre-service teachers’ quadrilateral classification. the findings that some mtes applied partial classification, instead of hierarchical classification, on quadrilateral when validating pre-service teachers’ quadrilateral classification show that the mtes have problems with their content knowledge which led them to fail in noticing pst’s quadrilateral classification. while they included volume 12, no 2, september 2023, pp. 261-274 271 some evaluative comments in their professional noticing practices, they lack in-depth interpretative analysis of student thinking, as shown by their invalid classification, where this finding is also reported by amador et al. (2021) who found that their participating preservice teachers rarely create links between student thinking and the broader concepts of teaching and learning. as many previous researchers suggested (e.g., avcu, 2022; erdogan & dur, 2014), the hierarchical classification of geometric figures requires the mastery of many aspects, some of which are the ability to identify the transitivity relation between concepts, asymmetry of the relations between quadrilaterals, and the asymmetry of the properties between quadrilaterals. according to erdogan and dur (2014), those three aspects can be seen when individuals label two quadrilaterals. in this study, the mtes tried noticing psts’ responses to the classification of quadrilateral by assessing the asymmetry of the relations or the properties between quadrilaterals, instead of the extent to which the transitivity relation emerges from psts’ responses. for example, the responses indicated in figure 2 and figure 3 reveal that all squares are rectangles, but not all rectangles are squares, by identifying the properties of rectangle and square through the venn diagram analysis. no indication was found in all the mtes’ responses that explicitly express the idea of using definition as the tool for transitivity analysis, like a response if a square is a rectangle and if a rectangle is a parallelogram, then a square is a parallelogram. thus, the mtes could identify the psts’ difficulties related to the use of prototypical images in choosing a family category among the given quadrilaterals (e.g., erdogan & dur, 2014), even when individuals might know a precise definition (fujita, 2012). responding to this study's findings, we would highlight that the hierarchical and partial classification of quadrilateral become essential knowledge for mte to assess their psts’ pck on quadrilateral classification. according to de villiers (1994), there is a rigid relationship between object definition and classification. for example, the trapezoid definition "a quadrilateral having at least one pair of opposite sides parallel" leads to a hierarchical classification in which the parallelogram is a particular case of a trapezoid; however, if the definition states that there is only one pair of opposite sides parallel, we have a partition classification, with the parallelogram belonging to a disjoint set of the trapezium. thus, mte should also notice the existence of whether the psts are working with an exclusive or inclusive definition of a trapezoid. 4. conclusion this research finds that mtes have different views on classifying quadrilateral. even though they have passed advanced education and are known as teacher educators, not all mtes notice pre-service teachers’ hierarchical classifications and define quadrilaterals inclusively. hence, 20% of mtes do not have any knowledge about pre-service teachers' ways of thinking on the hierarchical classification of quadrilaterals and failed to draw a guidance flowchart for validating pre-service teachers’ hierarchical classification of a quadrilateral. therefore, mtes’ knowledge of quadrilaterals, specifically the hierarchical classification of quadrilaterals, should be improved. references amador, j. m., bragelman, j., & superfine, a. c. 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(2010). mathematical knowledge for teaching teachers: the mathematical work of and knowledge entailed by teacher education. university of michigan. https://doi.org/10.3102/0013189x015002004 https://doi.org/10.1007/s10857-016-9352-0 https://doi.org/10.46517/seamej.v7i2.51 https://doi.org/10.33902/jpr.2021371325 https://doi.org/10.1007/s10857-007-9063-7 https://doi.org/10.1007/978-3-030-62408-8_17 https://doi.org/10.22342/jme.12.2.13537.223-238 https://doi.org/10.1023/b:jmte.0000009971.13834.e1 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p101-116 101 meta-analysis of computer-based mathematics learning in the last decade scopus database: trends and implications maximus tamur1*, sabina ndiung1, robert weinhandl2, tommy tanu wijaya3, emilianus jehadus1, eliterius sennen1 1universitas katolik indonesia santu paulus ruteng, indonesia 2linz school of education, stem education, johannes kepler universität linz, austria 3school of mathematical sciences, beijing normal university, beijing, china article info abstract article history: received jan 11, 2023 revised feb 10, 2023 accepted feb 26, 2023 computer-based mathematics learning (cbml) has gone global in the last decade and is making a substantial impact for educational purposes. but the fact is that in the scientific literature, it is found that studies aimed at testing these theoretical assumptions have inconsistent results. in this regard, this meta-analysis was conducted to determine the effect of cbml and to analyze categorical variables to consider the implications. data were retrieved from the scopus database using publish or perish between 2010 and 2023. this study examined 29 independent samples from 28 eligible primary studies with 1179 subjects. the population estimate was based on a random effects model, and the cma software was used as a calculation aid. the study's results provide an overall effect size of 1.03 (large effect). this indicates that applying cbml significantly affects students' mathematical abilities. the four categorical variables considered in the study are discussed to clarify research trends. furthermore, the research implications are outlined and contribute to future cbml implementation arrangements. keywords: computer-based mathematics learning, mathematical abilities, meta-analysis, scopus database this is an open access article under the cc by-sa license. corresponding author: maximus tamur, department of mathematics education, universitas katolik indonesia santu paulus ruteng jl. ahmad yani 10 ruteng, manggarai, east nusa tenggara 86511, indonesia. email: maximustamur@unikastpaulus.ac.id how to cite: tamur, m., ndiung, s., weinhandl, r., wijaya, t. t., jehadus, e., & sennen, e. (2023). meta-analysis of computer-based mathematics learning in the last decade scopus database: trends and implications. infinity, 12(1), 101-116. 1. introduction computer-based mathematics learning (cbml) has been widely implemented and included in the teaching system in almost every country. the use of cbml in the classroom is associated with students' academic abilities because teachers provide exciting learning opportunities and broader content (de mendivil et al., 2019; mclaren et al., 2017; xin et al., 2020; zhang & wang, 2020). applying cbml significantly influences students' mathematical abilities (kim et al., 2022; nurjanah et al., 2020; pereira et al., 2021; https://doi.org/10.22460/infinity.v12i1.p101-116 https://creativecommons.org/licenses/by-sa/4.0/ tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 102 shyshkina et al., 2018). it can be said that the use of cbml has become a trend in learning mathematics in the last decade. the use of cbml in learning is increasingly widespread because it presents content numerically, graphically and symbolically without the additional time burden of manually calculating complex computational problems (juandi, kusumah, tamur, perbowo, siagian, et al., 2021; juandi, kusumah, tamur, perbowo, & wijaya, 2021; tamur, kusumah, juandi, kurnila, et al., 2021; tamur, kusumah, juandi, wijaya, et al., 2021). integrating computer technology into learning mathematics will help students make connections in mathematics by making the learning process more realistic and effective (zaldívar-colado et al., 2017). cbml implementation will be more interesting, inventive, and exploratory (aungamuthu, 2013; foster et al., 2016; ochkov & bogomolova, 2015; tamur et al., 2022). these conditions allow students to be more active and successful in learning (das et al., 2021; tatar et al., 2014; timmers et al., 2013). cbml provides useful feedback to support students' representation, visualization, and critical abilities (granberg & olsson, 2015). applying cbml can improve students' mathematical ability (sma). however, in the scientific literature, studies testing these theoretical assumptions provide inconsistent findings. several studies provide findings stating the superiority of the cbml group (e.g., ishartono et al., 2022; jelatu et al., 2018; kariadinata et al., 2019; ningsih & paradesa, 2018; nurjanah et al., 2020; takači et al., 2015), whereas other studies have found that cbml implementation does not outperform the group that does not use it (e.g., del cerro velázquez & morales méndez, 2021; hamid et al., 2020). various studies that analyze the same problem sometimes provide findings that vary and contradict each other. as a result, making decisions or conclusions can be subjective (wicherts, 2020). the problem is that educators and other related parties need accurate information for making educational decisions and arrangements under what conditions the use of cbml achieves a high level of effectiveness in sma. the problems described above can be overcome by summarizing the various results of primary research. quantitative research procedures can be chosen to provide accurate policy-making information (higgins & katsipataki, 2015). in this regard, the meta-analysis study is the choice because of its role specifically to integrate the findings of the primary study and investigate the reasons for the variation in results to be considered in its implementation in the future. meta-analysis provides in-depth and accurate conclusions (fadhli et al., 2020; siddaway et al., 2019). thus, when there is a need to draw conclusive conclusions, it is necessary to take into account the results of various individual studies using meta-analysis methods (wicherts, 2020). previously, meta-analysis studies have been found in the literature regarding the effect of cbml implementation on sma (e.g., demir & basol, 2014; juandi, kusumah, tamur, perbowo, & wijaya, 2021; tamur, kusumah, juandi, kurnila, et al., 2021). however, these studies analyzed primary data from all databases, such as google scholar, eric, repository, library database, and others. in contrast to previous studies, this study analyzes data taken from the scopus database. apart from the fact that no studies specifically chose scopus as the only search database, scopus was also chosen because it applies consistent standards in selecting documents to be included in its index (phan et al., 2022). scopus also displays more documents than other top databases, such as the web of science, specifically for research reviews in education and social sciences (hallinger & chatpinyakoop, 2019; hallinger & nguyen, 2020). this study also fills a gap in the literature regarding a comparative picture of cbml implementation in indonesia and abroad. previous studies have also shown inconsistent trends in the variables studied in terms of year of publication, source of publication, and level of education (e.g., juandi, kusumah, volume 12, no 1, february 2023, pp. 101-116 103 tamur, perbowo, & wijaya, 2021; tamur et al., 2020; tamur, kusumah, juandi, wijaya, et al., 2021). this meta-analysis is necessary to evaluate the implementation of cbml to see the overall trend clearly and also consider the implications. thus, this study will directly contribute to cbml science and practice in the future, especially concerning educational settings. this research answers explicitly two questions, namely; (1) whether the mean effect size representing the intervention for each study on the effect of cbml on sma is significantly different from zero, and (2) whether there is a difference in the magnitude of the effect of cbml on sma based on categorical variables. 2. method this study focuses on analyzing the influence of the quantitative profile of the study group from the scopus database, which specifically investigates the effectiveness of cbml on students' mathematical abilities (sma). this work was conducted to determine the magnitude of the overall effect of cbml and to analyze the role of categorical variables in the differences in effect sizes of each study. to achieve this goal, a meta-analysis approach was applied. meta-analysis was developed to compare the results of various primary studies and its role is very vital for scientific development, namely providing a more objective method of drawing conclusions or decisions when various primary studies provide varying results (cooper, 2017; schmidt & hunter, 2015). in general, meta-analysis research begins with formulating research problems and hypotheses, followed by a literature search, then coding variables, then statistical analysis, and ends with the interpretation of findings (çoğaltay & karadağ, 2015). this work also follows these stages. 2.1. literature search the scopus database was chosen as the document search location in this study. furthermore, the publish or perish program is used to download a study on the application of cbml with the keyword combination used being geogebra mathematical ability; cabri mathematical ability; maple mathematical abilities; mathematical ability algebrator, matlab mathematical ability, matchcad, and wingeom. as an example, figure 1 presents the process of tracing studies related to implementing cbml on the scopus database using the pop application. the search results obtained 193 related studies. figure 1. the process of tracing data using the pop application. tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 104 2.2. literature inclusion criteria in this study, studies that were successfully identified using the pop application were then shared according to the following criteria: a. study in an english setting, and retrieved from scopus database between 2010-2023. b. provide statistical information to obtain effect size values. studies that do not meet this will be excluded from the analysis (e.g., dockendorff & solar, 2018; jacinto & carreira, 2017; zulnaidi et al., 2020). c. study with a quantitative approach and must involve a control group as a comparison. studies that used only one sample or used a qualitative approach were excluded from the analysis. furthermore, in this study, the data filtering process used the prisma (preferred reporting items for systematic review and meta-analyses) protocol, as shown in figure 2. figure 2. filtering data through prisma figure 2 presents the process of filtering data using prisma. finally, there are 28 studies, and because there are studies that investigate more than one independent sample, in the study conducted by apriatna et al. (2020) then, 29 effect sizes were examined. 2.3. coding all eligible studies were coded using a detailed coding scheme based on the coding manual and coding protocol. each preliminary study included in the analysis was coded. in this work, the research instrument is a coding sheet specially created to extract information from individual studies converted into numerical data. the coding was carried out by two volume 12, no 1, february 2023, pp. 101-116 105 students who had previously been specifically trained according to guidelines from cooper (2017). to determine reliability, a random sample of 5 from the eligible studies was duplicated and distributed among the coders. each coder was provided with a copy of the article, coding form and coding protocol. to assess the reliability between coders, the percent agreement or abbreviated pa is used. the pa method is the easiest and most intuitive approach to building reliability (syed & nelson, 2015). pa is simply the ratio of items that the two coders agree on to the total number of items assessed and is calculated using the following formula: 𝑃𝐴 = 𝑁𝐴 𝑁𝐴 + 𝑁𝑃 𝑥 100 where na is the total number of agreements and np is the total number of disagreements. an agreement rate of 85% or greater is predetermined to be considered high. from the calculation results, the level of both pas is 94.26%. the figure indicates there is a substantial to near-perfect match between the coders. thus the instrument to be used in this study is reliable. 2.4. statistical analysis the parameter used to estimate the population in a meta-analysis study is the effect size. in this study, the effect size is defined as the magnitude of the effect of cbml on sma. cma software stands for comprehensive meta-analysis used to transform the effect size of each study, including the overall effect size, p-value, q statistics, and confidence intervals. the program also draws funnel plots and research forest plots. hedges' g metrics were applied in this study, and the interpretation of effect sizes was based on the classification of cohen et al. (2017) i.e., less than 0.2 (negligible), 0.2 to 0.5 (small effect), 0.5 to 0.8 (medium effect), 0.8 to 1.3 (large effect), and more than 1.3 (very large effect). the random effects model was chosen as the estimation method because it does not assume that all the studies analyzed have the same true effect (pigott & polanin, 2020). in this study, the random effect model was determined after fulfilling the heterogeneity test. this test is carried out by observing the p-value. the null hypothesis, which states that all studies are the same (homogeneous), is rejected if the p-value <0.05. rejecting the null hypothesis indicates that effect sizes between studies or study groups may not measure the same population parameters (çoğaltay & karadağ, 2015). conditions show that differences in study categories affect study effect sizes. an analysis of publication bias was performed to prevent misrepresentation of the findings. publication bias reflects the fact that it is more likely that articles deemed statistically significant are published. in addition to the scientific facts that about 6% of researchers rarely try to publish research that is not significant (cooper, 2017). as a result, the aggregate effect size obtained may be exaggerated (juandi et al., 2022; park & hong, 2016). to anticipate this, funnel plots were examined to assess the possible amount of bias, and rosenthal's fsn statistic was used to assess the impact of bias (çoğaltay & karadağ, 2015). in this study, it is said to be resistant to publication bias if the distribution of effect sizes is symmetrical around the vertical line. however, if the fact is that the effect size scatter is not completely symmetrical, then a trim and fill procedure is used. then, if it is found that the observed effect and the virtual effect created according to the random effects model are the same then the study is resistant to the influence of publication bias. tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 106 3. result and discussion 3.1. results first, the results of research aimed at answering question one are presented. after screening the primary studies, 28 articles from the scopus database and 29 independent samples were analyzed in this study. figure 3 presents the research forest plot consisting of the study name, es (effect size), standard error, variance per the study, confidence intervals, z values, and p values. figure 3. research forest plot figure 3 shows the broad confidence levels and inconsistent response rates. descriptively this shows the effect size of each study is heterogeneous. statistically, this needs to be checked so that the estimation method is in accordance with the initial assumptions. table 1 presents a summary of the overall analysis to answer question 1 as well as to determine the estimation method. table 1. the meta-analysis results according to the estimation model model n hedges’s g standard error test of null q p decision z-value p-value fixed-effects 29 0.81 0.04 18,38 0.00 186.87 0.00 reject h0 random-effects 29 1.03 0.01 8,57 0.00 when table 1 is observed, it appears that the p value <0, which means that the distribution of effect sizes for each study is heterogeneous. this means that the estimation method for the population in this study is in accordance with the random effect model. table study name statistics for each study hedges's g and 95% ci hedges's standard lower upper g error variance limit limit z-value p-value priatna, 2017a 0,79 0,26 0,07 0,27 1,31 2,97 0,00 priatna, martadiputra & wibisono, 2018 0,92 0,27 0,07 0,40 1,45 3,44 0,00 munandar, usman, & saminan, 2020 1,97 0,32 0,10 1,34 2,61 6,11 0,00 juandi, & priatna, 2018 0,36 0,25 0,06 -0,13 0,86 1,43 0,15 nurafni, & ningrum, 2021 0,50 0,25 0,06 -0,00 1,00 1,95 0,05 takaci, stankov, & milanovic, 2015 0,68 0,11 0,01 0,46 0,89 6,24 0,00 septian, darhim, & prabawanto, 2020 2,33 0,38 0,15 1,58 3,08 6,09 0,00 rhamawati, 2019 2,32 0,37 0,13 1,60 3,04 6,32 0,00 saha, ayub, & tarmizi, 2010 0,60 0,28 0,08 0,05 1,14 2,15 0,03 maryono, rodiah, & syaf, 2020 1,10 0,27 0,07 0,57 1,63 4,05 0,00 ridha, pramiarsih, & widjajani, 2019 0,73 0,30 0,09 0,14 1,32 2,44 0,01 negara, wahyudin, nurlaelah, & herman, 2022 0,72 0,24 0,06 0,26 1,19 3,05 0,00 murni, sariyasa, & ardana, 2017 3,77 0,43 0,18 2,93 4,61 8,81 0,00 khalil, farooq, çak?ro?lu, & khali, 2018 0,82 0,32 0,10 0,19 1,45 2,53 0,01 zetriuslita, nofriyandi, & istikomah, 2019 0,01 0,31 0,10 -0,59 0,62 0,04 0,97 zulnaidi, & zamri, 2017 0,53 0,11 0,01 0,31 0,74 4,81 0,00 nurjanah, latif, yuliardi, & tamur, 2020 0,43 0,24 0,06 -0,03 0,90 1,82 0,07 kusumah, kustiawati, & herman, 2020 0,79 0,23 0,05 0,34 1,23 3,47 0,00 jelatu, sariyasa, & ardana, 2018 1,12 0,27 0,08 0,58 1,66 4,08 0,00 ningsih, & paradesa, 2018 1,53 0,29 0,08 0,97 2,10 5,32 0,00 apriatna, budiyono, & indriati, 2020a 0,60 0,26 0,07 0,10 1,11 2,35 0,02 apriatna, budiyono, & indriati, 2020b 0,63 0,26 0,07 0,13 1,14 2,45 0,01 kariadinata et al., 2019 1,35 0,31 0,10 0,75 1,96 4,38 0,00 muntazhimah, & miatun, 2018 0,97 0,32 0,10 0,34 1,60 3,02 0,00 ishartono et al., 2022 0,15 0,26 0,07 -0,35 0,65 0,59 0,55 velázquez, & méndez, 2021 0,32 0,22 0,05 -0,12 0,75 1,44 0,15 priatna, 2017b 2,85 0,33 0,11 2,19 3,50 8,56 0,00 zulnaidi et al., 2019 0,48 0,22 0,05 0,04 0,92 2,16 0,03 septian et al., 2020 1,57 0,34 0,12 0,91 2,24 4,63 0,00 1,01 0,12 0,01 0,78 1,25 8,58 0,00 -4,00 -2,00 0,00 2,00 4,00 fav ours a fav ours b plot forest research meta analysis volume 12, no 1, february 2023, pp. 101-116 107 1 also shows the results of hypothesis testing to answer the first research question. when table 1 is observed it appears that the p value for the test of null is less than 0 based on the random effects model. this means that the mean effect size representing the intervention for each study of the effect of cbml on sma is significantly different from zero. in other words, the overall results of the study clarify the superiority of the cbml group. the overall effect size based on the random effects model is 1.03 which is classified as a large effect according to cohen et al. (2017). it is then necessary to calculate whether this overall effect size is related to publication bias then the funnel plot of the study is observed. figure 4 presents the funnel plot of the study. figure 4. funnel plot of 29 independent samples when it is observed in figure 4, the study ess are spread not entirely symmetrically in the center of the funnel plot. therefore, it is necessary to observe whether these results resist publication bias's influence. for this reason, the trim and fill testing procedure is carried out as illustrated in figure 5. figure 5. trim and fill test the trim and fill test results as illustrated in figure 5 show that there is no difference between the observed effect sizes and the virtual effects created according to the randomeffects model. thus, there was no publication bias in this study or no studies were pruned or added due to publication bias. so the overall effect size calculated as 1.03 and categorized as large effect is not associated with publication bias. this value is not exaggerated. tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 108 second, the research results are presented to answer question two. it has previously been shown that the estimation method fits the random effects model. this shows that the effect size of each study is heterogeneous so that categorical variables that affect the relationship between cbml and sma must be investigated (arik & yilmaz, 2020). the summary of the analysis related to these catogri variables is illustrated in table 2. table 2. summary of category variable analysis category variables category n hedge's g heterogeneity decision (qb) df(q) p sample size 30 or less 18 0.92 4.32 1 0,03 reject h0 31 or over 11 0.73 educational stage junior high school (jhs) 17 0.87 6.99 2 0.03 reject h0 senior high school (shs) 5 0.76 college 7 0.58 publication sources journal 12 0.57 48.6 2 0.00 reject h0 proceeding 17 1.21 country indonesia 23 0.99 22.75 1 0.00 reject h0 foreign 6 0.57 3.2. discussion the first objective of this research is to determine the magnitude of the overall effect of applying cbml on sma. the analysis results gave the overall effect size of 1.03 and were accepted as a large effect. these results are supported by a previous meta-analysis conducted by tamur et al. (2020) regarding the effect of implementing mathematical software in indonesia. although the database and population used differ from this study, the overall effect size is almost the same, namely 1.16. in addition, the results of this study are almost the same as the findings of studies conducted by juandi, kusumah, tamur, perbowo and wijaya (2021) and juandi, kusumah, tamur, perbowo, siagian, et al. (2021) where the overall study effect sizes were 1.07 and 0.96, respectively. this shows a nearly similar overall trend with respect to cbml adoption. in this study, the number of subjects was 1127 people, and the average sample size was 44 people. related to that, the effect size of 1.03 can be interpreted as students who are ranked 22 in the experimental class are equivalent to students who are ranked 8 in the control class. this illustration illustrates the magnitude of the influence of cbml on sma. this condition is explained in the scientific literature that the application of cbml helps increase students' knowledge of mathematical concepts and procedures (zulnaidi & zamri, 2017). in cbml, students are motivated to learn because of technology's wide and interesting content (zulnaidi et al., 2020). cbml implementation produces concept organizing principles which are then used to think about various possible solutions (santos-trigo & reyes-rodriguez, 2016). furthermore, from the analysis of categorical variables, as illustrated in table 2, it can be seen that there are categorical variables in this study that clarify the effect size of the study. first, the analysis results show that the difference in sample size affects the study's effect size. it appears that a small sample size (30 or less) gives a larger effect size than the other categories. these results directly contribute to future classroom arrangements. the volume 12, no 1, february 2023, pp. 101-116 109 literature also supports this result (e.g., juandi, kusumah, tamur, perbowo, siagian, et al., 2021; juandi, kusumah, tamur, perbowo, & wijaya, 2021) that small sample size is defined as 30 or less great effect than in the other categories. the analysis results also show that differences in educational level affect the effect size of cbml on sma. the findings in this study illustrate that the implementation of cbml is more effective in jhs than in shs and college. this result is surprising because it differs from previous studies' findings (e.g., juandi, kusumah, tamur, perbowo, & wijaya, 2021). this difference may be due to the database used is not the same. however, further studies are needed to verify and clarify the consistency of the results regarding this variable. furthermore, table 2 presents the analysis results related to the publication source variable. the results of the analysis show that differences in publication sources also explain the variation in effect sizes between studies. statistically, the mean effect size of the study group from proceedings was greater than that of the journal. this indicates that this study is not associated with publication bias, as supported by the trim and fill test. in this study, there was no indication that journal editors chose to publish only significant articles (cooper, 2017) due to the fact that the study group originating from proceedings was even more significant. this study also considers country differences as a categorical variable. in the literature, six studies were found from abroad that examined cbml for sma, and the rest were from indonesia. there is a significant difference in the mean effect size between categories. this result is not surprising because the implementation of cbml abroad has been going on for a long time, as sugandi and delice (2014) reported. these results were influenced by the hawthorne effect, as investigated by tamur, kusumah, juandi, wijaya, et al. (2021) that students feel enthusiastic about learning because of the novelty of the treatment. if the same treatment is given continuously, the effect will decrease. however, this difference in results is interesting for further investigation through further research. finally, given the importance of implementing cbml, teaching teachers how to manage technology-enabled learning (dockendorff & solar, 2018). continuous training for teachers is needed to integrate technology effectively (kartal & çınar, 2022). these results have implications for the importance of mathematics teacher training and its impact on developing technological pedagogical content knowledge. in this study, many studies were excluded because the categories were not measured. hartatiana et al. (2017) for example conducted an experiment regarding the effectiveness of cbml by only considering gender differences. although there is statistical information that supports the effect size transformation, it does not lead to the purpose of this study. thus, further studies that specifically consider gender differences as a categorical variable need to be carried out. furthermore, of the 193 identified studies, only 28 met the inclusion requirements. of the number that do not meet the analysis requirements, it examines more about analyzing student abilities related to cbml implementation (e.g., mendrek et al., 2018), development studies (e.g., khalil et al., 2018) and review research (e.g., juandi, kusumah, tamur, perbowo, siagian, et al., 2021; tamur et al., 2020; wang & tahir, 2020). from the distribution of data extracted, it was identified that several studies specifically investigated differences in initial ability as a categorical variable (e.g., kariadinata et al., 2019; muntazhimah & miatun, 2018), whereas more studies did not measure it. the direct implication of this condition is that in the future, it is necessary to investigate whether these categorical variables influence the effectiveness of cbml on students' mathematical performance. tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 110 4. conclusion this meta-analysis analyzed 29 independent samples from 28 primary studies. based on the random effects model, it was found that the application of cbml had a large effect on sma. the variation in the effect size of each primary study was moderated by the categorical variables observed in this study. however, these findings are only based on the scopus database searched through the pop application. the fact is that there are still many related studies that cannot be downloaded, especially publications from the ieee, which must go through an affiliate and be paid specifically. further cooperation procedures are needed to obtain more related studies. acknowledgements the authors would like to appreciate the technical assistance from two students of the mathematics education study program at the catholic university of indonesia, santu paulus ruteng, who were involved as variable coders. references apriatna, e. j., budiyono, & indriati, d. 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(2017). the effectiveness of the geogebra software: the intermediary role of procedural knowledge on students’ conceptual knowledge and their achievement in mathematics. eurasia journal of mathematics, science and technology education, 13(6), 2155-2180. https://doi.org/10.12973/eurasia.2017.01219a https://doi.org/10.12973/eurasia.2017.01219a tamur et al., meta-analysis of computer-based mathematics learning in the last decade scopus … 116 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p55-68 55 analysis of student's ability to solve mathematical literacy problems in junior high schools in the city area yulyanti harisman1*, dwita elfri mayani2, armiati1, hamdani syaputra1, mohd hasril amiruddin3 1universitas negeri padang, indonesia 2universitas negeri yogyakarta, indonesia 3universiti tun hussein onn malaysia, batu pahat, johor, malaysia article info abstract article history: received jan 24, 2023 revised feb 27, 2023 accepted feb 28, 2023 mathematical literacy problems require counting and solving mathematical problems in daily life. this study aims to analyze the ability of junior high school students to solve mathematical literacy problems. this qualitative research uses a case study approach with the subject of 15 junior high school students in the city area. the instrument in this study is six mathematical literacy questions oriented to pisa test questions and interview guidelines. the data analysis technique in this study is thematic analysis. this study's results show three groups of students' abilities in solving mathematical literacy problems based on their initial abilities: time stone, power stone, and mind stone. keywords: mathematical initial abilities, mathematical literacy ability, pisa content quantity this is an open access article under the cc by-sa license. corresponding author: yulyanti harisman, department of mathematics education, universitas negeri padang jl. prof. dr. hamka, air tawar bar., padang city, west sumatra 25171, indonesia. email: yulyyuki@gmail.com how to cite: harisman, y., mayani, d. e., armiati, a., syaputra, h., & amiruddin, m. h. (2023). analysis of student's ability to solve mathematical literacy problems in junior high schools in the city area. infinity, 12(1), 55-68. 1. introduction mathematics is a universal science that benefits all aspects of human existence, forms the basis of contemporary technological advancement, plays a part in many academic fields, and can enhance cognitive abilities (ashim et al., 2019; mardhiyana, 2015). students' ability in mathematics is measured by their ability to count and reason logically and critically when solving problems (fathani, 2016; indrawati & wardono, 2019; masjaya & wardono, 2018). "mathematical literacy," which is characterized as a person's capacity to formulate, utilize, and comprehend mathematics in a variety of life circumstances, is the ability to solve mathematical problems (anwar, 2018; fajriyah, 2018; fathani, 2016; sari & wijaya, 2017; setiawan et al., 2014). https://doi.org/10.22460/infinity.v12i1.p55-68 https://creativecommons.org/licenses/by-sa/4.0/ harisman et al., analysis of student's ability to solve mathematical literacy problems … 56 mathematical literacy is ensuring that all students comprehend mathematics and how to relate it to the actual world to use mathematical information and make intelligent decisions that impact their life, job, and society (botha & van putten, 2018; pillai et al., 2017). any mathematics literature will make students critical thinkers in understanding concepts and applying them to real-world problems (abdullah & richardo, 2017; anwar, 2018; fathani, 2016; permanasari, 2016; rachmantika & wardono, 2019; sukmawati, 2018). mathematical literacy skills are needed to solve problems mathematically that are closely related to the context of life (janah et al., 2019; malasari et al., 2017). the existence of this mathematical literacy ability can help students solve problems because, in using it, they are required to think systematically, conceptually, and causally (baharuddin et al., 2022; hwang et al., 2018; sari et al., 2017; setiani et al., 2018). every student must be able to solve issues in the form of story problems as part of their mathematical literacy (purnama & suparman, 2020). since the student utilizes such skills to solve all his life difficulties involving numerous complicated parts and problems, they are not simply employed to solve mathematical concepts, addressing whether learning only requires cognitive aspects. thus, it is thought that pupils should be able to grasp this skill. but in fact, the research results by sari and wijaya (2017) show that student literacy levels are still poor and that pupils do not possess the three information literacy skills: identifying the needed information and locating and assessing its quality, and effectively sharing information candidly. according to research results from the 2011 trends international mathematics and science study (timss), this assertion is accurate (janah et al., 2019), which showed that the achievement of indonesian students in mathematics was at the level of 45 out of 50 participants (noviana & murtiyasa, 2020). the programme for international student assessment is used to gauge students' proficiency in mathematics (pisa) (anggoro et al., 2019; dores & setiawan, 2019; hewi & shaleh, 2020b; noviana & murtiyasa, 2020; ridzkiyah & effendi, 2021). several countries that are a part of the organization for economic cooperation and development (oecd), headquartered in paris, france, developed the pisa survey (hewi & shaleh, 2020a). since 2000, indonesia has taken part in the pisa assessment program. this international initiative, which assesses reading, math, and scientific literacy, is conducted every three years. the involvement of the indonesian nation in the pisa research is being undertaken to evaluate indonesian pupils' literacy proficiency, which is far from satisfactory compared to other countries (mahdiansyah & rahmawati, 2014). pisa data from 2000 revealed that indonesian students' proficiency in mathematics was still only moderate, with an average score of 367, placing indonesia 39th out of 41 participants. meanwhile, the 2003 pisa results, followed by 40 countries, put indonesian students in 38th place with an average score of 360. these two outcomes are not significantly different from the pisa outcomes from the ensuing years (2006, 2009, 2012, 2015, 2018), which put indonesia at the bottom. pisa creates a variety of questions based on the following four contents: change and relationship, shape and space, quantity, and uncertainty and data (dores & setiawan, 2019). according to pisa, there are six stages of pupils' proficiency in mathematics (setiawan et al., 2014), as described in table 1. table 1. mathematics literacy ability level according to pisa level student abilities 1 students can use their knowledge to address common problems and problems with a broader context. 2 students can interpret problems and solve them with formulas. volume 12, no 1, february 2023, pp. 55-68 57 level student abilities 3 students are competent at following instructions and selecting problemsolving techniques. 4 students may choose and incorporate many representations, work well with models, and then relate those representations to the real world. 5 in addition to tackling complex problems, students can work with models for difficult circumstances. 6 students can utilize their reasoning to generalize, draw conclusions, and discuss the outcomes of their research while still solving mathematical problems. due to the pisa results and the low mathematical literacy ability of indonesian students, researchers did a study to examine the mathematical literacy ability of junior high school students who were pisa-oriented. some researchers choose the content of numbers because the operation of numbers is an essential aspect of learning mathematics. if the student's ability in the number of materials is low, it will affect the sub-material in other pisa content. this is supported by noviana and murtiyasa (2020), which assert that the direct approach to describing and measuring various items in quantity includes evaluating correlations and changes, gathering and analyzing data, and gauging certainty. this allows modeling of the situation to test space and shape, relationships and change, uncertainty and data (noviana & murtiyasa, 2020). additionally, compared to other content, the outcomes of mathematical literacy in terms of content amount are still relatively low. the study's findings by patriana et al. (2021) indicated that all pisa content requires low levels of mathematics proficiency. the same thing was conveyed by hasnawati (2016), which showed that space and shape had the lowest average score for student accomplishment, with an average score of 36.57, and that the most significant domain for student achievement was change and relationship, with an average score of 37.75 for each topic. research mahdiansyah and rahmawati (2014) also demonstrates that pupils' achievement in mathematical literacy is still somewhat low, at 24.9, in terms of content amount. achievement levels were 32.8, 26.8, and 25.7, respectively, in the content areas of uncertainty, data, change and relationship, and space and shape. in light of this, it can be said that the pisa content quantity is still relatively low. as a result, this study aims to examine junior high school students' aptitude for solving mathematical literacy issues. 2. method the qualitative research design for this study employed a case study research methodology. fifteen students from five different lubuk sikaping schools participated in this study in west sumatra and were observed as they attempted to solve mathematical literacy challenges. purposive sampling, which is the method of choosing samples with specific considerations, is used to choose the subject of the study. students with various skills are selected (high, medium, and low). the teacher-administered midterm exam reveals the pupils' aptitude. the information employed in this study is quantitative, derived from student math literacy test results, and qualitative, derived from interview responses. six literacy test questions based on pisa-oriented math problems on quantity and content serve as the study's research tool. this study's topic of choice is numerical data. because the operation of numbers is a crucial component of learning mathematics, researchers focus on the substance of numbers. this is supported by research noviana and harisman et al., analysis of student's ability to solve mathematical literacy problems … 58 murtiyasa (2020). it claims if a student's aptitude for the number subject is low, it will impact the sub-material in other pisa subjects. questions on pisa adapted from previously published research articles, notably noviana and murtiyasa's study, cover all levels. the gathered mathematical literacy questions for the amount of content are shown below (noviana & murtiyasa, 2020), which can be seen in table 2. table 2. the mathematical literacy problem of quantity content no mathematical literacy problems 1 you make your salad dressing. this is the recipe for 100 ml of sauce. salad oil 60 ml vinegar 30 ml soy sauce 10 ml for a 150 ml batch of sauce, how many milliliters of salad oil are required? (level pisa: 1, context: personal, process: formulate) 2 jenn works in a place that rents out video games and dvds. the annual membership price in this store is 10 zeds. as can be seen in the following table, members pay less to rent dvds than non-members do: non-member rental fee for one dvd dvd rental fee for members 3,20 zeds 2,50 zeds what is the bare minimum members must rent to make up the cost of their membership dues? could you show me your work? (level pisa: 2, context: personal, process: formulate). 3 the carpenter will need the following supplies to complete a set of bookcases: four long wooden panels, six short wooden panels, twelve little clips, two large clips, and fourteen screws are now in the carpenter's inventory. additionally, he has 510 screws, 20 large pins, and 200 little clips. how many sets of bookcases can a carpenter construct? (level pisa: 3, context: educational and occupational) 4 a simple pizza with cheese and tomato as toppings is available in pizzerias. additionally, you can customize your pizza by adding other toppings. bacon, ham, mushrooms, olives, and more topping options are available. ross wished to have two different topping combinations on his pizza. how many possible combinations does ross have? (level pisa: 4, context: educational and occupational) volume 12, no 1, february 2023, pp. 55-68 59 no mathematical literacy problems 5 as stated in the table, a milk factory in bandung can produce milk daily with a fixed pattern within 10 days. mr amin is an employee of the dairy factory inspecting table damage. day to number of productions there were lots of table malfunctions. 1 1500 30 2 1400 28 3 1550 31 4 1500 30 5 1600 32 6 1600 32 how much mr amin found production and table damage on the 10th day? (level pisa: 5, context: occupation) 6 the following picture is a javanese batik motif. for fabrics with an area of 6 cm2, 1 piece of white flower consisting of 4 petals is painted, then for a material of 12 cm2, 4 parts of a white flower are painted, and so on. several white flowers = n2 and area of fabric = 6n if n is the number of rows of white flowers. if the fabric used is getting more comprehensive with the same white floral motif as the previous pattern, which one increases faster: the number of white flowers or the area of the fabric? prove and explain your answer! (level pisa: 6, context: occupation) after fifteen students were asked to answer the above six problems, they were interviewed about their answers. the interview strategy is known as the semi-structural strategy. this is based on research from harisman et al. (2021), this used a technique for conducting interviews known as the semi-structural method. the section rubrics in table 3 were utilized to evaluate the student's responses (noviana & murtiyasa, 2020). table 3. maximum score for each question question indicators maximum score 1 and 2 formulating the situation mathematically 10 3 and 4 reasoning 15 5 and 6 solving problems 25 the data were evaluated and graded based on the evaluation criteria stated in table 3 and using mentally accurate replies. the results of the interview analysis were then used to determine the student's conduct from the data. harisman et al., analysis of student's ability to solve mathematical literacy problems … 60 3. result and discussion the math literacy skills of the junior high school students at lubuk sikaping are described in this section. the labels h for high ability, m for medium class, and l for low ability are given to students. s-1 stands for the first school, s-2 for the second, s-3 for the third, s-4 for the fourth, and s-5 for the fifth. table 4, which lists the names and abilities of students in each school. table 4. student labels of student for each school student school origins school one (s-1) school two (s-2) school three (s-3) school four (s-4) school five (s-5) fa (h) da (h) rz (h) da (h) hb (h) ai (m) ro (m) ct (m) mk (m) by (m) dt (l) an (l) fz (l) ag (l) ri (l) labeling is done to group students more efficiently according to knowing mathematical literacy skills in the final item. student representatives who have the same response to the first question and the same responses to the other questions are given the data description. the description concludes with a summary of the study's findings for each institution student. based on the labels already provided above, the mathematical literacy abilities of the students will be described. here is a summary of the student's mathematical formulations of the situation (level 1 pisa). figure 1 shows the results of student work that the student has a relatively high ability to formulate situations mathematically. translation: given recipe for making 100 ml of sauce -salad oil 60 ml -vinegar 30 ml -soy sauce 10 ml question how much is salad oil needed to make 150 ml of dressing? answer oil salad 60 ml = 0,6 vinegar 30 ml = 0,3 soy sauce 10 ml = 0,1 oil salad = 0,6 x 150 = 6 x 15 = 90 ml so, the salad oil needed is 90 ml figure 1. student answers hs-3 figure 1 shows that the student has written down the available information on the question in total and can make conclusions about the question. the students' answers are all correct, indicating that they have read the questions thoroughly so that they can answer the questions from the questions. the following are the results of the interview that match the statement above: p : what information do you get on the question? hs-3 : recipe for making 100 ml of sauce p : what is being asked about it? volume 12, no 1, february 2023, pp. 55-68 61 hs-3 : the amount of salad oil for 150 ml of sauce p : how are you going to solve the problem? hs-3 : multiplied as usual. the next step is the analysis of the answers to the first question given by different students from different schools, as shown in figure 2. translation: worth comparison many (ml) oil salad (ml) 100 60 150 𝑥 𝑎1 𝑏1 = 𝑎2 𝑏2 100 60 = 150 𝑥 100 𝑥 = 9000 𝑥 = 9000 100 𝑥 = 90 so, the salad oil needed is 90 ml figure 2. student answers hs-1 figure 2 demonstrates that the student has noted the information about the question and can draw conclusions about it. this is evident from the answer sheet; students can deduce the data from the tables and respond to questions using a grade comparison. the pupils' answers are accurate, showing that they have carefully read, comprehended, and been able to respond to the question. the outcomes of the hs-1 interview process are listed below. p : what information do you get on the question? hs-1 : recipe, mam p : what is being asked about it? hs-1 : plenty of salad oil to make 150 ml of salad sauce. p : how do you solve the problem? hs-1 : by using a comparison method, mam the interview results and the student's responses indicate that hs-1 pupils have reasonably excellent mathematical literacy skills, demonstrating that they comprehend problems and have effective problem-solving techniques. additionally, figure 3 shows the responses from hs-4 students for other diverse solutions. translation: oil salad = 60 ml + 30 ml = 90 vinegar = 30 ml + 15ml = 45 soy sauce sauce = 10 ml + 5 ml = 15 + 150 figure 3. student answer hs-4 figure 3 demonstrates how poorly the pupils comprehend the issue. this is evident from the answer sheet: students cannot discern questions from the questions and do not record general information, so the children do not comprehend the problem. the interview's findings that support the previous assertion are as follows. harisman et al., analysis of student's ability to solve mathematical literacy problems … 62 p : what information do you get on the question? hs-4 : how to prepare sauce powder p : what is being asked about it? hs-4 : the amount is mam p : how do you solve the problem? hs-4 : summing them up, mam hs-4 learner answers, hs-4 does not understand the problem, so it cannot answer the question correctly. the next question is done with the same treatment as the first question. low-skilled students tend not to write down in detail what is asked and the strategies used in solving the problem. moderately capable students tend to identify issues well, but some students cannot answer correctly and make many miscalculations. students with high mathematics skills tend to do better because they can accurately calculate and identify problems. based on the above description, it is known that students with low, medium and high abilities have different levels of ability. although it does not meet all indicators of mathematical literacy, students can already understand and solve the problem's meaning. this aligns with research conducted by kafifah et al. (2018), which states that some students can interpret and solve problems and even communicate the correct answers. this is in line with how the draft assessment analytical framework defines mathematical literacy, which is the capacity for people to formulate, use, and comprehend mathematics in various circumstances. this shows that acquiring knowledge, articulating, solving, and understanding issues based on logic using existing concepts and facts and following suitable methods are all aspects of mathematical literacy. the study's findings aim to identify the skills held by fifteen junior high school pupils. after reviewing the assignments and creating interview transcripts for each student, in this paper, the students' mathematical literacy skills are collected into three groups, namely: the time stone, the power stone, and the mind stone. time stone is a term for students who have low initial ability but have moderate literacy skills; power stone is a term for students who have the medium early ability but have high literacy skills; and finally, mind stone is a designation for students who have increased initial capabilities but have moderate literacy skills. the results of the above analysis showed 15 students from 5 different schools with different initial abilities also entered the category, including 3 students who had low initial knowledge and low mathematical literacy ability as well, 2 students in the time stone category, 4 students with the medium ability and moderate mathematical literacy ability, 1 student in the power stone category, 1 student with high initial knowledge and high literacy ability, and 3 students in the mind stone category. from the analysis results, it can be concluded that students with high initial abilities do not necessarily have high literacy skills. this follows the results of kafifah et al. (2018) that the mathematical literacy ability of students with high, medium, and low initial abilities is still below average, so it can be suggested to increase literacy questions regarding mathematical literacy skills in students. researchers in the future will be able to assess students' mathematical literacy skills using the findings of this analysis, and teachers will be able to enhance teaching strategies in the classroom using the results as a guide. the teachers should be aware of students' learning preferences and modify their teaching techniques (noviana & murtiyasa, 2020; putra et al., 2016; syawahid & putrawangsa, 2017). mathematical literacy skills must also be developed during learning (afriyanti et al., 2018; madyaratria et al., 2019; mardiana, 2018). the one of the essential life skills is mathematics literacy. it is a fundamental ability that is equally important to have as reading. as a result, mathematics instruction in schools volume 12, no 1, february 2023, pp. 55-68 63 must focus on fostering mathematical literacy and improving each student's capacity to use and apply mathematics to real-world issues or circumstances (dahm & de angelis, 2018; north, 2017; sumirattana et al., 2017; tokada et al., 2017). according to wati et al. (2019), if students possess mathematical literacy abilities, they will be able to recognize mathematical elements in the context of real-world problems and recognize known variables, math models, and problem-solving techniques, as well as design and put into practice solutions-seeking strategies and determine facts, procedures, algorithms, and mathematical models (kariman et al., 2019). math results back into real-life contexts and assessing mathematical solutions in real-life contexts (khanifah et al., 2019; mutia & effendi, 2020; zahroh et al., 2020). students' literations ability depend on teachers' and student background (harisman et al., 2020). 4. conclusion following work review and the creation of recordings of each student's interview outcomes, this paper grouped students' mathematical literacy abilities into three groups: the time stone, power stone, and mind stone. students with low initial abilities but moderate literacy skills are referred to as time stones; students with medium initial abilities but high literacy skills are referred to as power stones; and students with high initial abilities but moderate literacy skills are referred to as mind stones. the power stone is the initial ability of medium students but has high literacy skills. the last one, the mind stone, is for students with high initial abilities but moderate literacy abilities. references abdullah, a. a., & richardo, r. 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(2020). gerakan literasi matematika dalam peningkatan kemampuan pemecahan masalah matematis siswa [mathematical literacy movement in improving students' mathematical problem solving abilities]. delta-pi: jurnal matematika dan pendidikan matematika, 9(2), 165-177. https://doi.org/10.33387/dpi.v9i2.2293 https://doi.org/10.1016/j.kjss.2016.06.001 https://doi.org/10.31004/cendekia.v6i1.1093 https://doi.org/10.1088/1742-6596/895/1/012077 https://doi.org/10.26877/imajiner.v1i5.4456 https://doi.org/10.33387/dpi.v9i2.2293 harisman et al., analysis of student's ability to solve mathematical literacy problems … 68 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p109-122 109 the developing math electronic module with scientific approach using kvisoft flipbook maker pro for xi grade of senior high school students aulia fonda 1 , sumargiyani 2 1,2 universitas ahmad dahlan, jl. kapas no.9, umbulharjo, yogyakarta, indonesia 1 auliafonda@gmail.com, 2 sumargiyani04@yahoo.com received: december 07, 2017 ; accepted: june 01, 2018 abstract this study aims to determine the feasibility of learning media electronic module (e-module) based on kvisoft flipbook maker pro on derivative material with scientific approach using research and development research methodology. the study was conducted in two schools by taking 4 students as a student response in small classes and 20 students in a large class. the research instrument used in the form of evaluation formative form of teaching materials for material experts, media experts, and student responses. from the result of the assessment of expert material instrument obtained an average score of 101.25 with very good criteria, the assessment of expert media instruments obtained an average score of 116 with good criteria as well as on student response instruments obtained 86.44 with very good criteria. these results indicate that the e-module of mathematics with the scientific approach of the derived material for the xi class of the even semester high school is worthy of use in the process of learning in the classroom. keywords: e-module, derivative, scientific, kvisoft flipbook maker. abstrak penelitian ini bertujuan untuk mengetahui kelayakan media pembelajaran elektronik modul (e-modul) berbasis software kvisoft flipbook maker pro pada materi turunan dengan pendekatan scientific menggunakan metodologi penelitian research and development. pada materi turunan dengan pendekatan scientific menggunakan metodologi penelitian research and development. penelitian ini dilakukan di dua sekolah dengan mengambil 4 siswa sebagai respon siswa pada kelas kecil dan 20 siswa pada kelas besar. instrumen penelitian yang digunakan berupa lembar evaluasi formatif bahan ajar untuk ahli materi, ahli media, dan respon siswa.dari hasil penilaian instrumen ahli materi diperoleh skor rata-rata 101,25 dengan kriteria sangat baik, penilaian instrumen ahli media diperoleh skor rata-rata 116 dengan kriteria baik serta pada instrumen respon siswa diperoleh 86,44 dengan kriteria sangat baik. hasil tersebut menunjukkan bahwa e-modulmatematika dengan pendekatan saintifik materi turunan untuk kelas xi sma semester genap layak digunakan dalam proses pembelajaran di kelas. kata kunci: e-modul, turunan, saintifik, kvisoft flipbook maker how to cite: fonda, a., & sumargiyani, s. (2018). the developing math electronic module with scientific approach using kvisoft flipbook maker pro for xi grade of senior high school students. infinity, 7(2), 109-122. doi:10.22460/infinity.v7i2.p109-122. fonda, & sumargiyani, the developing math electronic module with scientific … 110 introduction education is an effort to develop the potential of students. education is a learning process that can produce expected behavioral changes. the development of science and technology increasingly encourages renewal efforts in the utilization of technology results in the learning process. teachers are required to be able to use the tools that can be provided by the school, and do not rule out that the tools are in accordance with the times. teachers can at least use cheap and efficient tools which, though simple, are necessary in order to achieve the intended teaching objectives (arsyad, 2011). learning is a process of interaction between several components including learning subjects, learning objects learned, and learning media. learning media is a means of communication used to convey messages or learning materials, to attract students 'interest on learning materials and improve students' understanding of the material presented (nuroifah, 2015). learning by using media in teaching and learning process has two important roles, namely: (1) media as teaching aid and (2) media as learning resource used by self by learners independently, so that learning media have various benefits that can be used during teaching and learning process (setyono, afri, & deswita, 2017). currently teachers in indonesia who are literate ict ranged from 10% -15%, this data shows that of the more than 2.7 million teachers in indonesia, it is estimated that at most 15% of teachers are able to exploit the potential of ict in the learning process. student learning outcomes show a significant difference between learning without media and learning using media. therefore, the use of learning media in teaching and learning process is highly recommended to enhance the quality of learning including computer media. learning mediabased learning places the participants to interact directly with the learning materials in accordance with the capabilities and interests of the participants (fadli, suharno, & musadad, 2017). learning media that can be developed one of them is teaching materials. teaching materials can be developed for teachers to be used by students such as modules. in the previous research has been developed electronic module with statistics for class vii smp / mts with scientific approach expressed feasible use in learning process. (marla & suparman, 2015). e-module creation can be developed with some existing applications. previous studies have developed a kvisoft flipbook maker-based learning medium that is considered feasible with a percentage of media validation results kvisoft flipbook maker obtaining 86.67%. mathematics is one of the subjects studied at all levels of education. at the level of primary and secondary education mathematics is included in the basic group that should be mastered by learners because of the importance of mathematics at the level of primary and secondary education is then the subjects of mathematics ranks first in this case the number of lessons (kintoko & sujadi, 2015). for students the material that includes elusive material in mathematics is derivative. derived material is one material that has a very wide range of applications both in mathematics itself, as well as in other branches of science such as in the fields of science, technology, economics and so on (megariati, 2014). learning derivative material still has not achieved the expected achievement of student learning outcomes is still low. therefore, improvement efforts are needed that can change the conditions in the learning (rokhman, 2014). many facts in the field still indicate that mathematics learning is only seen as a monotonous and procedural activity, that is the teacher explains the material, gives an example, assigns the student to do the exercise questions, checks the student's answers in passing, then discusses the solution of the problem which is then exemplified by the students (ngilawajan, 2013). teachers have the duty and responsibility optimally able to carry out the learning activities well marked by the high student activeness.kakaktifan students in question is the involvement of students in carrying out activities ranging from discussion of concepts, volume 7, no. 2, september 2018 pp 109-122 111 the process of finding solutions to the conclusion of the concepts learned (sinurat, syahputra, & rajagukguk, 2015). based on the results of interviews of teachers of sma negeri 8 yogyakarta and sma negeri 5 yogyakarta said that the 2013 curriculum print book issued by the government is still limited, no teacher develops modules, especially in electronic form on mathematics subjects, students rely more on lks and powerpoint from teachers in learning process, and lack of reference exercise questions for students. in the derivative and integral material some students are also not very interested because the built-in mindset says that the material is complicated by the number of descriptions, each form is different in completion and in further discussion the derivation is divided into trigonometric derivatives which are also very varied formulas. moreover in the curriculum of 2013 students do not have mathematical textbooks that match and these materials require additional references. students also need to practice a lot of questions in accordance with the 2013 curriculum with a scientific approach so that students are more active in the learning process. based on the mentioned problems, the success of learning activities to improve student learning outcomes is investigated in the following research questions: (a) the design of e module learning media using kvisoft flipbook maker pro software on derived material and (b) the feasibility of the emodule media. a study aimed at producing electronic module instructional media design (e-module) on derivative material for high school xi class students and to know the eligibility of e-module electronic learning media in derivative material for high school xi grade students that can improve student learning outcomes theoretical is expected to contribute to the development of science in supporting the world of education, especially in the learning of mathematics and the benefits can practically improve the skills in developing an interesting learning media and fun, especially in math lessons. method the research development used is the method of research and development (r & d). research development is a research method to develop a new product or refine an existing product and can be accounted for. this study involved 4 students of class xi mipa 5 and 20 students of class xi mipa 7 in sma negeri 8. for in sma negeri 5 yogyakarta research involves 4 students of class xi ipa 2 and 20 students class xi ipa 6. instruments used in the form of observation sheet, questionnaires, and interview guides. data collection techniques used in the form of observation, giving questionnaires, and interviews. while the data analysis techniques are: a) qualitative data, in the form of criticism and input from material experts and media experts obtained from the validation. in addition, there is also e-module feasibility data obtained from questionnaire analysis of feasibility test of material experts, media experts and student response questionnaire to e-module.b) quantitative data in the form of score of emodification feasibility test result by material expert, media expert and also score of questionnaire result of student response to the product. questionnaire assessment is developed and developed based on an evaluation component that includes 1) content feasibility, 2) language, 3) presentation, 4) kegrafikan. based on the above criteria, a questionnaire grid for media expert feasibility test, feasibility test of the material expert and student response to e-module with several indicators, among others: fonda, & sumargiyani, the developing math electronic module with scientific … 112 table 1. indicator indicator material expert media expert student response feasibility of content 1. compliance with ki, kd 2. compliance with teaching materials 3. conformity with the scientific approach 4. the truth of material substance 5. benefits for the addition of knowledge insight 6. conformity with values, morality, and social language 1. readability 2. clarity of information 3. compatibility with good and true indonesian language rules 4. effective and efficient use of language (short and clear) 1. readability 2. clarity of information 3. compatibility with good and true indonesian language rules 4. effective and efficient use of language (short and clear) 1. readability 2. clarity of information 3. compatibility with good and true indonesian language rules 4. effective and efficient use of language (short and clear) presentation 1. completeness of information 2. order of the dish 3. giving motivation, attraction 4. interaction (provision of stimulus and response) 5. the linkage in everyday life 1. clarity of purpose 2. order of the dish 3. giving motivation, attraction 4. interaction (provision of stimulus and response) 5. completeness of information 1. order of the dish 2. giving motivation, attraction 3. interaction (giving stimulus and response) 4. completeness of information 5. clarity of purpose volume 7, no. 2, september 2018 pp 109-122 113 indicator material expert media expert student response grapic 1. use of letters: type, color, and size 2. layout or layout 3. illustrations, pictures, photos 4. design view 5. video conformity 1. use of letters: type, color, and size 2. use of spaces 3. layout or layout 4. navigation key 5. compatibility in music 1. use of letters: type, color and size 2. layout or layout 3. illustrations, pictures, photos 4. design tampill 5. navigation key 6. animation, music, and video data that has been obtained through a questionnaire by expert assessment of products and students in the form of qualitative value will be converted into quantitative values likert scale. table 2. likert scale from the data that has been collected, we calculate the average by the formula: ̅ ∑ this development research focused on the development of electronic module or e-module using kvisoft flipbook maker pro software derived by scientific approach packaged in cd (compact disc). development of this media through several stages to obtain the feasibility of media e-module that can be used (figure 1). figure 1. step of research and development method (r&d) information score ss (very angree) 5 s (agree) 4 cs (enough agree) 3 ts (disagree) 2 sts (very disagree) 1 design validation product design data collection potential and problems trial usage product revision product trial product revision final product product revision fonda, & sumargiyani, the developing math electronic module with scientific … 114 the data obtained from both media experts, material experts and students is transformed into qualitative values based on ideal assessment criteria. the ideal scoring criteria are shown in table 2 below: table 3. covertion category quantitative data to qualitative data interval score criteria ̅ very good ̅ ̅ good ̅ ̅ enough ̅ ̅ not good ̅ very poor information: ̅ (max ideal score min ideal score) (max ideal score – min ideal score) where = empirical score = ideal mean = ideal standart deviation in data analysis to find the feasibility of this product, the highest score is 5 and the lowest score is 1. results and discussion results the development of e-module learning media using kvsoft flipbook maker pro software has gone through several stages of revision in accordance with the steps of r & d method. determination of feasibility test using 3 questionnaire instrument that is questionnaire of feasibility of material expert, media expert, and student response. questionnaire used as a quantitative e-modification feasibility test data. questionnaire obtained and calculated, then adjusted to the criteria above categories. from questionnaires material experts, media experts, and student responses are obtained the following calculations: table 4. result calcuting questionnaires feasibility of materials expert no expert score quantitative data criteria 1. expert 1 114 very good 2. expert 2 96 good 3. expert 3 99 good 4. expert 4 96 good average 101.25 very good based on the results of product quality assessment, it shows that the e-module is judged in terms of material by expert 1 material experts are included in the criteria are very good and include good criteria by expert 2, teacher of mathematics class xi sma negeri 8 yogyakarta stated e-module including both criteria and teacher of mathematics study class of sma negeri volume 7, no. 2, september 2018 pp 109-122 115 5 yogyakarta stated e-module including good criteria. the average results of the assessment in terms of the material included in the category is very good. table 5. result calcuting questionnaires feasibility of media expert no expert score quantitative data criteria 1. expert 1 116 good 2. expert 2 116 good 3. expert 3 116 good average 116 good based on the results of product quality assessment, indicating that the e-module is assessed in terms of media by lecturers of media experts included in the criteria of good, teachers of mathematics study of sma negeri 8 yogyakarta stated e-module including both criteria and teacher of mathematics study of sma negeri 5 yogyakarta stated e-modules include good criteria. average results of assessment in terms of media include in good category. table 6. result calcuting questionnaires feasibility of student response no. school average quantitative data criteria 1. sma negeri 8 yogyakarta 87,52 very good 2. sma negeri 5 yogyakarta 85,37 very good average 86,44 very good based on the questionnaire results of student responses, indicating that the e-modules that have been developed included in the very well category. table 7. results calculation questionnaire and questionnaire of student response no. aspects of assessment average quantitative data criteria 1. material expert 101,25 very good 2. media expert 116 good 3. student response 86,44 very good based on the results of the questionnaire assessment and student response questionnaire, it shows that the e-module is included in the criteria is very good for use in the learning process by the material experts and student responses as well as good criteria for media experts. discussion trial data in developing electronic module (e-module) math derivative material for xi grade even semester sma based on curriculum 2013 with scientific approach using kvisoft flipbook maker pro software. the results of this product is packaged in the form of cd (compact disk). this development is structured by the steps of the use of research and development (sugiyono, 2011). fonda, & sumargiyani, the developing math electronic module with scientific … 116 potentials and problems especially in math lessons and the availability of mathematics e module xi class in high school that some students still think mathematics difficult to understand, there is no teacher who uses and develop e-module as a source of learning mathematics students, as for computers and internet network in the school has not been maximally utilized as a student learning media with a scientific approach. from description above can be arranged a map of needs in the manufacture of e-module. the researcher developed a mathematical e-module derived material for an even semester grade xi class based on the 2013 curriculum with a scientific approach. the development of e-module derived material begins by designing e-module products. stages of product design are emodule writing in accordance with the guidance of development guide and in accordance with the writing module (arsyad, 2011). making this product using microsoft word 2010, corel draw suite x5, geogebra, nitro pdf, macromedia flash and kvisoft flipbook maker pro. the resulting product is then revised by material and media experts then tested to students. here's the display of the results of each design, namely: cover the cover view of the e-module contains the author's name, title, supporting material of derived material, subjects, and usage goals. while the cover on the back contains the author's profile. the front and rear cover cover of the e-module can be seen in figure 2 and figure 3. figure 2. front cover figure 3.back cover title page the cover cover of the e-module contains the title, author name, mentor name, editor name, cover design name, software used and name of agency. display page title can be seen in figure 4. figure 4. title page volume 7, no. 2, september 2018 pp 109-122 117 foreword contains gratitude for the completion of the e-module of mathematics material class xi grade even grade semester based on the 2013 curriculum with a scientific approach and a brief explanation of the contents of the e-module. the introductory view can be seen in figure 5. figure 5. foreword e-modul table of content the table of contents helps the users of the e-module to find the desired page and is useful for knowing all the things contained in the e-module. the table of contents view can be seen on figure 6. figure 6. table of content preliminary introduction contains brief descriptions, requirements, core competencies, basic competencies, activities, learning objectives, scientific approaches, benefits, and usage instructions. the preliminary view can be seen on figure 7. figure 7. preliminary e-modul fonda, & sumargiyani, the developing math electronic module with scientific … 118 concept maps the e-module concept map contains the learning activity structure that e-module users will learn. the concept map view can be seen on figure 8. figure 8. concept maps learning activities in this learning activity is divided into three parts namely learning activities 1, learning activities 2, and learning activities 3.display each learning activity can be seen on figure 9(i), 9(ii), and 9(iii). figure 9(i). learning activites 1 figure 9(ii). learning activites 2 volume 7, no. 2, september 2018 pp 109-122 119 figure 9(iii). learning activites 3 evaluation this evaluation is made using macromedia flash, so that e-module users are facilitated in doing and calculating the scores of evaluations that have been done. the evaluation view can be seen on figure 10. figure 10. ealuation answer keys and glossary this final section contains answer keys, glossary and bibliography.can be seen on figure 11(i) and 11(ii). figure 11(i). answer keys fonda, & sumargiyani, the developing math electronic module with scientific … 120 figure 11(ii). glossary the trial procedure is by presenting e-module in computer and students enthusiastically using e-module and actively doing question and answer to the researcher. the researcher guides the process of product testing with the accompanying math teacher. the researcher also provided a little explanation of the material presented on the e-module. furthermore, the researcher distributed the assessment sheet in form of formative evaluation instrument instrument and the students fill out the assessment sheet. this assessment sheet aims to determine the student's response to e-modules that have been developed. overall, the result of the research shows that e-module can be used. the result of questionnaire obtained is processed into a quantitative data in the form of average score calculation. from table 8 it can be seen that the e-module developed by the researcher entered the category very well so that the e-module is feasible to be used in the learning process both accompanied by the teacher and used as learning independently. conclusion the feasibility of mathematical e-module products using kvisoft flipbook maker pro software derived material for class xi sma based on the 2013 curriculum with scientific approach developed included in very good category based on the calculation result of average composite score of material experts of 101.25 and good category of the media experts of 116 and the category is very good from the average score of student response is 86.44. so that the mathematics e-module material derived class xi sma semester evenly based on the 2013 curriculum with scientific approach worthy of use in the process of learning in the classroom. references arsyad, a. (2011). media pembelajaran. jakarta: rajawali pers depdiknas. 2003. fadli, a., suharno, s., & musadad, a. a. (2017). deskripsi analisis kebutuhan media pembelajaran berbasis role play game education untuk pembelajaran matematika. in prosiding seminar nasional teknologi pendidikan. kintoko, k., & sujadi, i. (2015). pengembangan media pembelajaran matematika berbantuan komputer dengan lectora authoring tools pada materi bangun ruang sisi datar kelas viii smp/mts. jurnal pembelajaran matematika, 3(2). marla, a., & suparman. (2015). pengembangan e-modul matematika materi statistika untuk kelas vii smp/mts dengan pendekatan saintifik. admathedu, 355-361. volume 7, no. 2, september 2018 pp 109-122 121 megariati, m. (2014). peningkatan hasil belajar matematika pada materi turunan fungsi menggunakan teknik probing prompting di kelas xi ipa 1 sekolah menengah atas negeri 2 palembang. jurnal pendidikan matematika, 5(1). ngilawajan, d. a. (2013). proses berpikir siswa sma dalam memecahkan masalah matematika materi turunan ditinjau dari gaya kognitif field independent dan field dependent. pedagogia: jurnal pendidikan, 2(1), 71-83. nuroifah, n. (2015). pengembangan media pembelajaran berbasis aplikasi android materi sistem ekskresi siswa kelas xi sma negeri 1 dawarblandong mojokerto. jurnal mahasiswa teknologi pendidikan, 1(1). rokhman, n. (2014). multimedia pembelajaran turunan bernuansa konstruktivisme dan problem solving. indonesian digital journal of mathematics and education. 1(1):112. setyono, t., afri, l. e., & deswita, h. (2017). pengembangan media pembelajaran matematika dengan menggunakan macromedia flash pada materi bangun ruang kelas viii sekolah menengah pertama. jurnal ilmiah mahasiswa fkip prodi matematika, 2(1). sinurat, m., syahputra, e., & rajagukguk, w. (2015). pengembangan media pembelajaran matematika berbantuan program flash untuk meningkatkan kemampuan matematik siswa smp. jurnal tabularasa, 12(02). sugiyono (2011). metode penelitian pendidikan (pendekatan kuantitatif, kualitatif, dan r&d). bandung: alfabeta. fonda, & sumargiyani, the developing math electronic module with scientific … 122 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p291-306 291 infinity generalizations and analogical reasoning of junior high school viewed from bruner's learning theory lilis marina angraini1*, vahid norouzi larsari2, ilham muhammad3, nia kania4 1universitas islam riau, indonesia 2charels university, czechia 1universitas pendidikan indonesia, indonesia 4universitas majalengka, indonesia article info abstract article history: received may 16, 2023 revised jul 24, 2023 accepted jul 27, 2023 published online aug 1, 2023 inductive reasoning has an important role in mathematics learning. it includes making generalizations and analogical reasoning. while a generalization explains the relationship between several concepts applied in more general situations, analogical reasoning compares two things. this research is qualitative and descriptive. it reviews and describes the mathematical reasoning abilities of junior high school students based on bruner's learning theory. it was conducted at one of the junior high schools in pekanbaru in the eighth grade in the 2022/2023 academic year, involving 70 students. the students were divided into three categories of prior mathematical knowledge: low, medium, and high. the instruments used to obtain data on how mathematical reasoning abilities relate to bruner's learning theory in this study were (1) a test of mathematical reasoning abilities and 2) an interview guide. the results show that the average mathematical reasoning abilities of the eighth graders in this study were very high for the material on arithmetic sequences and series and low for the material on geometric sequences and series. however, the eight grade students' average generalizing and analogical reasoning abilities were quite good for both materials. keywords: analogical reasoning, generalized reasoning, sequences and series this is an open access article under the cc by-sa license. corresponding author: lilis marina angraini, department of mathematics education, universitas islam riau jl. simpang tiga, pekanbaru, riau 28288, indonesia. email: lilismarina@edu.uir.ac.id how to cite: angraini, l. m., larsari, v. n., muhammad, i., & kania, n. (2023). generalizations and analogical reasoning of junior high school viewed from bruner's learning theory. infinity, 12(2), 291-306. 1. introduction mathematics is one of the subjects that are taught from an early age even up to the higher education level. this is because mathematics has an important role and benefits in everyday life (angraini, 2019). someone with an ability in mathematics can form a systematic mindset, reason, make conjectures, make decisions carefully, thoroughly, have curiosity, be creative, and be innovative (erdem & soylu, 2017; habsah, 2017; hidayat & https://doi.org/10.22460/infinity.v12i2.p291-306 https://creativecommons.org/licenses/by-sa/4.0/ angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 292 husnussalam, 2019; hidayat et al., 2018; hutajulu et al., 2019; sukirwan et al., 2018). in addition, mathematics is a tool used to support knowledge in the social, economic, and scientific fields. one example of the application of mathematics is used in personal or business financial management, such as calculating expenses, income, savings, interest, investments, and budget planning. mathematics has special characteristics, so mathematics learning needs to be handled specifically as well. one of the special characteristics of mathematics is its emphasis on deductive processes that require logical and axiomatic reasoning. deductive thinking is essential in the context of mathematics for constructing proofs, establishing the truth of mathematical statements, and ensuring logical coherence. it forms the foundation of mathematical rigor and precision, enabling mathematicians to confidently explore, develop, and communicate mathematical ideas (hidayat & aripin, 2023; ningsih et al., 2023). this also shows that mathematics emerges from thoughts related to ideas, processes, and reasoning (bernard & chotimah, 2018; habsah, 2017; herbert et al., 2015; maharani, 2014; nu’man, 2012). the process of thinking about mathematical ideas requires an understanding of problems related to the material being considered and the ability to reason. the results of the 2018 programme for international student assessment (pisa) indicate that the quality of education in indonesia is rated the 75th out of 80 nations, with a decrease in pisa score in each subject area, including a drop from 386 to 379 in mathematics. in addition to these findings, the trends in international mathematics and science study (timss) ranked indonesia the 44th out of 49 nations. the 2015 timms scores for mathematics achievement were 54% poor, 15% intermediate, and 6% high. the pisa and timms findings indicate that mathematics education in indonesia is of a very poor standard. the timss results and decreased pisa scores in every subject area, including mathematics, indicate that indonesian students have difficulty applying mathematical reasoning to solve complex mathematical problems. from this data, it can be concluded that there are serious problems in teaching and learning mathematics in indonesia. low mathematical reasoning ability can hinder students' development in understanding more complex mathematical concepts and can have a negative impact on their academic achievement in various fields. the pisa and timms results are only one indicator that reflects the overall situation of mathematics education in indonesia. however, there are still many factors that can affect students' mathematical abilities, such as curriculum, teaching methodology, teacher qualifications, student participation rates, socio-economic factors, and learning culture. the reasoning is the main characteristic of mathematics that is inseparable from the activities of studying and solving mathematical problems (jeannotte & kieran, 2017; norqvist et al., 2019). reasoning abilities are critical to the understanding of mathematics. this is because mathematics is a science that has axiomatic deductive characteristics, which requires thinking and reasoning skills to understand it. the mathematical reasoning ability is the process of thinking mathematically in obtaining mathematical conclusions based on facts or available or relevant data, concepts, and methods (lestari & jailani, 2018; matapereira & da ponte, 2017). according to jäder et al. (2017) and hidayat et al. (2022) there are six general skills in reasoning, namely, (1) identifying similarities and differences, (2) problem-solving, (3) argumentation, (4) decision-making, (5) testing hypotheses and conducting scientific investigations, and (6) using logic and reason. the introducing students to reasoning has several advantages: (1) if students are given the opportunity to use their reasoning skills in making predictions based on their own experiences, they will remember it more easily; (2) if students are required to use their reasoning, it will encourage them to make conjectures; volume 12, no 2, september 2023, pp. 291-306 293 infinity and (3) it helps students understand the value of negative feedback in deciding an answer (marasabessy, 2021; mukuka, balimuttajjo, et al., 2020; mukuka, mutarutinya, et al., 2020). bruner's theory is closely related to mathematics learning (gading et al., 2017; wen, 2018). brunner (2019) stated that learning mathematics entails gaining an understanding of the mathematical ideas and structures included in the material being studied, as well as searching for correlations between these concepts and structures. students must establish order by manipulating materials that correspond to intuitive regularities they already possess. hence, they participate cognitively in the process of learning. additionally, bruner indicated that the greatest approach to learning is to comprehend ideas, meanings, and connections via an intuitive process. bruner proposed that learners construct their own knowledge by discovery learning (joshi & katiyar, 2021). in particular, in learning that encourages students to actively seek and acquire information from their experiences, students naturally offer outcomes to themselves and seek answers to issues via their own efforts, resulting in the production of meaningful knowledge (inde et al., 2020; rahmayanti, 2021; tanjung et al., 2020). according to brunner (2019), if the teacher allows students to discover a rule (including concepts, theories, and definitions) through examples that describe/represent the rules that are the source, the learning process will go smoothly and creatively; in other words, students will be led inductively to understand a general truth. inductive reasoning has an important role in the development of mathematics. by observing several cases, one can draw a conclusion as a generalization or abstraction of the cases. however, this conclusion is still provisional until it can be proven. if it has not been proven, then the pronouncement is only a conjecture. from this, it can be seen that the big role of inductive reasoning is to build mathematical knowledge because mathematical discoveries often occur through observations assisted by inductive reasoning. mathematics does have a deductive nature, as it relies on logical reasoning and the construction of deductive proofs to establish the truth of mathematical statements. however, used inductive reasoning in mathematics for several reasons: inductive reasoning allows students to explore new concepts and patterns. inductive reasoning helps students generate hypotheses that can guide their mathematical investigations. inductive reasoning is also valuable in mathematics for identifying potential counterexamples. inductive reasoning plays a role in developing mathematical intuition and creativity (english, 2013; jablonski & ludwig, 2022). inductive reasoning consists of making generalizations and analogical reasoning. a generalization is an explanation of the relationship between several concepts that are applied in more general situations. conclusions drawn from inductive generalizations can take the forms of either rules or predictions based on those rules. meanwhile, analogical reasoning compares two different things. an inductive analogy not only shows similarities between two different things; it also draws conclusions on the basis of those similarities. analogies can help students understand a material through comparisons with other materials to look for similarities in nature between the materials being compared (english, 2013; jablonski & ludwig, 2022; piaget, 1999). durak and tutak (2019) and sumartini (2015) explained that in mathematics, mathematical reasoning is the process of mathematically thinking in order to arrive at mathematical conclusions based on facts or accessible or relevant data, ideas, and procedures. research results by durak and tutak (2019) and sumartini (2015) shows that the increase in the mathematical reasoning abilities of students who receive problem-based learning is better than students who receive conventional learning. gifted students have better reasoning skills in applying statistical concepts, interpreting data, or solving statistical problems compared to ordinary grade students. reasoning is very important in mathematics angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 294 learning because it constitutes a goal of mathematics learning, in addition to other goals related to understanding concepts that teachers already know, such as numbers, comparisons, geometry, and algebra. analogical reasoning is a cognitive process that involves identifying similarities and making connections between different situations or concepts. studying analogical reasoning in junior high school students is important for fostering creativity and critical thinking. analogical reasoning promotes creative thinking and the ability to see relationships and patterns between seemingly unrelated concepts. exploring analogical reasoning in junior high school students can contribute to the development of their critical thinking skills and creativity. generalization refers to the ability to recognize and apply patterns or principles across different contexts. understanding generalization in junior high school students is important for deepening conceptual understanding. generalization requires students to identify commonalities and underlying principles among various examples or situations. by studying how students generalize mathematical concepts, educators can gain insights into the depth of their conceptual understanding. this study aims to look at students' mathematical reasoning abilities, especially in terms of the generalizing and analogical reasoning abilities of the eight grade students in the materials on arithmetic and geometric sequences and series, in relation to bruner's learning theory. 2. method this research is qualitative and descriptive in nature. this study describes the mathematical analogical reasoning and generalizing abilities of junior high school students based on bruner's learning theory. the data analysis technique used is interpretative analysis. interpretive analysis techniques involve interpretation and in-depth understanding of the data that has been collected. researchers try to understand the context, meaning, and relationships that emerge from the data being analyzed. this understanding can be based on bruner's learning theory and relevant frameworks, and supported by emerging findings from the data analysis. this research was conducted on the eight grade students of junior high schools 34 pekanbaru in the 2022/2023 academic year. there were a total of 70 the eight grade students at junior high schools 34 pekanbaru, all of whom were taken as sample in this study, to obtain more in-depth information about the mathematical reasoning abilities of junior high school students for further research development. the students were divided into three categories of prior mathematical knowledge based on the scores gained from a test on previous materials, namely, low, medium, and high categories. in this study, the mathematical analogical ability refers to the process of drawing a conclusion on the basis of similarities by comparing two different things. this conclusion can later be used to explain or as a basis for reasoning. the analogical ability was measured using these indicators (english, 2013): (1) the student could recognise patterns (from images or numbers) and (2) the student could ascertain how the visual patterns or numbers relate to one another. the capacity to derive general inferences from the primary structures observed is in this study referred to as mathematical generalizing ability. patterns, general principles, and specific examples are observed in accordance with some underlying rules. the indicators of the mathematical generalizing ability used in this study were as follow (english, 2013): (1) the student could produce general rules and patterns and (2) the student could use the generalizations that they had made to solve problems. volume 12, no 2, september 2023, pp. 291-306 295 infinity in this study, bruner's theory of the learning process was implemented in the following operational steps: (1) determining learning objectives; (2) identifying the characteristics of the students (prior mathematical knowledge); and (3) conducting an assessment of student learning outcomes based on the theory of learning in three stages, namely, the enactive, iconic, and symbolic stages (jablonski & ludwig, 2022). the enactive stage is the learning stage where students are given the opportunity to manipulate concrete objects directly. the iconic stage is the learning stage where students manipulate concrete objects into images. the symbolic stage is the learning stage where students manipulate the images obtained from the previous stages into mathematical symbols. the instruments used to obtain data on how mathematical reasoning abilities relate to bruner’s learning theory in this study were (1) a test of mathematical reasoning abilities and 2) an interview guide. interviews were conducted directly between the researcher and the respondent, in which the researcher asked questions and received verbal answers from the respondent. interviews were conducted in a semi-structured manner (some questions were predetermined but there was flexibility in asking additional questions) which were conducted after the administration of the questionnaire. the data obtained were calculated using statistical tests, and the results are to be explained in depth. the following are two of the questions in the mathematical reasoning abilities test taken by students (see figure 1). figure 1. the questions of the mathematical reasoning abilities 3. result and discussion this study aims to look at the mathematical generalizing and analogical reasoning abilities of junior high school students for the materials on arithmetic and geometric sequences and series from the perspective of bruner's learning theory. to gain information on the students' mathematical reasoning abilities for the materials, a test was administered with results provided in table 1. angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 296 table 1. data on students' generalizing and analogical reasoning abilities descriptive statistics students n 70 means 70.67 sd 7.76 max 90 min 60 table 1 show that average score of the generalizing and analogical reasoning abilities of the eight grade students of junior high schools 34 pekanbaru was quite high, but none of the students achieved a score of 100. the highest score obtained was 90. table 1 shows that overall, eighth grade students at smp 34 pekanbaru have good generalization and analogy thinking skills, as indicated by a high average score. this indicates that the majority of students have succeeded in applying mathematical thinking skills that involve generalizations and analogies in problem solving. despite the high average scores, no student achieved a perfect score of 100 on the generalization and analogy thinking skills test. this shows that even though students have good abilities, there is still room for improvement and further development in their mathematical thinking skills. the highest score achieved by the students was 90, which shows that there are some students who have very good mathematical thinking skills and are close to a perfect score. however, there is potential for students to reach even higher levels of ability. the average score of generalizing and analogical reasoning abilities was then calculated based on the prior mathematical knowledge, which was measured from the score of a test on previous materials (see table 2). table 2. average generalizing and analogical reasoning abilities based on prior mathematical knowledge ability n means high 23 77 medium 24 70 low 23 65 table 2 show that average generalizing and analogical reasoning abilities of students with high prior mathematical knowledge surpassed those of students with moderate and low prior mathematical knowledge by far. this shows that, descriptively, prior mathematical knowledge could also distinguish students' generalizing and analogical reasoning abilities. meanwhile, from the questionnaire data obtained on the students' mathematical reasoning abilities, the following were found: 1) the students strongly agreed that they were able to solve the exercise questions on arithmetic sequences and series given by the teacher; 2) the students had a high level of confidence in learning the arithmetic sequences and series material; 3) the students fairly agreed that they were able to solve the exercise questions on volume 12, no 2, september 2023, pp. 291-306 297 infinity geometric sequences and series given by the teacher; and 4) the students had a fair level of confidence in learning the geometric sequences and series material. during the learning process on the arithmetic and geometric sequences and series materials, the teacher implemented bruner's learning theory. bruner is best known for his "discovery learning" concept. he said that mathematics learning will be successful if the teaching process is directed to the concepts and structures made in the subject being taught, in addition to the relationships between concepts and structures. he also explained several operational steps to implementing his theory of the learning process: (1) determining learning objectives; (2) identifying the students’ characteristics (initial abilities), in addition to selecting the subject matter, determining the topics the students must learn inductively, developing the learning material in the form of inductive examples, illustrations, assignments, and so on for students to learn, and arranging lesson topics in the order from the simplest to the most complex, from the concrete to the abstract; and 3) conducting an assessment of the students’ learning processes and outcomes. in this study, the analogical reasoning ability was measured using three indicators: (1) the student could identify patterns (from images or numbers), (2) the student could ascertain the link between patterns or numbers; and (3) the student could estimate the rules behind the patterns. meanwhile, the generalizing ability had two indicators: (1) the student could generate general rules and patterns and (2) the student could use the generalizations they had made to solve problems (durak & tutak, 2019; english, 2013; jablonski & ludwig, 2022; sumartini, 2015). the following is the achievement of the generalizing and analogical reasoning abilities of the students for the arithmetic sequences and series material (see table 3). table 3. average scores of generalizing and analogical reasoning abilities of the arithmetic sequences and series material ability n means high 23 100 medium 24 95 low 23 90 the instrument used to measure the generalizing and analogical reasoning abilities of the students for the arithmetic sequences and series material consists of three questions. almost all students, whether they had a low, medium, or high ability, were able to answer these questions. as explained in brunner (2019) learning theory, if the teacher permits students to find a rule (including ideas, theories, and definitions) for themselves via examples that describe/represent the source rules, the learning process will go easily and creatively. in other words, students are guided inductively to comprehend a general truth. bruner's learning theory has advantages in a number of ways: (1) it determines if learning is worthwhile using discovery learning; (2) the student's newfound information will be retained for a very long time and it will be easy to recall; (3) problem-solving requires a great deal of discovery learning, which is preferred because it allows the student to display the information they have learned; (4) if generalizations are generated by the student for themselves rather than being supplied in final forms, transfer may be improved; (5) the use of discovery learning may help foster learning motivation; and (6) it enhances the student’s logical thinking and capacity for independent thought. angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 298 table 4. average scores of generalizing and analogical reasoning abilities of the geometric sequences and series material ability n means high 23 54 medium 24 45 low 23 40 the instrument used to measure the students' generalizing and analogical reasoning abilities for the geometric sequences and series material consists of two questions. almost all of the students, whether they had a low, medium, or high ability, were unable to answer these questions (see table 4). the questionnaire data collected support this finding: a) the students fairly agreed that they were able to solve the exercise questions on geometric sequences and series given by the teacher; b) the students had a fair level of confidence in learning the geometric sequences and series material; c) the students were bored in solving the problems on geometric sequences and series; and d) the students had a hard time learning the geometric sequences and series material. the following are some examples of students’ answers to the generalizing and analogical reasoning abilities test (see figure 2). figure 2. students’ answers to the mathematical reasoning abilities test the following are the results of interviews between researchers and students regarding student test answers: r : how do you determine the value of y. s : to determine the value of y, we need to find the pattern of the relationship between the number of seats and rows in theater a and theater b. in theater a, the seating capacity is 720 seats. in theater b, we know that there are 20 rows of seats, with 25 seats in the front row. however, it should be noted that many of the rows behind are 5 seats more than the front row. r : what can you conclude from this information? volume 12, no 2, september 2023, pp. 291-306 299 infinity s : if each row of seats in theater b has 25 seats plus a few extra seats, then we can calculate the number of seats in theater b with the row formula and an arithmetic series. r : right. now, we need to find out how many seats there are. based on the information that there are 5 more seats in the back row than the front row, what can you conclude? s : in this context, we know that there are 20 rows of seats as “n”, with 25 seats in the front row as “a”, the number of rows behind is 5 seats more than the front row as “b”. r : very well. so, what is the value of y based on your explanation? s : based on my calculations, the y value is 1450 seats. r : how would you solve problem number 2? s : to solve this problem, i will look for the pattern of growth in plant height from day to day. in the data given, the plant height on the first day is 2. however, i need to find a general pattern to determine the plant height on the n day. r : good. what can you conclude from this data? s : i see that in the data, the plant height for each day is not given. however, if i observe the height of the plants from the first to the second day, there is an increase. i can assume that there is a steady pattern of improvement every day. r : very well. with these assumptions, how can we determine the height of the plant on the n day? s : if there is a steady pattern of increase in plant height each day, we can assume that plant height follows a geometric progression. in a geometric series, each term is divided by the previous term to give a fixed ratio. r : good. so, how can we determine the height of the plant on day n in this geometric series? s : in a geometric series, we need to determine the growth ratio of the height of the plants. if the ratio is r, then the plant height on the nth day can be determined by the geometric sequence and series formulas. r : good. now, let's apply the formula to this problem. the plant height on the first day is 2. can you determine the plant height on the n day? s : yes, the plant height on day n is like the answer i wrote. r : very well. now, let's look at the second question. how would you determine the length of the plant on day 7? s : to determine the length of the plant on day 7, i need to determine the value of u7. r : good. can you determine the value of u7? s : yes, if i replace the value of n with 7 in the geometric sequence and series formula, the answer is as i wrote it. furthermore, interviews were conducted with students who had low prior mathematical knowledge: r : how do you feel about the material? s : to be honest, i found it difficult to understand some of the concepts. r : can you explain in more detail about the difficulties you are facing? s : i have difficulty understanding the meaning of the questions in the test. sometimes the questions are complex and i am at a loss as to what to do. i am often confused about angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 300 where to start. i sometimes use the wrong formula or forget the formula i am supposed to use. r : i saw that. is there anything else you think may have affected your test results? s : yes, i feel that i am often in a rush when working on questions. i want to solve it quickly, but sometimes i forget the formula. i also have difficulty distinguishing formulas in arithmetic sequences and series and geometric sequences and series. i often make mistakes in applying the right formula. r : from our discussion, it seems that several factors such as understanding the question, understanding the approach, using the right formula, and speed of work influence your test results. s : i think so. according to study by guarino et al. (2021), it is possible to understand children's challenges with analogy problem solving by evaluating visual attention during analogy problem solving and a measure of inhibitory control, which is thought to be crucial to analogical thinking. there is a connection between visual attention, inhibitory control, and a number of behavioral performance indicators. according to richland et al. (2006), the interplay between improvements in relational knowledge, the ability to integrate different relationships, and inhibitory control over featural distraction determines how analogical reasoning changes with age. because a variety of abilities, such as originality, creativity, and inductive reasoning, are crucial for future scholastic success and professional success. theoretically analogous thinking actually aids students in comprehending abstract notions that are then articulated or analogized to become tangible in studying mathematics. additionally, by connecting concepts that were previously distinct into a single idea, pupils are able to learn new information or concepts. following that, factors that must be taken into account when answering problems involving analogical reasoning include first ensuring that students have mastered any prerequisite knowledge or ideas. as a result, students can reduce conceptual blunders in their earlier learning and recognize concepts and problem-solving techniques that are included in the proper source problem that will aid in addressing the target problem. lee and lee (2023) found potential connections between answers and their expected generalization reasoning challenges for pupils. the issue with generalization reasoning is that it frequently foresees problems with educational actions intended to support students' mathematical knowledge. additionally, students overcome issues relating to their own selfefficacy and confidence by using generalization reasoning. yao (2022) identified two categories of representation transformations that helped go from empirical to structural generalizations: a structurally beneficial treatment and a mathematically significant conversion. mathematically significant conversions frequently result in generalizations that show why the generalizations are accurate, as opposed to structurally beneficial treatment. yao and manouchehri (2019) state that to help learners become more proficient at constructing mathematical generalizations, it is vital to better understand the forms that the constructive process might take in various mathematical contexts. the study reported here aims to offer an empirically grounded theory of forms of generalization middle students made as they are engaged in explorations regarding geometric transformations within a dynamic geometry environment. based on their sources, participants' statements about the properties of geometric transformations were categorized into four types: context-bounded properties, perception-based generalizations, process-based generalizations, and theorybased generalizations. although these forms of generalizations are different in their volume 12, no 2, september 2023, pp. 291-306 301 infinity construction process, with appropriate pedagogical support generalizations of the same type and different types of generalizations can build on each other. dgs mediated the construction of these forms of generalizations based on how learners used it. according to vamvakoussi (2017), it is important to take into account the usefulness of prior mathematical knowledge. actually, critics of conceptual shift perspectives on learning claim that they overemphasize the negative consequences of existing knowledge while ignoring students' creative ideas that can serve as the foundation for additional learning. teachers should frequently employ analogies as instructional mechanisms to teach concepts and procedures, selecting sources that are differentially generated to correspond to the analogy's content purpose (richland & begolli, 2016; richland et al., 2004). whether an analogy is used in response to a student's request for assistance depends on the source and target construction. students frequently participate in the parts of the analogy that take the least amount of analogical thought, but teachers typically maintain control of each comparison by providing the majority of the comparison. according to costello (2017), case-based learning and other constructivist teaching strategies can help students gain the analytical and problem-solving abilities necessary for the modern workplace. according to dias et al. (2020), the pedagogical approach could help children's intellectual growth by facilitating their learning through exploration, reaching all students, and encouraging the development of stochastic notions. the inhibitory control factor that has been posited to contribute to the protracted development of analogical and also dgs mediated the construction of these forms of generalizations based on how learners used it. in this study the researcher also conducted interviews with several students regarding the results of the material test for geometric sequences and series, 3 students who were interviewed as representatives of each ability level stated that: 1) students had difficulty understanding the meaning of the questions; 2) students have difficulty determining a principle that will be applied to solve the problem; 3) students are mistaken in determining the right concept to solve the problem; 4) students are in a hurry to work on the questions; 5) students do not memorize formulas properly and correctly; 6) students cannot distinguish formulas in arithmetic sequences and series from geometric sequences or series and students are not careful in understanding the questions asked. based on this, it is better for future research related to analogical reasoning and generalization to relate prior mathematical knowledge, inhibitory control, dynamic geometry environment in improving analogical reasoning and generalization abilities. these problems arose because solving problems on geometric sequences or series requires a higher level of thinking than solving problems on arithmetic sequences or series. learning with bruner's learning theory also has several weaknesses: 1) discovery learning requires high intelligence on the student’s part (if you are not smart enough, the learning will be less effective), and 2) theoretical learning takes time, and if it is not run in a guided or directed manner, it may result in chaos and uncertainty about the subject being studied. the paragraphs provide a comprehensive discussion on various aspects related to analogical reasoning and generalization in mathematics education. here are the key points derived from the provided information: a) importance of inhibitory control and visual attention: inhibitory control and visual attention play crucial roles in analogy problem solving. the ability to focus attention and control distractions is essential for successful analogical thinking. b) interplay between relational knowledge, integrative abilities, and inhibitory control: improvements in relational knowledge, the ability to integrate different relationships, and inhibitory control over distractions contribute to the development of analogical reasoning. these factors influence how analogical reasoning changes with age. angraini, larsari, muhammad, & kania, generalizations and analogical reasoning … 302 c) benefits of analogical thinking in mathematics education: analogical thinking helps students comprehend abstract notions by connecting concepts and making them tangible. it allows students to learn new information or concepts by connecting previously distinct ideas into a single idea. d) challenges in generalization reasoning: potential challenges in generalization reasoning, which may affect students' mathematical knowledge. students can overcome these challenges by using generalization reasoning, which also contributes to their selfefficacy and confidence. e) forms of generalization in mathematics, four types of generalizations: context-bounded properties, perception-based generalizations, process-based generalizations, and theorybased generalizations. these forms of generalizations have different construction processes but can build on each other with appropriate pedagogical support. f) importance of prior mathematical knowledge: prior mathematical knowledge is crucial when addressing problems involving analogical reasoning and generalization. it helps students reduce conceptual errors from earlier learning and recognize relevant concepts and problem-solving techniques. g) role of analogies and instructional strategies: teachers can employ analogies as instructional mechanisms to teach concepts and procedures. analogies should be selected based on their differential generation to correspond to the content purpose of the analogy. h) constructivist teaching strategies: the benefits of case-based learning and other constructivist teaching strategies in developing analytical and problem-solving abilities in students. i) challenges in solving geometric sequences and series problems: based on interviews conducted with students, difficulties in understanding questions, determining applicable principles, and mistaking concepts were identified. other challenges included rushing through questions, improper memorization of formulas, and difficulty distinguishing between arithmetic and geometric sequences or series. considerations for future research: future research should consider the relationship between prior mathematical knowledge, inhibitory control, and the dynamic geometry environment to enhance analogical reasoning and generalization abilities. in conclusion, the discussion highlights the importance of inhibitory control, visual attention, and prior mathematical knowledge in analogical reasoning and generalization. it emphasizes the role of analogies, instructional strategies, and constructivist approaches in facilitating students' mathematical learning. the challenges faced by students in solving geometric sequences and series problems provide insights for further research and instructional improvement. 4. conclusion the study's conclusion indicates that the average generalizing and analogical reasoning abilities of the eight grade students in this study for the arithmetic and geometric sequences and series materials as a whole was quite good. the average generalizing and analogical reasoning abilities of the students for the arithmetic sequences and 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(2019). middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. the journal of mathematical behavior, 55, 100703. https://doi.org/10.1016/j.jmathb.2019.04.002 https://doi.org/10.31764/jtam.v4i1.1736 https://doi.org/10.1007/s11858-017-0857-5 https://doi.org/10.1016/j.jmathb.2022.100964 https://doi.org/10.1016/j.jmathb.2019.04.002 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p155-164 155 students’ mathematical representation ability through implementation of maple lusiana 1 , yunika lestaria ningsih 2 1,2 universitas pgri palembang, jl. jend a. yani lorong gotong royong no. 9/10 ulu, palembang, south sumatera, indonesia 1 luu_sii_ana@gmail.com, 2 yunika.pgri@gmail.com received: june 25, 2018 ; accepted: august 31, 2018 abstract this research aims to investigate the students’ mathematical representation ability through the implementation of maple in learning definite integral and those with ordinary learning. this research used a quasi-experimental design with a pretest-posttest control group design. the population of this study was students of 2 nd semester of mathematics education program at pgri palembang university academic year 2017/2018. the sample in this study were students of class ii.a and ii.b, class ii.a as a control group and class ii.b as experiment group. data were collected through a test of mathematical representation ability. data in this study were analyzed through descriptive quantitative. the result of this study showed that students’ mathematical representation ability through implementation maple in learning definite integral better than those who get the ordinary learning. keywords: definite integral, students’ mathematical representation ability, maple. abstrak tujuan penelitian ini adalah untuk mengetahui perbedaan kemampuan representasi matematis antara mahasiswa yang mendapatkan pembelajaran berbantuan maple pada topik integral tentu dengan mahasiswa yang mendapat pembelajaran biasa. penelitian ini menggunakan metode quasi eksperimen dengan desain pretest-posttest control-group design. populasi dalam penelitian ini terdiri dari mahasiswa semester ii program studi pendidikan matematika universitas pgri palembang tahun akademik 2017/2018. sampel dalam penelitian ini adalah mahasiswa semester ii.a dan ii.b, dengan kelas ii.a sebagai kelas kontrol dan kelas ii.b sebagai kelas eksperimen. data dikumpulkan melalui tes kemampuan representasi matematis. analisis data dilakukan dengan deskriptif kuantitatif. hasil analisis data menunjukkan bahwa kemampuan representasi matematis mahasiswa melalui pembelajaran berbantuan maple lebih baik dari mahasiswa yang mendapat pembelajaran biasa. kata kunci: integral tentu, kemampuan representasi matematis, maple. how to cite: lusiana, l., & ningsih, y. l. (2018). students’ mathematical representation ability through implementation of maple. infinity, 7(2), 155-164. doi:10.22460/infinity.v7i2.p155-164. mailto:luu_sii_ana@gmail.com mailto:email-author-3@ymail.com lusiana & ningsih, students’ mathematical representation ability through … 156 introduction integral calculus is a subject which should be taken by students of mathematics education on the early semester. integral is a part of calculus concept that is important to be understood by students (serhan, 2015). according to oberg (2000) student’s understanding on this topic determines student understands figure in advanced mathematics problems out that involves integral in it such as advanced calculus, differential equation, and so on. furthermore, in learning integral most of students can do a calculation of integral procedurally, but they experience the trouble of settling integral issue conceptually and an implementation of definite integral (oberg, 2000). besides that, the trouble and weakness of students’ in learning integral were students difficult to determine upper limit and lower limit of an integral (nursyahidah & albab, 2017), students confused in answering the integral exercise (yuliana, tasari & wijayanti, 2017), students are not able to implement mathematical representation on definite integral topic (gonzales-martin & camacho, 2004), and students didn’t understand with symbolic and verbal definitions of definite integral (grundmeier, hansen, & sousa, 2006; serhan 2015). according to student’s analysis in definite integral topic which had been expressed, generally it can be argued that the student’s difficulty in learning the definite integral concept is related to mathematical representation ability. goldin (rangkuti, 2014) said, “representation is a configuration (format or arrangement) which is able to describe, represent, and symbolize something in one way.” kinds of representations are often used in describing mathematics ideas such as diagram, graph, table, mathematical statement, written text, or even combination of those (hutagaol, 2013). mathematical representation ability is an ability which is expected to be reachable in mathematics learning. nctm stated there are five abilities which are aim in this mathematics learning, those are problem-solving, reasoning, communication, connection and representation (hutagaol, 2013). mathematical representation ability is considered crucial, because it’s closely connected to communication ability and mathematical problem-solving (sabirin, 2014). mathematics concept representations in kinds of formats are used by someone in communicating the understood mathematics concept. in solving mathematical problem, mathematical representation, including picture and table would be able to help anyone who simplifies mathematical problem formerly was considered difficult. therefore, it is an important to analyze the mathematical representation ability of students when they learning the definite integral. according to serhan (2015) students must developed their ability to make connections between different representations. the students’ ability in make more than one representations of the definite integral concept, show the students understanding of it. in additional, huang (2015) stated that the concept of integrals can be presented in different representation. the graphical representation is used to solve the large area under a curve, and symbolic representation is used to solve the integration problem. many studies were aimed to overcome the student’s weaknesses in understanding integral. tall (1993) expressed that computer program can be used to help student in understanding calculus concept. i additional, nctm stated that students can explore and identify the concepts in mathematics and its relation using computer technology (ningsih & paradesa, 2018). one of software which is able to be used for helping a learning of integral calculus is maple. maple is software that was developed by waterloo maple inc to solve mathematics volume 7, no. 2, september 2018 pp 155-164 157 problem. according to garvan (2001), maple program has such a great potential that is useful in mathematics learning both at school and college. this program had been being used by many students, educators, mathematicians, statisticians, and scientists to do numeric and symbolic computation. as for the advantages of maple program was mentioned by garvan (2001) such as : (1) can do numeric computation exactly, (2) can do numeric computation for such a big number, (3) can do symbolic computation well, (4) has a plethora of default instruction in library and packages to mathematics processing widely, (5) has plot facility and animation for chart both in twodimensional (2d) and three-dimensional (3d), and (6) has programming language facility which can be used to write function, package, interactive window, etc. based on a study which had been established, using of maple in mathematics learning was proven able to give lots of benefits. the development of teaching materials in integral calculus maple-based that was established by paradesa, zulkardi & darmawijoyo (2013) has potential effect towards student’s ability of understanding in mathematical concept. maple integrating on integral calculus can enhance student’s mathematical concept understanding ability (awang & zakaria, 2013; ningsih & paradesa, 2018). maple which had been used as medium of calculus vector was successful in enhancing student’s understanding abiliy (noor et al., 2018). considering the advantages of using maple in learning, this study will investigate the implementation of it towards the students’ mathematical representations ability. therefore, the hypothesis of this study is “students’ mathematical representation ability through implementation maple better than those who get the ordinary learning”. this study limited on definite integral topic. the mathematical representation ability that was reviewed on this study is limited on three indicators, which are (1) visual representation: students can serve data or information from a representation into chart, (2) symbol representation or mathematical expression, students can write mathematics model or mathematics formula, and (3) verbal representation: students can give the explanation about steps of problem solving. method this study used quasi-experimental design with pretest-posttest control-group design. the population of this study was students of 2 nd semester of mathematics education program at universitas pgri palembang in aacademic year 2017/2018. sample in this study were students of class ii.a and ii.b, class ii.a as control group (cg) with 42 students, and class ii.b as experiment group (eg) with 42 students. students in eg get the implementation of maple, meanwhile the cg get the ordinary learning. data were collected through the essay test of mathematical representation ability. this test is including 3 (three) problems which containing the 3 (three) indicators of mathematical representation ability. example of the test, can be seen in figure 1. lusiana & ningsih, students’ mathematical representation ability through … 158 let the area under the curves, , and a. sketch the graph of the area b. what is the formula to calculate large of the area c. find the large of area, and give the explanation figure 1. the example of test point a) is indicator the visual representation, students have to sketch the graph based on the function correctly. point b) is problem for symbol representation or mathematical expression; students should make a formula to determine the large of the area. the last point is indicator the verbal representation, students have to explain by their own words how to find the large of area. the maximum score for each indicator is 4 and the lowest is 0. before conducting the research, two groups had been ensured that they had similar capability in mathematical representation. it was measured with homogeneity variants of pretest. then, data of students’ mathematical representation ability posttest were analyzed through descriptive quantitative. the normality and homogeneity test was conducted as the prerequisite test for inference statistic. the normality test use kolmogorov-smirnov method, and lavene method use for the homogeneity test. results and discussion results this study was established on month february-april, 2018. definite integral material in this study is limited on definite integral topic, the large of area and the volume of rotary objects. the result of students’ mathematical representation ability pretest can be seen on table 1. table 1. the result of pretest eg cg ̅ 22.02 20.91 s 5.81 6.42 max 31.25 40.63 min 6.25 9.38 according to table 1, known that average of pretest score of students’ mathematical representation ability between eg and cg is not too different. the result of pretest homogeneity can be seen in table 3. eg and cg have homogeneous variants. it means that two groups have the same initial of students’ mathematical representation ability. volume 7, no. 2, september 2018 pp 155-164 159 table 2. the result of homogeneity pretest groups n f sig conclusion eg 42 0.462 0.499 homogeneous cg 42 after the learning was held, two groups took the posttest. the result of posttest is described on table 3. based on table 3, known that the average posttest score of students’ mathematical representation ability in eg is 54.76 higher than the cg score. table 3. the result of postest eg cg ̅ 54.76 35.12 s 10.03 9.33 max 84.38 65.62 min 34.38 15.62 the result of normality and homogeneity of posttest as the prerequisite test can be seen on table 4 and table 5. table 4. the result of normality test groups n k-s sig conclusion eg 42 0.699 0.714 normal cg 42 1.200 0.112 normal table 5. the result of homogeneity test groups n f sig conclusion eg 42 0.504 0.480 homogeneous cg 42 based on the normality test, data of students’ posttest are in normal distribution. therefore, data testing is continued with variants homogeneity by using lavene test. based on table 5, the significant score was 0.480 which means higher than 0.05 so that posttest data variants on eg and cg are homogeneous. hypothesis testing for normal and homogeneous data of students’ mathematical representation ability are continued by doing t test. as for the tested-hypothesis in this study is students’ mathematical representation ability that got implementation of maple learning is better than those who got ordinary learning. the result of t test can be seen on table 6. table 6. the result of average differences test groups n t sig conclusion eg 42 9.291 0.000 different cg 42 based on table 6 shows the significant score of t test is 0.000, this score is lower than 0.05 which causes the research hypothesis of this study is acceptable. in other words, it shows that lusiana & ningsih, students’ mathematical representation ability through … 160 students’ mathematical representation ability that got implementation of maple learning on eg is better than those who got ordinary learning on cg. according to the analysis result of students’ response on posttest, students’ mathematical representation ability for each indicator can be seen on table 7. table 7. average score of students’ mathematical representation ability indicator on eg no measured aspects average score 1 visual representation : serve data and information from a representation into graphical representation 61.01 2 symbol representation or mathematical expression : a. build mathematics model b. write the mathematics formula 30.36 60.42 3 verbal representation : write an explanation about steps of problemsolving 54.96 table 7 shows that indicator which experiences the highest score is indicator 1, which the visual representation ability. on the contrary, indicator which experiences low score is indicator 2. in indicator symbolic representation, students are not able to make mathematical model correctly. the average score is only 30.36. discussion the result of this study showed that students’ mathematical representation ability through implementation of maple in learning definite integral topic is better than those who got ordinary learning. maple using in integral calculus learning gives a plethora of benefits on students. the advantage of maple in making graph helps them to make a correct graph and understand it. besides, three-dimensional animation graph in maple makes students more understand the volume of rotary objects, so that the right method can be applied in figuring problems out. for example on indicator 1, students have to make a representation into a graph correctly. pretest result shows that students are still not good enough at making graph of the large area on flat field that is limited by two curves. the mistakes that had been done by students on this indicator are, they cannot determine the right coordinate-point, they cannot determine cuttingpoint of two curves, and they cannot determine the upper and lower limit of an integral. this agrees with the finding of previous research study (nusyahidah & albab, 2017). the example of students’ response can be seen on this figure 2. volume 7, no. 2, september 2018 pp 155-164 161 figure 2. students’ eg response on pretest then, after the implementation of maple was applied, students’ response to this indicator increases. they’re already able to make graph well, determine cutting-point of two curves appropriately, and able to determine the upper and lower limit of integral. the example of students’ response can be seen on this figure 3. this finding show that students visual representation in make graph is increased. students are more attractive and supported to make the correct graph with maple. this condition is in line with the statement of previous research studies (kilicman, hassan & husain, 2010; ningsih & paradesa, 2018). figure 3. students’ eg response on posttest meanwhile, for indicator 2, part a) students are able to make the correct formula for finding the large area. the average score is 60.42. students seem like to solve the problem based on the formula that had been given. this fact shows that students were tends to solve the problem using the analytical thinking (huang, 2015). for indicator 2, part b) students are not able to make the correct mathematical model. students felt difficult to understanding the mathematics symbols in learning integral, as stated by grundmeier, hansen & sousa (2006). in indicator 3, students are not able to make the explanation of the problem solving based on their own words. students felt difficult in describe their answer; many of them just calculated the integral not give the way to find the answer. this difficulty is also similar with serhan (2015). lusiana & ningsih, students’ mathematical representation ability through … 162 conclusion based on the result of data analysis, it can be concluded students’ mathematical representation ability through the implementation of maple in learning definite integral topic is better than those who got ordinary learning. the average score of students’ mathematical representation ability on experiment-group (eg) is 54.76 higher than control-group (cg). the average score of students mathematical representation ability in this study is still have to be increased. based on findings, students are able to make a graph representation, even though the score is not big enough. in the future study this ability is still need to be developed. additionally, in this study, students still have so much weakness when they tried to build mathematical model from real life issues for definite integral topic and to give the explanation. the upcoming study will analyze how to make that ability better. acknowledgments this study is funded by universitas pgri palembang, under dipa penelitian lppkmk in academic year 2017/2018. researchers would like to thank to rector universitas pgri palembang, dean of fkip universitas pgri palembang, lecturers, and students who are involved in this project. references awang, t. s., & zakaria, e. 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(2000). an investigation of undergraduate calculus students ’ conceptual understanding of the definite integral. retrieved from https://scholarworks.umt.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=116 51&context=etd paradesa, r., zulkardi, z., & darmawijoyo, d. (2013). bahan ajar kalkulus 2 menggunakan macromedia flash dan maple di stkip pgri lubuklinggau. jurnal pendidikan matematika, 4(1), 95-109. rangkuti, a. n. (2014). representasi matematis. forum paedagogik, 6(1), 110-127. sabirin, m. (2014). representasi dalam pembelajaran matematika. jurnal pendidikan matematika uin antasari, 1(2), 33-44. serhan, d. (2015). students' understanding of the definite integral concept. international journal of research in education and science, 1(1), 84-88. tall, d. (1993). students’ difficulties in calculus. in proceedings of working group, 3, 13-28. yuliana, y., tasari, t., & wijayanti, s. (2017). the effectiveness of guided discovery learning to teach integral calculus for the mathematics students of mathematics education widya dharma university. infinity journal, 6(1), 01-10. https://scholarworks.umt.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=11651&context=etd https://scholarworks.umt.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=11651&context=etd lusiana & ningsih, students’ mathematical representation ability through … 164 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p243-260 243 systematic literature review on the recent three-year trend mathematical representation ability in scopus database rizki wahyu yunian putra1*, sunyono1, een yayah haenilah1, hasan hariri1, sugeng sutiarso1, nurhanurawati1, nanang supriadi2 1universitas lampung, indonesia 2universitas islam negeri raden intan lampung, indonesia article info abstract article history: received mar 20, 2023 revised apr 28, 2023 accepted may 25, 2023 published online jul 6, 2023 mathematical representation is essential as the gateway to mastering mathematical literacy. this literature study aims to determine the growth of students' mathematical representation abilities in the last three years. this literature study presents a literature review on the development of mathematical representations, including media, strategies, and measurement instruments to serve as the basis for future mathematical representations. the literature study method used is the slr (systematic literature review), utilizing a review procedure that refers to the preferred reporting items for systematic reviews and meta-analyses (prisma) framework in the period 2020 to 2022 because the growth in literature studies on mathematical representations is very significant and latest. data were collected by reviewing 24 scopus-indexed articles and proceedings from the scopus database. the study's results reveal that from the 24 selected literature, it can be interpreted if the mathematical representation ability can be improved and fulfills the indicators of mathematical representation itself by providing innovation in media, strategies, and instruments in learning mathematics. the innovations provided can be in the form of technology integration (geogebra), no longer conventional strategies (rme strategy), and instrument indicators of the representation itself. thus the ability of students' mathematical representations is no longer included in the low category in solving mathematical problems. keywords: mathematical representation, systematic literature review this is an open access article under the cc by-sa license. corresponding author: rizki wahyu yunian putra, doctoral student, department of education, universitas lampung jl. prof. dr. ir. sumantri brojonegoro no.1, bandar lampung, lampung 35141, indonesia. email: rizkiwahyuyp@radenintan.ac.id how to cite: putra, r. w. y., sunyono, s., haenilah, e. y., hariri, h., sutiarso, s., nurhanurawati, n., & supriadi, n. (2023). systematic literature review on the recent three-year trend mathematical representation ability in scopus database. infinity, 12(2), 243-260. 1. introduction contextual problems or problems related to real situations are usually used as a basis for developing mathematical literacy problems because their structure emphasizes the development of various contexts (lestariningsih et al., 2018). mathematical literacy https://doi.org/10.22460/infinity.v12i2.p243-260 https://creativecommons.org/licenses/by-sa/4.0/ putra et al., systematic literature review on the recent three-year trend mathematical … 244 problems can be used to measure individual mathematical literacy abilities, which in pisa are based on aspects of mathematical content and context. based on the results of the program for international student assessment (pisa), the ability of indonesian students in the field of mathematical literacy is generally still low. the 2015 pisa results show that mathematics achievement in indonesia is ranked 63 out of 72 countries with a score of 386 (oecd, 2017). where students have representation skills in mathematical activities to solve mathematical literacy problems. therefore, it is important to develop mathematical literacy problems for students to support their abilities in the representation (lestariningsih et al., 2018). the ability of mathematical representation can assist students in building concepts and expressing mathematical ideas, as well as making it easier for students to develop their abilities (herdiman et al., 2018). mathematical representation is the ability to restate mathematical problems through selecting, interpreting, translating, and using objects (graphs, tables, pictures, diagrams, formulas, equations) to express problems (oecd, 2003). one of the competency standards that is the focus of the literature study is the ability to represent mathematically. this problem is related to other literature studies that enrich the ability of mathematical representation. mathematical representation ability is the ability to restate notations, symbols, tables, pictures, graphs, diagrams, equations, or other mathematical expressions into other forms. mathematical representations consist of visual representations, verbal (written text), mathematical equations, or expressions (lestari et al., 2020). representation ability is very important for students to have, with representation ability, can make it easier for students to solve mathematical problems (sari et al., 2022). a problem that is considered complicated and complex can become simpler if the strategy and use of mathematical representations are used in accordance with the problem. therefore, the selection of the student's representational model plays a very important role in making appropriate and accurate mathematical problem-solving strategy decisions (ramanisa et al., 2020). in addition, representation is seen as an important part of mathematical activities and means capturing mathematical concepts because students' success in solving problems cannot be separated from the role of the representation (hwang et al., 2007). this is because the ability of students' accurate mathematical representations can make them expand their abilities in solving mathematical problems. representation is an expression of mathematical ideas or ideas displayed by students in their efforts to find a solution to the problem they are facing. mathematical representations have forms that include visual, verbal, and symbolic representations. visual representation, namely making pictures to clarify problems and facilitating their completion, verbal representation, specifically expressing mathematical ideas, writing down the steps for solving mathematical problems, writing interpretations of a representation, and symbolic representations, namely making mathematical models, solving problems involving mathematical expressions (nctm, 2000 ). a literature study related to the importance of representation in mathematics, among others, states that students' representational abilities are the key to success in understanding mathematical concepts and solving problems (birgin, 2012; villegas et al., 2009). based on this description, the researcher argues that a comprehensive review is needed regarding how research descriptions of mathematical representations in learning. for this reason, a literature study is carried out in the form of a systematic review of mathematical representations. lusiana and suryani (2014) also stated that slr can be a theoretical background for future research, useful as a reference, research material, or answering questions related to interesting topics by understanding previous studies. data volume 12, no 2, september 2023, pp. 243-260 245 collection was in the form of the results of a literature study related to mathematical representations which were then extracted, with questions on the articles being analyzed. a systematic literature review is a rational, transparent, and reproducible way to analyze the existing literature. a systematic literature review is a form of secondary study and includes different approaches to building, exploring, and summarizing accessible evidence related to a particular research question (munn et al., 2018). in addition, slrs can also help provide a better understanding and monitor research trends (lame, 2019; suherman et al., 2021). systematic literature review (slr) is a survey-based quantitative descriptive approach (tamur et al., 2023). the survey was conducted on secondary data in the form of basic research results on students' mathematical reasoning abilities. the stages of the research include data collection, data analysis, and drawing conclusions (tamur & juandi, 2020). the data collected is in the form of primary research that has been published in national journal articles, data collected from electronic databases registered and indexed by google scholar, semantic scholar, eric, and direct urls of national journals. next, extraction of all the articles found was performed. the purpose of this research is to systematically review the literature related to mathematical representation ability by referring to type spider with the question model: sample, phenomenon of interest, design, and evaluation (higgins et al., 2021; newman & gough, 2020). therefore the formulation of this research problem includes: rq1 what research topics are most studied by researchers? rq2 what research methods are often used in the study of mathematical representations? rq3 what learning media and teaching materials are used to develop mathematical representations? rq4 what learning strategies are used to develop mathematical representations? rq5 what instruments are used to measure the mathematical representation? the following is a summary of table 1 of type spider in systematically reviewing the literature related to mathematical representation abilities: table 1. summary of spider sample phenomenon of interest design evaluation toddlers, preschoolers, students, teachers, prospective teachers, boys, girls research problems or objectives: development and validation instruments for measuring mathematical representations, development of representational pedagogy, components of mathematical representations: reflection, motivation, and representational actions. problem-solving strategies using visual heuristic tools, react strategy, android challengebased learning, rme, collaborative learning, pmri and discovery learning, fourdimensional research, pedagogical approach, ddr, case studies, quasi-experiments, exploratory descriptive, task design, and phenomenological qualitative visual representation, development of abstract mathematical concepts, solving mathematical word problems, translating from symbolic to verbal form, changing representations of basic functions, complex modeling and problem solving tasks, interdisciplinary models of mathematics and science, theory of representation development, development instruments and measurement validation representation development, representation learning strategy. putra et al., systematic literature review on the recent three-year trend mathematical … 246 by using the type spider literature study it is easier to analyze the main literature which is then extracted and the data analyzed with the aim of answering questions related to media, strategies, and instruments from mathematical representations. 2. method this literature study uses the systematic literature review (slr) method using a review procedure that refers to the preferred reporting items for systematic reviews and meta-analyses (prisma) framework which is used to answer the problem formulation, the stages of the prisma framework: planning, implementation and reporting (higgins et al., 2021; newman & gough, 2020). journals obtained from the scopus database with the keywords used are mathematical representations. the literature selection process refers to the inclusion and exclusion criteria. inclusion and exclusion criteria were used in the primary literature selection. the purpose of inclusion and exclusion criteria is used to minimize ambiguity and reduce the possibility of bias in the literature study. these criteria are shown in table 2 (higgins et al., 2021; mcdonagh et al., 2014; newman & gough, 2020). table 2. inclusion and exclusion criteria inclusion criteria exclusion criteria the literature is in the form of academic studies— philosophical, theoretical, educational practices about mathematical representations. the population/research subjects consisted of toddlers, pre-school children, college students, teachers, prospective teachers, boys and girls. literature published in the last 3 years in a reputable journal, scopus. literature presents a comparison of contexts, learning methods, measurement of mathematical representations. the literature is supported by research methodologies and empirical data with strong validity. literature published in english. incomplete literature. literature published more than last 3 years. literature in the form of books the search flow and the amount of literature identified in the prisma framework are shown in figure 1. the literature selection process was carried out in four steps, namely searching for keywords and then selecting literature based on title and abstract, inclusion and exclusion criteria as well as the complete text which will produce the referenced literature. volume 12, no 2, september 2023, pp. 243-260 247 figure 1. prisma procedure based on the image above framework prisma was obtained from 2119 journals. then filter by title and abstract obtained from 924 journals. then with the filter according to the inclusion and exclusion criteria, 157 journals were obtained. then with a full-text filter, 40 were obtained. of the 40 journals, 24 were relevant. at the end of the selection, 24 main articles were selected to be analyzed further. the main literature that has been selected is then extracted to collect data that contributes to answering the predetermined research questions. table 3 represents the iterative data extraction to answer research questions with 5 properties. table 3. properties of data extraction mapped to research questions property research questions research questions research topics and trends rq1 research methods/design rq2 research media rq3 research strategy rq4 research instruments rq5 browsing to search the main literature on several journal databases refers to the procedure in the developed slr (higgins et al., 2021), then compiled as a guide as a compass in searching and analyzing literature to conduct a review and minimize the possibility of bias (see figure 2). putra et al., systematic literature review on the recent three-year trend mathematical … 248 figure 2. systematic literature review procedure (higgins et al., 2021) after the data is extracted, it is then classified, assessed, compared, analyzed, combined and concluded as a whole. the data extracted and analyzed from the main articles are used to answer the questions of the literature study. for the record, the analysis of this article uses statistical techniques and searches using search engines. this slr was carried out aiming to analyze literature related to mathematical representation according to statistical techniques and searches using search strings, but not based on manual reading of articles published in various journals. so this slr may miss several articles regarding mathematical representations in several journals. 3. result and discussion 3.1. results the results of the research contained in this literature review are in the form of article data tabulations documented related to the ability of mathematical representation articles (see table 4). shift and select studies extract data from the studies assess the quality of the studies combine the data (synthesis/meta-analysis) discuss and conclude overall findings systematic literature review dissemination identify the issue and determine the question wirte a plan for the review (protocol) search for studies volume 12, no 2, september 2023, pp. 243-260 249 table 4. research related to mathematical representations authors & year journal research result bakar et al. (2020) eurasia journal of mathematics, science and technology education the discoveries show that small kids are able photographic artists and their visual portrayal pictures support how they might interpret the idea of expansion. according to the findings of this study, the development of mathematical visual representations is crucial for facilitating the understanding of abstract mathematical concepts. fauziyah and jupri (2020) journal of physics: conference series according to the findings, the test instruments used to measure students' mathematical representation and communication skills were inadequate. most understudies find it hard to close in view of blunders in understanding the importance of numerical statements, in actuality, settings. learning activities need to be created based on these findings to achieve the expected indicators, which will help students improve their communication skills and mathematical representations. nurrahmawati et al. (2021) international journal of evaluation and research in education (ijere) students were still unable to correctly make representations when translating from symbolic form to verbal form (problems in daily life) using a particular equation system, according to the findings of the data analysis. students still cannot draw a complete graph when asked to translate into graphical form, and because their errors are misinterpretations and implementations, they cannot maintain semantic alignment. between the source portrayal and the objective portrayal. on the basis of this, a lesson plan that can help students differentiate between representations needs to be developed. sproesser et al. (2022) international journal of stem education according to these findings, lessons on function should include multiple representations and representational changes to help students develop rich function concepts and flexible problem-solving skills, satisfy curriculum requirements, and respond to didactic considerations. teaching functions, in particular, needs to be more balanced by incorporating tasks with and without situational context and representational adjustments when necessary. these discoveries ought to propel educators, especially those showing non-scholastic streams, to give situational setting a more conspicuous job in their illustrations about capabilities to cultivate their understudies' learning and fabricate spans among science and certifiable circumstances. putra et al., systematic literature review on the recent three-year trend mathematical … 250 authors & year journal research result tytler et al. (2023) international journal of science and mathematics education making maps, understanding them, measuring and modeling data, sampling data, and using scales are all parts of the mathematical process. this examination offers new experiences into how educators support understudy learning in these two subjects, through the phases of onboarding, testing portrayal, building agreement, and carrying out and extending authentic frameworks. santia et al. (2021) journal of physics: conference series subjects used verbal and symbolic representations to calculate, find and fix errors, and justify their answers when solving unstructured problems, according to the findings. however, only the first subject used visual representations to find and fix errors. when compared to well-structured problems, subjects reveal less information needed to solve unstructured problems. sagita et al. (2021) journal of physics: conference series the following conclusions are drawn from the analysis and discussion that took place in the preceding chapter. there were two types of learning barriers: learning barriers related to learning barriers and learning barriers related to 3d material. transferring data or representations from representations to diagrams, graphs, or table representations; (3) obstacles to learning how to draw geometric shapes to help solve problems. 4) related to obstacles to learning. make use of illustrations to solve problems. 5) related to obstacles to learning how to make images with geometric patterns. calculating the percentage of teacher and student responses to teaching materials yielded a value of 94.66 percent with very positive criteria and 82.96 percent with positive criteria for teaching materials. kaitera and harmoinen (2022) lumat: international journal of mathematics education, science and technology the findings demonstrated that the teaching strategy, which places an emphasis on discovering various approaches to solving math problems, has the potential to enhance students' test performance, problem-solving abilities, and ability to explain their reasoning in assignments. the discoveries of this review recommend that educators can uphold the improvement of critical thinking systems by empowering class conversation and utilizing, for instance, a visual heuristic instrument called critical thinking keys. lestariningsih et al. (2020) journal of physics: conference series based on the findings of the analysis, expert review, and small group discussions, the research demonstrates that the developed problems meet valid and practical criteria. what's more, the issues grew additionally have potential impacts in view of volume 12, no 2, september 2023, pp. 243-260 251 authors & year journal research result field test examination which shows understudies are engaged to deliver portrayals in taking care of issue. rahayu et al. (2021) journal of physics: conference series five inaccuracies in (1) students' ability to present concepts in various mathematical representations were identified as learning barriers in the study as didactic and epistemological barriers. 2) recognizing the concept of orthogonal projections in geometry between points on lines; 3) build mathematical types of room to explain issues; ( 4) recognizing students' visual representations of the skill of locating a point or line that is perpendicular to a line; 5) using the pythagorean theorem and algebraic concepts to perform calculations utomo and syarifah (2021) journal of international education in mathematics, science and technology the examination results show that visual portrayal happens in both high, medium and low limit classes. students with high and low abilities perform symbolic representation tasks at the understanding level, whereas capacity students perform symbolic representation tasks at the problem-solving level. additionally, students with high abilities write conclusions as they write topics. students with moderate writing ability complete their questions by writing conclusions. the lowskilled students wrote down what they understood, and questions about the problems were posed to them. septian et al. (2020) journal of physics: conference series students who use geogebra in integral fields increase their mathematical representation abilities more effectively than students who use conventional learning, according to high and low categories of mathematical representation abilities. in the high and low prerequisite ability categories, geogebra in the integral field is superior to conventional learning for prerequisite skills. susilawati (2020) journal of physics: conference series when compared to students in the control group, the android-based challenge-based learning students' mathematical representation skills improved. in the high, medium, and low categories, students with android-based challenge-based learning and conventional learning based on prior knowledge of mathematics perform identically when it comes to mathematical representation. students engage in a challenge-based learning interaction with the android app, and expository learning based on prior math knowledge is used to assess the mathematical representation abilities of high, medium, and low students. challenge-based learning on android makes conflict resolution, discovery, and social interaction easier. putra et al., systematic literature review on the recent three-year trend mathematical … 252 authors & year journal research result fiantika (2021) journal of physics: conference series the findings demonstrated that mental rotation made use of feature coding, imaginary angle stimulus rotation, and fast stimulus matching, as well as symbolic number sense, ordinal magnitude sense, and line division in math skills. pedersen et al. (2021) mdpi the outcomes uncover a reasonable connection between the numerical points covered and the kind of portrayal utilized, and further propose that specific parts of illustrative capability are rethought when dt is utilized. in order to facilitate representational competence in relation to the utilization of dt, we offer five recommendations for the creation of math tasks. finally, we inquire as to whether the dt only initiates a new activity or a new representation. saskiyah and putri (2020) journal of physics: conference series students' ability to solve problems involving mathematical expressions and load problem situations based on the provided data or representations had improved, according to the findings. drawing geometric shapes to clarify the issue and facilitate its resolution is a rare indicator. giving a genuine setting through pmri and cooperative growing experiences helps understudies in creating marks of numerical portrayal. sirajuddin et al. (2020) journal of education for talented young scientists the study found that algebraic symbol representation, image representation, and geometric representation are the three types of representations that the subject raises when expressing algebra. a large portion of the members created mathematical emblematic portrayals and a few experienced troubles in delivering pictorial portrayals and mathematical portrayals. the same patterns, namely perception, appearance, strategy, and reexamination, were also observed in the production of geometric representations by the researchers. taqwa and rahim (2022) journal of physics: conference series the outcomes showed that understudies' capacity to comprehend vector ideas with numerical portrayals was superior to understudies' verbal portrayals. the understudies' mean scores in the verbal and numerical portrayal designs were 33.91 and 59.17 separately. the paired sample t test yielded the following results: t count = -12.96 and sig. = 0.00. according to these findings, students' comprehension of vector concepts in verbal and mathematical representations differ significantly. the aftereffects of this study show that's how understudies might interpret vector ideas actually relies upon the portrayal of the inquiries in light of the fact that their comprehension isn't lucid. based on these findings, vector learning must focus not only on mathematics but also on connecting the volume 12, no 2, september 2023, pp. 243-260 253 authors & year journal research result meanings of vector operations in different representations and the meanings of vectors in different representations. post and prediger (2022) journal of mathematics education research case studies show that teaching methods can vary greatly: for students with advanced comprehension, translating condensed concepts from a given text into other representations—visual area models, fractional symbolic representations, and three language variations—seems sufficient. some students need the teacher's help to break down a complex concept like "parts" into several concept elements like "part," "whole," and "part-whole relations," and to explicitly link the concept elements in some representation for the concept elements rather than just translating them. different. theories about teaching practice with multiple representations and professional development may benefit from these findings. björklund and palmér (2022) educational studies in mathematics this study examines and expands the potential of interactive book reading as an educational tool for quality improvement and adds to our understanding of how to teach numbers to toddlers. rahayu and kuswanto (2021) journal of technology and science education the exploration discoveries show that mikimom is powerful in further developing understudies' decisive reasoning abilities and numerical portrayal with a score of 0.287 (enormous impact size) and 0.179 (medium impact size). hakim et al. (2020) journal of physics: conference series students who use mobile learning have better representational learning outcomes than students who use traditional learning, according to the study's findings. awantagusnik et al. (2021) aip conference proceedings the results of the study show that shaden is able to use more than one representation to solve contextual problems, but students are still found to be able to use only one mathematical representation, namely verbal representation. jewaru et al. (2021) aip conference proceedings the results showed that students who understood vectors were easy physics concepts, but most students had problems with vector concepts. students seem to respond correctly to the mathematical representation test, but there are still many errors in the physical representation. after analyzing the data in the 24 selected articles (see table 4), we will focus on discussing the three main pillars that support mathematical representation abilities: theory, strategy, and measurement. in addition, we will also reveal the types of research gaps that are the keys to opening up research opportunities in the future. putra et al., systematic literature review on the recent three-year trend mathematical … 254 rq1: mathematical representation research topics some of the main descriptions obtained from the results of this analysis are that research on mathematical representations currently focuses on topics: 1) contextual learning through pmri, rme, and ethnomathematics theories; 2) development and validation of mathematical representation measurement instruments such as teaching materials using geogebra or; 3) development of learning strategies oriented to mathematical representations in didactic designs. the tendency of mathematical representation research focuses on mathematical literacy activities. this is also a more specific development in generating mathematical representations in students. researchers of mathematical representations for students need to highlight how various roles such as contextual learning through pmri, rme, or ethnomathematics whose teaching materials with geogebra and didactic designs can lead to psychological studies which then raise questions about how the coherent sequence (sequence) is appropriate in developing representations. mathematical. rq2: development of mathematical representation research methods after analyzing 24 articles through a bibliometric perspective, a review of the contents began. based on the method aspect of the article, it shows the variation in the use of research methods. some articles that use a qualitative approach such as case studies, observations, interviews, and phenomenology as much as 79%. several other articles use a quantitative approach, especially those relating to the development of measurement instruments as much as 17%. whereas in other contexts, mathematical representation has been widely studied using a mixed method approach, for example developing a measurement instrument for mathematical representation with a scale and deeply interviewing as much as 4%. rq3: development of mathematical representation-oriented learning media the results of the analytical review found several theories that underlie the development of media in learning that is oriented toward mathematical representations. from the several articles that have been analyzed, we found at least 3 of the latest learning media that are offered to develop mathematical representation skills offered, namely teaching materials such as visual heuristic tools, mikimom, mobile learning using geogebra, and the carom comic game. rq4: mathematical representation-oriented learning strategies from the several articles that have been analyzed, we found at least 6 of the latest learning strategies offered to develop mathematical representation abilities as can be known, namely: problem-solving strategies using visual heuristic tools, react strategies, android challenge-based learning, realistic mathematics education (rme), collaborative learning, pmri, and discovery learning. based on tracing research developments regarding mathematical representations, it concerns contextually oriented learning which is then in the form of ethnomathematics which is developed with a didactical design. rq5: measurement of mathematical representation after discussing learning theories and strategies, the review of the development of mathematical representation measurement instruments aims to reveal the various instruments that have been developed and currently exist. because another element of learning mathematical representation that is also important to analyze is measurement. most volume 12, no 2, september 2023, pp. 243-260 255 of the instruments are like creating geometric shapes to clarify problems and facilitate solving, solving problems involving mathematical expressions, and creating problem situations based on the data or representations provided. 3.2. discussion this systematic literature review reveals how far research has progressed regarding mathematical representations over the last three years. several research results show that the development of mathematical representations has reached various areas of study, including engineering, computer science, and physics. so that studies on mathematical representations can look further at how this ability is generated in the field of engineering. in line with this, the development of learning strategies that are oriented toward mathematical representations has also developed rapidly and is increasingly varied. where learning can be done through contextual learning such as react strategies, problemsolving strategies using visual heuristic tools, android challenge based learning, realistic mathematics education (rme), collaborative learning, pmri, and discovery learning (hidayat et al., 2023; nuraida & amam, 2019; prahmana et al., 2020). besides that, the development of media or teaching materials that support and facilitate the learning of mathematical representations has also developed a lot, such as visual heuristic tools, mikimom, mobile learning using geogebra, and the carom comic game. apart from this, learning designs also need to be developed as in didactical design research (ddr) (rosita et al., 2019; sari & darhim, 2020). this can be a reference in determining learning and research on mathematical representations in the future, namely deepening how communication and representational skills can be improved, where students need to be more accustomed to being introduced to math problems with the help of appropriate methods, using representations in other fields of mathematics, developing translation skills in learning when understanding concepts and solving problems, varying the frequency of solutions beyond characteristics, teachers can be oriented to guide student transduction, increase the number of subjects and extend the study period, implementation of teaching materials, understand the natural development of problem solving skills and strategies and whether kind of "out-of-the-textbook" approach in math class,developing and paying attention to student representation in solving mathematical literacy problems or contextual problems, hypothetical didactic designs, assessing students' visual skills, teaching materials using geogebra, providing real contexts through pmri and collaborative learning processes, misunderstandings in the concept of dividing equations, providing real contexts through pmri and collaborative learning processes, misunderstandings in the concept of sharing equations, expanding the data corpus, theoretical assumptions about the conditions needed to learn mathematics, and preparing and checking in advance all mobile devices that will be usedteaching materials using geogebra, providing real context through pmri and collaborative learning processes, misunderstandings in the concept of sharing equations, providing real contexts through pmri and collaborative learning processes, misunderstandings in the concept of sharing equations, expanding the corpus of data, theoretical assumptions about the conditions necessary for learning mathematics, as well as prepare and check in advance all mobile devices that will be usedteaching materials using geogebra, providing real context through pmri and collaborative learning processes, misunderstandings in the concept of sharing equations, providing real contexts through pmri and collaborative learning processes, misunderstandings in the concept of sharing equations, expanding the corpus of data, theoretical assumptions about the conditions necessary for learning mathematics, as well as prepare and check in advance all mobile devices that will be usedas well as prepare and putra et al., systematic literature review on the recent three-year trend mathematical … 256 check in advance all mobile devices that will be usedas well as prepare and check in advance all mobile devices that will be used. 4. conclusion based on the explanation above, it is found that the application of mathematical representations, among others, has limited competence related to changes in the representation of basic functions then the generative nature of the interdisciplinary model of mathematics and science has significant implications for curriculum and practice reviews. however, students' ability to solve mathematical word problems still shows that students are still experiencing difficulties so the creation of mathematical visual representations is very important to facilitate the development of abstract mathematical concepts. students' mathematical representation abilities have increased, including solving problems involving mathematical expressions and loading problem situations based on the data or representations provided. in translating from symbolic form to verbal form (problems in everyday life) that follow a given system of equations, students are still not able to make representations correctly. when students are asked to translate into graphical form, students still cannot draw a complete graph and the mistakes made by students are misinterpretation and implementation, so they cannot maintain semantic harmony between source representation and target representation. verbal and symbolic representations are used by subjects to calculate, detect and correct errors, and justify their answers in solving unstructured problems but visual representations are used only by the first subject to detect and correct errors. then also the react strategy can be applied to develop representational abilities, reasoning, and mathematical dispositions that involve students actively. as well as challenge-based learning based on android facilitates processes of conflict, discovery, social interaction, and students' reflective processes so that mathematical representation abilities increase while the material is easy to understand and exciting and mobile learning has better representational learning achievements than students who use conventional learning. the mathematical representation ability of students who use geogebra in integral fields is better than students who use conventional learning and the ability of mathematical representation in high and low categories. prerequisite abilities using geogebra in integral fields are better than those carrying out conventional learning in the high and low prerequisite ability categories. this means that representation is important to study further because it affects the ability of unstructured problems. the limitation of this research is that it is only limited to reviewing scopus-indexed journals, even though it is not only scopus indexed but there are also many other journals such as wos, sinta, scimogo rank and publish or perish journals. references awantagusnik, a., susiswo, s., & irawati, s. 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(2018). the effectiveness of problem-based learning with social media assistance to improve students’ understanding toward statistics. infinity, 7(2), 97-108. doi:10.22460/infinity.v7i2.p97-108. mailto:pakyulikids@unwdha.ac.id mailto:feryfirmansah@unwidha.ac.id yuliana, & firmansah, the effectiveness of problem-based learning with social media … 98 introduction the development of technology and information grows fast nowadays. it is indicated through the appearance of many social media (gunduz, 2017), such as whatsapp, bbm, line, whechatt, telegram, etc. this phenomenon contributes positive effect in educational development, such as online learning via social media. according to stephenson (2001), online based learning has 4 main features, namely the use of social media for dialog and learning material as well, the support of online technology, students’ authority in handling the learning process through task, learning result, or assessment. online learning efficiency, effectiveness, and makes the students keep learning (mitcell & honore, 2008). teaching and learning process will be easier to conduct. it can be conducted wherever and whenever without any time and place burden. online discussion will be the best learning type since many people including college students use social media to communicate. unfortunately, they haven’t used social media to learn together including the students and the lecturers of widya dharma university. the use of technology and information system is highly needed in understanding lecture material. for example, the existence of calculator, excel, and spss application really supports applied statistics learning. the applications do not only help in calculating the data, but also in displaying and analyzing the data. in applied statistics, statistics methods are learnt. they are hypothesis test, analysis of variance, analysis of regression, and analysis correlation. the students should master them well especially when they are doing research. therefore, applied statistics is a crucial subject to understand by the students of mathematics education program (sungkono, yuliana, & syaifuddin, 2014). applied statistics subject is still problem for the students of mathematics education program. it can be seen from the result of their applied statistics test in which there are more than 60% of students haven’t passed the passing grade. besides, the result of interview for the students who are doing thesis indicates that the students haven’t mastered statistics material for their research. moreover, the students said that statistics subject is not impressing subject for them. on the other words, they cannot tell what they have learnt about. they couldn’t correlate statistics with their daily life. besides, their skill and understanding in using statistics is low. they study statistics only from the lecturer without any further discussion with their friends. the learning process is teacher-centered so the students feel uncomfortable to ask questions. consequently, their understanding is just partial and they couldn’t apply statistics in their research well. nowadays, there are many innovative and active teaching methods. innovative teaching methods can effectively improve students’ learning achievement (yuliana, tasari, & wijayanti, 2017; sethi, sethi, & jeyaraj, 2017). the methods can be implemented appropriately based on the need of the class. besides, the development of technology and information grows fast. it can be used to support student learning process. learning doesn’t absolutely need a meeting in a room. but, it can be conducted in different places. volume 7, no. 2, september 2018 pp 97-108 99 many college students use gadgets even they follow the trend. this phenomenon is a kind of benefit in learning process (nawi, hamzah, ren, & tamuri, 2015). the students will learn better if they learn something near to them. learning method that is based on the problem will engage them to think critically. the technology–based learning and problem-based learning will well collaborated in learning process. research about the use of latest information technology was done by naidoo & kopung (2016). it used social media whatsapp to develop mathematics learning. moreover, yuwono & syaifuddin (2017) collaborated pbl and smartphone media to teach mathematics. problem based learning is a learning method that involves students to solve the problem. the steps of this method (rubiah, 2016) are (1) appearing a problem and ensuring that the problem is contextual; (2) organizing the subject toward the problem; (3) giving the students responsibility to conduct the learning process; (4) making small groups; (5) asking the students to present what they have learnt. problem based learning (pbl) has a crucial role for the students in the process of exploring important and meaningful questions, investigating a problem solution, and developing a deep-integrated understanding of content and process (frank, lavy, & elata, 2003; hmelo-silver, 2004). in this research, researcher is interested to combine pbl with social media whatsapp in teaching mathematics. the teaching steps are (1) the lecturer makes a whatsapp grup discussion. the member is all students. (2) the use of whatsapp eases the lecturer to deliver the learning objectives and lesson plan to the students. (3) the students then can explore the learning materials from any references. (4) to motivate the students, some materials are shared in the group by the lecturer. (5) the lecturer also gives some tasks to accomplish. (6) then, the students discuss in the group. the role of lecturer here is helping the students in organizing the task, motivating the students to collect related information, testing, and solving the problem. moreover, the lecturer helps the students in preparing their findings report. (7) later on, the result is presented in front of the class. social media that is used in the research is whatsapp. it is collaborated with problem base d learning (hmelo-silver & borrow, 2006). the students make a group discussion by whatsapp. they can get information about lesson plan, learning material, and problem discussion. the discussion can be conducted everywhere and every time. they don’t need to wait until class meeting. problem based learning with social media assistance emphasizes on the process of learning. it is reflected by keeping the students active. in accordance with ningsih & rohana (2016), learning which accentuates more on the process is meaningful for learner. the reality of the students’ condition about their understanding toward statistics subject and the technology and information development motivates the teacher to do research about how to find the most effective way of teaching to improve students’ understanding toward statistics material by pbl with social media assistance. the understanding toward the material can be identified through the learning result after joining the class. method students of mathematics education program are the subject of this research. by employing cluster random sampling, control and experimental class are determined. control class is 21 students taught using conventional method, while experimental class is 18 students treated yuliana, & firmansah, the effectiveness of problem-based learning with social media … 100 using problem based learning with social media assistance. the illustration of pbl learning steps with social media assistance can be seen as follows: figure 1. the illustration of pbl learning steps with social media assistance the research uses documentation method, questionnaire method, and test method to collect the data. the documentation method is used to identify the problem. the questionnaire method is used to know students’ responses toward the learning process. the questionnaire method is used to measure the validity of the instrument as well. the statement of questionnaire consists of two alternative answers: yes and no (yogesh, 2006). to know the students’ response toward the use of problem based learning with social media assitance, questionairre is distributed in experimental class. there are 16 statements that have been validated by 3 validators. the result of validity test from the three validators state that the questionairre could be used for the research. by three collecting data techniques above, data is gotten. then, they are classified as qualitative and quantitative data. qualitative data involves lesson plan, teaching material, test items, and questionnaire. meanwhile, quantitative data is the score of pre-test and post-test. the result of pre-test is used to know the early condition before treatment for both control and experimental class. the students’ understanding after treatment is measured through post-test. the result of pre-test and post-test is analyzed by using t-test with significant α = 0.05 and 95% confidence interval (keselman, othman, wilcox, & fradette, 2004). lilliefors method is used to do normality test as prerequisite test of t-test (usman, 2016). meanwhile, homogeneity test uses comparison of 2 variances is f-test (yuan & bentler, 1999). all computation in the analysis was done through spss for windows computer application (chatfield, 1995; pallant, 2010). results and discussion results learning documents that are developed in the class are syllabus, lesson plan, learning material, learning media, and test items. the instruments are validated by three experts judgment and its result is the learning documents can be used in the research. the lecturer’s and students’activity during the learning process is described in table 1. making a groups discussion via wa sharing contextual problem (via whatsapp) organizing the problem (via whatsapp) giving the students responsibility making small groups in solving the problem presenting the invention sharing the lesson plan, learning objectives, and learning sources/references (via whatsapp ) reflection and evaluation (via whatsapp) whatsapp) http://journals.sagepub.com/author/keselman%2c+hj http://journals.sagepub.com/author/othman%2c+abdul+r http://journals.sagepub.com/author/wilcox%2c+rand+r volume 7, no. 2, september 2018 pp 97-108 101 table 1. the lecturer’s and students’activity during the learning process in experimental class phase the lecturer activities phase 1 students orientation towards the problems explaining the learning objectives sharing the learning sources motivating the students to actively involve in solving the problem phase 2 organizing the students helping organize the problem phase 3 guiding individual’s research and group encouraging the students to collect information and problem solving method phase 4 developing and serving the report helping the students prepare the report encouraging the students to distribute the task and work together phase 5 analyzing and evaluating the process of problem solving evaluating the learning result and asking the groups to present their report before treatment, each class was given pre-test. the result of pre-test can be seen in table 2 as follows. table 2. the students’ pre-test data experiment class control class the students 18 21 maximum 88 85 minimum 30 49 mean 62.61 65.05 standard deviation 15.996 10.45 the score more than 60 55.56% 66.67% based on the pre-test data, maximum score and average score are high. nevetheless, the percentage of students in both experimental and control class which is more than 60 is 55.56% and 66.67%. pre-test data on table 1 is analyzed its homogeneity and normality. the result of f-test is p value = 0.054. it is more than significant (α) 0.05. it means that the data is homogenious. meanwhile, the result of test is p value 0.146 for experimental class and 0.20 for control class. the p value from eachclass is more than significant 0.05. it means, each class is normal. two prerequisite tests are fullfilled. therefore, t-test can be conducted. the result can be seen on table 3. yuliana, & firmansah, the effectiveness of problem-based learning with social media … 102 table 3. the result of t-test for pre-test t df p value mean difference standard error dif lower bound upper bound equal variances assumed 0.571 37 0.572 2.437 4.269 -6.213 11.086 equal variances not assumed 0.553 28.477 0.585 2.437 4.407 -6.583 11.456 the result of test is p value 0.572 more than significant 0.05. it means, both classes have same capability before treatment. having identified the early condition of control class and experimental class, both classes were given treatment. each class was given 10 meetings. then, both classes were measured its capability using test. the data description of post-test is shown in table 4. table 4. the students’ post-test data experimental class control class the students 18 21 maximum 85 85 minimum 30 25 mean 61 48.57 standard deviation 13.09 17.13 the score more than 60 72.22% 19.05% based on the data on table 2 and table 4, the average score of pre-test is higher than the average score of post test. the coverage material of pre-test and post-test is a little bit different. the material of post-test is wider than that of pre-test. it is the factor why the average score of pre-test is higher than the average score of post-test. in experimental class, the students who get score more than 60 is 72.22%. it is categorized as high improvement. it contras, it is only 19.05% who get more than 60 in control class. based on the result of pre-test and post-test, there isi decrease for the mean score in both experimental class and control class. however, there is an increase 16.66% for the students who get more than 60 in experimental class. if the score 60 is used as passing grade so there are 13 students who can pass it and only 5 students who get under it. in contrast, there only 4 students can pass the passing grade and 17 students fail to achieve it. it shows there is a decrease for control class. the data describes the result of post-test in experimental class better than the result fo pre-test. moreover, the post test score in experimental class is better than that in control class. post-test data in table 4 is analyzed its homogeneity and normality. the result of f test is p value = 0.161. it is more than significant (α) 0.05. it means that the data has the same variance or the data is homogeneous. meanwhile, the result of lilliefors test is p value 0.096 for experimental class and 0.200 for control class. p value for both classes is more than significant 0.05. it means both class are normal. volume 7, no. 2, september 2018 pp 97-108 103 the prerequisite test has been fulfilled. then, to identify the efectiveness of teaching method, t-test two tails is carried out. the result of t-test is p value 0.019 and the mean score different between control class and experimental class is 12.5. p value is less than significant 0.05. it mean there is difference between the students taught using problem based learning with social media assistance and those who are taught using conventional method. the value of difference is positive. it means the learning result of experimental class is better than control class. in summary, problem based learning with social media assistance is more effective to teach statistics than conventional method. the result can be seen in table 5. table 5. the result of t-test for post-test t df p value mean difference standard error dif lower bound upper bound equal variances assumed 2.511 37 0.017 12.429 4.949 2.401 22.456 equal variances not assumed 2.564 36.564 0.015 12.429 4.847 2.603 22.254 based on the result of t-test, p value is 0.017. the difference mean of post-test score between experimental class and control class is 12.429. the result of p value is less than significant (α) 0.05. it means, there is difference of learning achievement for the class that is taught using problem based learning with social media assistance and that is taught using conventional method. the mean difference of learning achievement between experimental class and control class is positive so the learning achievement of experimental class is better than control class. based on the pre-test score, the students’ ability in both classes is in the same level. later on, the experimental class was taught by problem based learning with social media assistance. the post-test score of experimental class shows that the students’ understanding toward statistics is better than that of control class. it is supported by the percentage of minimal passsing grade on the data in table 4. the percentage of minimal passing grade score in experimental class is better than that in control class. based pre-test data and post-test data means that problem based learning with social media assistance is more effective to teach statistics than conventional method. this statement is supported by students’s reponse result throught questionairre. the questionnaire has been stated valid and reliable so it is ready to use in the research. the result is 61% items get positive response from the students. the percentage of each item is more than 80%. it means problem based learning with social media assistance can improve students’ interest in joining lecture. discussion in experimental class, the students are taught using pbl with social media assistance. first, whatsapp group discussion is made. this group will encourage the students to intensively communicate with other students and the students with the lecturer as well (hrastinski, 2009). communication between the students in experimental class runs better than that in control class. besides, the use of whatsapp lets the students efficiently communicate in sharing knowledge and information (hamzah, 2015; kartikawati & pratama, 2017; bouhnik & deshen, 2014). the material that is distributed by the lecturer and the students through whatsapp can be received fast by the students without any time and place burden. this kind of yuliana, & firmansah, the effectiveness of problem-based learning with social media … 104 knowledge sharing is better than in control class. control class receives knowledge and information just in lecture time. in experimental class, the students are divided into some small groups so that the lecturer c an focus on the learning management (nicholl & lou, 2012). pbl with social media assistance method lets the students study in small groups that takes place outside in the classroom. each group discusses to solve problem. the researcher modify the learning process of pbl to avoid boredom. in line with lee & bae (2008) state the one of factors that determines the success of pbl is modifying it. the conclusion of the research is in line with the research that was done by padmavathy & mareesh (2013). it is the implementation of problem based learning with social media assistance to increase students’ motivation and interest to learn statistics. so the students have better understanding toward statistics. figure 2. the students are discussing learning material in group the role of teacher in class tends to be facilitator in which he/she just ensures that the students learn something and help the students when they need it. the teacher gives some real life problems (botty et al, 2016) through questions, teamwork, and how to come to its answer. students work in small groups of three or four people. in the group, the students present their discussion result. the students also learn how to make a conceptual knowledge not only theory memorization (mustaffa et al, 2014). it makes statistics more impressing for the students. as the consequence, their understanding is better. it can be seen from the result of their statistics test. volume 7, no. 2, september 2018 pp 97-108 105 figure 3. the student is presenting discussion result the benefits of problem based learning with social media assistance to teach statistics show that this method is more effective than conventional method. the same conclusion is written by kazemi & ghoraishi (2012). it can be seen from the result of post test with t-test analysis. the result of t-test two tails is p value equals to 0.019, while the mean difference of post test in experimental class and control class is 12.5. p value is less than significant 0.05. it means there is difference learning result between experimental class and control class. because the difference value is positive, the conclusion the learning result of experimental class is better than control class. it is similar with research conclusion carried out by kazemi & ghoraishi (2012). conclusion in line with the validation result, problem based learning with social media assistance that is developed by the researcher is implemented like how it is stated in lesson plan. the learning method also got positive response from the students. it can be seen by the result of questionnaire. in addition, the result of t-test shows that problem based learning with social media assistance is more effective to teach statistics than conventional method. acknowledgments the research could be well accomplished because the support of widya dharma university. the research was funded by technology research and high education ministry in 2017 budgeting. the researcher thanks so much for the fund and support. references botty, h. m. r. h., shahrill, m., jaidin, j. h., li, h. c., & chong, m. s. f. 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(2017). project-based learning approach: improvements of an undergraduate course in new product development. production, 27(spe). http://scholar.google.com/scholar?cluster=3037957875855789605&hl=en&oi=scholarr http://scholar.google.com/scholar?cluster=3037957875855789605&hl=en&oi=scholarr http://scholar.google.com/scholar?cluster=3037957875855789605&hl=en&oi=scholarr https://doi.org/10.3102/10769986024003225 yuliana, & firmansah, the effectiveness of problem-based learning with social media … 108 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p237-254 237 development of teaching materials for elearning-based statistics materials oriented towards the mathematical literacy ability of vocational high school students in in supianti1, poppy yaniawati1*, siti zuraidah md osman2, jasem al-tamar3, niki lestari4 1universitas pasundan, indonesia 2universiti sains malaysia, malaysia 3kuwait university, kuwait 4smk negeri 2 baleendah, indonesia article info abstract article history: received aug 8, 2022 revised sep 13, 2022 accepted sep 14, 2022 the aim of this study is the development of teaching materials for statistical materials based on edmodo-assisted e-learning and how they impact students' mathematical literacy skills. the method used is the dick & carey development model. the research population is class xii catering, one of the vocational high schools in bandung city, with a sample of class xii catering 1. the instrument used a mathematical literacy test, validation sheet, student response questionnaire, and interview guidelines. the data collected were analyzed descriptively qualitatively through triangulation, q-cochran statistical test, and n-gain test. the results of this study indicate that the design of teaching materials for statistical materials based on edmodo-assisted elearning is very feasible to use in learning mathematics. furthermore, the results of the application of teaching materials have a positive effect on mathematical literacy skills with a reasonably good category. therefore, teaching materials must further develop animation, use communicative language, and utilize the latest technology. keywords: e-learning, edmodo, mathematical literacy ability this is an open access article under the cc by-sa license. corresponding author: poppy yaniawati, department of mathematics education, universitas pasundan jl. sumatera no. 41 bandung, west java 40117, indonesia email: pyaniawati@unpas.ac.id how to cite: supianti, i. i., yaniawati, p., osman, s. z. m., al-tamar, j., & lestari, n. (2022). development of teaching materials for e-learning-based statistics materials oriented towards the mathematical literacy ability of vocational high school students. infinity, 11(2), 237-254. 1. introduction today's education cannot be separated from the use of technology. most teachers and students already use technology in schools. according to information from 258 students, 50.8% often, 45% never, and 4.2% never mathematics teachers in west java apply ict in learning (supianti, 2018). online learning can be said to be an electronic learning system https://doi.org/10.22460/infinity.v11i2.p237-254 https://creativecommons.org/licenses/by-sa/4.0/ supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 238 called e-learning which uses a website and can be accessed anytime, anywhere. e-learning is a learning activity that uses electronic services, for example, telephone/mobile phone, video, audio, computer, laptop, tablet, internet access, and so on (alifia & pradipta, 2021; fisher et al., 2019; supianti et al., 2021; wahyuni & sugiharta, 2019). thus, the internet network as a supporting facility for e-learning must be fulfilled. students can use e-learning to learn independently. the role of teachers in modern learning systems is much as a facilitator, which results in a paradigm shift that prepares students to learn independently (yaniawati, 2012). in line with that, learning mathematics using e-learning is better than conventional (supianti, 2013). e-learning-based learning provides opportunities for students to learn freely and without being pressured, accessible in the sense of finding learning resources, free from embarrassment as in conventional learning when they cannot answer questions from the teacher, fail to learn, and so on (supianti, 2018). online learning or elearning is practical because it can be used anywhere and anytime (irfan et al., 2020). easy, safe, and simple e-learning media can help students set strategies. e-learning media is interactive, and the information delivered is more real-time (sugianto et al., 2022). media in the learning process delivers the source of the message and the recipient of the message and stimulates thoughts, feelings, attention, and willingness so that they are encouraged and involved in the learning process (hamid et al., 2020). learning media will overcome the limitations of time, space, and senses. the function of learning media is also to influence the climate, conditions, and learning environment (pratama & ismiyati, 2019). edmodo is a secure and free social media-based digital classroom that helps teachers manage virtual classes to connect students with other students (balasubramanian et al., 2014). thus, the edmodo application is suitable for teaching and learning activities because it supports what teachers and students need in the learning process. the edmodo application has a school environment-based network because it helps teachers in virtual classes according to learning conditions in the classroom, based on natural class divisions in schools, where classes contain assignments, quizzes, and final assignments in each lesson (putranti, 2013). learning through the edmodo platform needs to be applied to be more varied, active, interactive, and independent (pratama & ismiyati, 2019). the learning objectives will be achieved with suitable learning media, and the student's abilities are expected to increase. the ability that is expected to increase is the ability of mathematical literacy. mathematical literacy is a person's ability to formulate, use, and interpret mathematics in any context, including mathematical reasoning, mathematical concepts, procedures, facts, and tools to describe, explain and predict an event (sari, 2015). even johar emphasized that knowledge and understanding of mathematical concepts are essential, but more importantly, activating students' mathematical literacy skills to solve everyday problems (yudiawati et al., 2021). mathematical literacy is an individual's ability to effectively use his mathematical knowledge to solve real problems in everyday life (anwar, 2018). effectiveness in question is in solving one's problems, starting with understanding problems, formulating them, and using mathematical knowledge to solve them and interpret them. the mathematical literacy of students in indonesia is still below the oecd average. the average international score to determine the ability of mathematical literacy is 500 (level 3), and indonesian students' average mathematical literacy score is 375 (level 1). based on data from the national center for education statistics that the mathematical literacy ability of indonesian students in pisa 2015 it was still low, namely 37.9% at level 1, 30.7% at level 1, 19.6% at level 2, 8.4% at level 3, 2.7% at level 4, 0.6 at level 5, and no one was able to reach the value of 0.6%. in pisa 2015, the mathematical literacy results of indonesian students were 380 and 490, which are the average scores of all countries that took the mathematical literacy test (kafifah et al., 2018). thus, students' mathematical literacy skills volume 11, no 2, september 2022, pp. 237-254 239 need to be improved, with the development of teaching materials for statistics materials based on e-learning assisted by edmodo oriented towards mathematical literacy skills. therefore, based on the introduction, the research objectives were: (1) to analyze the design of the development of teaching materials for statistics materials based on e-learning assisted by edmodo; and (2) to analyze the mathematical literacy abilities of students who used these teaching materials. 2. method this type of research uses r & d (research and development) with the model dick & carey (gafur, 2012) through 10 stages in the figure 1. figure 1. dick & carey's learning design model (gafur, 2012) the figure 1 showed the first stage in this research is to identify the competencies and general learning objectives of the teaching materials developed, namely statistical material. the second stage is to identify the objectives of developing teaching materials and the strategies used in presenting the developed teaching materials. the third stage identifies the characteristics of the initial ability of the class xii culinary management smkn 2 baleendah as many as 36 students. the initial ability of students is seen from the average daily test results which show the average value is still below the minimum completeness criteria. the fourth stage is to formulate special abilities that must be mastered specifically during and after completing learning. the fifth stage is developing relevant research instruments, namely the development of 6 items of essay questions on mathematical literacy abilities. the sixth stage is finding researches and student characteristics that are carried out based on relevant learning theories. the seventh stage is choosing materials based on the strategies that have been set and the abilities to be improved. the material in the teaching materials includes competence, main content and practice questions. identify instructional goals conduct instructional analysis identify entry behaviours write performance objectivies develop criterion referenced test develop instructional strategy develop/ select intructionals materials develop/ conduct formative evaluation revice instructions develop/ conduct summative evaluation supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 240 the eighth stage is a formative evaluation with validation, limited trials and field trials. validators in this study are material & media experts, and student response questionnaires. limited trials were conducted on 6 students who had obtained statistical material selected by purposive sampling. field trials were conducted on class xii culinary administration students at smk negeri 2 baleendah, bandung regency. the sample was taken by 30 students and given learning using teaching materials for statistics based on elearning assisted by edmodo. the sampling technique is purposive sampling, which is tailored to research needs. namely, all students have gadgets. the study went through a limited trial with ten subjects and a field trial with 30 subjects using the one group pretestposttest research design. the instruments used in this study are in the form of tests and nontests. the test-shaped instrument consists of 6 mathematical literacy test questions in the form of a description, while the non-test is in the form of validation sheets, student response questionnaires, and interview guidelines. the collected data were analyzed using qcochran, n-gain, and triangulation by comparing validation results, questionnaires, and interviews. the ninth stage makes improvements to the evaluation results in stages three to seven. the tenth stage is the refinement of teaching materials based on input at the summative evaluation stage. 3. result and discussion 3.1. result 3.1.1. e-learning-based teaching materials the development of e-learning-based teaching materials is carried out through 10 stages of dick & carey with the following results: the first stage identifies instruction goals, teaching materials have not been accompanied by animation and the existing e-learning, learning media have not varied, the second stage of conduct instructional analysis, relevant knowledge is statistical material because this application plays a vital role in daily life. the third stage of identifying entry behaviors, students have not been able to manage and regulate themselves in thoughts, feelings and behaviors, so that students' self-regulated learning is low, the fourth stage of writing performance objectives, simple, easy and appropriate learning media is the use of the edmodo application, the fifth stage of developing criterionreferenced tests, it is necessary to develop relevant question items that improve mathematical literacy skills, the sixth stage of develop instructional strategy, teaching materials are arranged according to the components of e-learning, the seventh stage of develop and select instructional materials, the display of teaching materials prepared based on the provisions of the development of teaching materials, the eighth stage of design and conduct formative evaluation of instructional, statistical teaching materials have been designed equipped with contextual problems, exciting animations, drawings with examples as shown in figure 2. volume 11, no 2, september 2022, pp. 237-254 241 figure 2. display of teaching materials then the teaching materials were validated by six material experts and three media experts. the validation results by material experts obtained 90.78% on very valid criteria, the media experts obtained 88.52% on very valid criteria, and student response questionnaires obtained 76.47% on the criteria were entirely valid. thus, the teaching materials developed are said to be feasible and can be used in learning. the results of the validation of teaching materials by material experts and media experts, along with student response questionnaires, are listed in table 1. supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 242 table 1. data from the validation of material experts no criterion average grades (percentages) v1 v2 v3 v4 v5 v6 1 aspects of content eligibility 76.67 90 96.67 85 96,67 95 2 aspects of presentation feasibility 78 90 100 96 94 94 3 aspects of linguistic feasibility 80 86.67 93.33 100 93.33 95.56 4 aspects of mathematical literacy assessment 80 86.67 91.11 95.56 91.11 93.33 average 78.67 88.33 95.28 94.14 93.78 94.47 average validity 90.78 validity level very valid, or can be used without revision based on table 1, the difference in scoring was obtained for v1. the criteria with the most significant value were the feasibility of language and mathematical literacy. for v2, the most significant criterion was the content and presentation feasibility. followed by v3, the most significant criterion was the feasibility of the presentation. for v4, the most significant criterion is linguistic eligibility. for v5, the biggest criterion is content eligibility, and for v6, the most significant criterion is linguistic eligibility. next, the data from the validation results of media experts are presented in table 2. table 2. media expert validation data no criterion average grades (percentages) average v1 v2 v3 1 display 74.67 89.33 96 86.67 2 use 80 83.33 93.33 85.55 3 utilization 90 96.67 93.33 93.33 average validity 81.56 89.78 94.22 88.52 validity level very valid, or can be used without revision table 2 shows that the most significant criterion of teaching materials is utilization, with an average percentage of 93.33%. the difference between display and utilization was 6.66%, and between utilization and usage was 7.78%. next is the student response questionnaire presented in table 3. table 3. data on the results of the recapitulation of student response questionnaires no aspects statement number average value (percentage) 1 interest 1-6 75.67 2 material 7-11 74.40 3 language 12-14 79.33 average validity 76.47 validity level “quite valid, or usable but needs minor revisions” table 3 shows that the teaching materials assessed by students have a good language aspect because the scores obtained are more significant compared to the aspects of interest and material. the difference between the three aspects is not much different. in the ninth stage of revised instruction, improvement of teaching materials based on criticism and suggestions, input and suggestions are included indicators as triggers from making the content of the discussion, determining quiz material to the end on the post-test question into direct evaluation, whether through the media students can achieve the expected essential competencies. the teaching materials are then revised according to these inputs and suggestions. in the tenth stage of design and conducting summative evaluation, an evaluation is carried out to improve the effectiveness of teaching materials. volume 11, no 2, september 2022, pp. 237-254 243 based on the q-cochran test shows the validation of material experts with statistical results ρ = 0.010< α = 0.05 means that there are differences in the validation results of elearning-based teaching materials assisted by edmodo between material experts validators 1, 2, 3, 4, 5 and 6. as for the q-cochran test, media experts showed results of ρ = 0.007< α = 0.05, meaning that there are differences in the validation results of e-learning-based teaching materials assisted by edmodo between media expert validators 1, 2, and 3. 3.1.2. mathematical literacy skills based on the pretest and post-test results of students' mathematical literacy ability tests, the average student score before learning using edmodo-assisted e-learning-based statistics teaching materials is 30.23. then, using statistics teaching materials based on elearning assisted by edmodo obtained an average score of 74.30. based on the minimum completion criteria set by the school of 75. so in the results of the pretest, students have not been able to reach the minimum completion criteria, while in the post-tests results, students have almost reached the minimum completion criteria. of the 30 students who took part in the learning, 19 students reached the minimum completion criteria, and 11 students had not reached the minimum completion criteria, so 63.33% of students had reached the minimum completion criteria. based on the completeness of learning, it is said to be complete with good categories if the average is 60% 79%. based on the n-gain value, a result of 0.6 with moderate interpretation was obtained, meaning that there was an increase in mathematical literacy ability with moderate criteria. thus, the results of the analysis of pretest and posttest data obtained by students have achieved complete learning with good categories, and an increase in their mathematical literacy skills is moderate. a comparison diagram of the pretest and post-test values of the mathematical literacy test is shown in figure 3. figure 3. the value of pretes and postes of mathematical literacy ability one example of the results of superior student work and low related to mathematical literacy test questions (level 1) with indicators of students being able to build their knowledge by making their data and processing the data into a frequency distribution table is presented in figure 4. 0 20 40 60 80 pretes postes pretes postes supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 244 question class xii busana 2 is carrying out mathematics learning with the teacher. the task given is identifying information and using his knowledge to solve the problem. with the instruction, make a set of scores of 80 students, with the lowest test score being 35 and the highest score being 99, where log 80 = (1.9031). present in a group frequency distribution table. superior student answers translate: count of data = 80 range = r = 99 – 35 = 64 determine the number of classes: k = 1 + 3.3 log 80 k = 1 + 3.3 (1.9031) k = 1 + 6.28 k = 7.28 k = 7 class length = 64/7 = 9.1 = 10 asor student answers translate: 2. the income that students and their wood shop often earns is 141,912 rupiah figure 4. postest problem number 1 based on figure 4, the answers of superior students have been able to make their data with their knowledge. solve problems with already known formulas so that students can create a frequency distribution table from data that has been created by themselves. thus, students are considered capable of completing level 1 mathematical literacy skills. as for the answers, the students did not do the questions according to the instructions. the students did not write down the requested data and could not solve the problem using formulas, so the students were not able to make a frequency distribution table. thus, students are considered unable to complete level 1 mathematical literacy skills. volume 11, no 2, september 2022, pp. 237-254 245 3.2. discussion in the first stage of the development of teaching materials for statistics materials based on e-learning assisted by edmodo was found that in the process of teaching and learning mathematics, only package books are available in the library. in the learning process, teachers always use the old method, namely the one-way learning process, so the teacher explains or lectures more. the lecture method is straightforward for verbalism to occur. more visual students become at a loss, teachers find it challenging to conclude between students who already understand or not, and students become passive in the learning process (helmi, 2016). so student activity becomes less because students only listen, without opening up opportunities for students to think more broadly than what the teacher conveys. thus, students experience difficulties when learning mathematics, especially when faced with math problems that are not routine. these difficulties are further increased by learning from home due to the covid-19 pandemic. teachers and students often use whatsapp groups to give assignments or just provide youtube links already available on the internet and ask students to learn about it. so, it can be concluded that the learning media used has not been maximized. schools need to consider efforts and approaches to improve the quality of e-learning and the learning outcomes achieved (al-smadi et al., 2022). especially for mathematics, they were learning that students find challenging and cause students' grades to always be below the criteria of minimal completion. so that the design of teaching material products that are attractive, easy to understand, and can be studied anywhere, which is currently indispensable for students to learn remotely using e-learning-based teaching materials that are already well done and students feel interested. e-learning can be beneficial in mastering the material, which is expected to increase awareness that learning is essential and fun in mathematics subjects, will achieve maximum achievements and positively influence mathematics learning and even increase learning outcomes (utami & cahyono, 2020). in the second stage, the material in this study is statistics. statistics is the subject matter of mathematics related to data collection and concluding the results of observations in the field. statistics has become the basis for researchers, research, or observations in various fields of science (listiati, 2022). statistics material is significantly related to daily life. statistics will be needed by students, both in the field of work and in completing their studies. wahyuningrum (2020) states that by studying statistics, a person can explain the relationship between variables, make decisions for the better, overcome changes and make plans and predictions. therefore, the basic competence used is to determine and analyze the size of the centralization of data and the dissemination of data presented in frequency distribution tables and histograms. apart from that, by studying statistics, many students will be trained to think systematically, conscientiously, and understand more mathematical symbols, because many are faced with formulas and problems that are not routine. studying statistics will benefit students in their daily lives because statistics can be applied in everyday life. statistics are often used in everyday life, such as in the research of a study, and applied in disciplines such as astronomy, biology, economics, and industry (janna, 2020). the third stage is to identify the initial abilities of the students and the strategies used. at this stage, it is found that students' initial ability in mathematics is still low, both in terms of affective and cognitive. so, the right learning strategy uses the edmodo application, which can be accessed on mobile phones and computers and helps students learn independently from home. learning by using phones is used to access learning, materials, instructions, and questions related to student learning, when and wherever they want to learn (yunianta et al., supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 246 2019). edmodo's application can help students achieve affective and cognitive abilities for the better. the edmodo application is a digital application that looks very easy and simply because it is similar to social media applications. of course, students are used to social media. edmodo is a free and secure learning platform available on www.edmodo.com. this website looks similar to facebook but is much more private and safer for the learning environment (ompusunggu & sari, 2019). using the feature in edmodo encourages student involvement in responsible learning (balasubramanian et al., 2014). edmodo has complete learning features, including flowing features, a calendar, classes, discovery, and messages. thus, using edmodo can help teachers and students in learning activities carried out online can be more directed and organized. edmodo can help students in learning because edmodo is developed based on classroom management and social media principles. edmodo also makes it easier for teachers to track student abilities. grades and assignments are automatically stored in the system and easy to access (ekayati, 2018; hanifah et al., 2019; kristianti, 2016; putranti, 2013; wahyuni & sugiharta, 2019). the fourth stage is to formulate the developed procedure. at this stage is an explanation of the procedures for accessing edmodo. the first step that must be done is to create an account for both teachers and students and fill out the registration form and valid data. then the teacher arranges the edmodo account, such as forming classes according to the many classes taught and students joining the provided classes (ekayati, 2018). these steps are not an obstacle for students because students are used to accessing applications on the internet. however, when installing the application, some students experienced problems because their cellphone memory was full, so they could not install edmodo. however, this can be handled because creating an account on edmodo does not always have to install an application. edmodo can also be accessed from google directly. conditions like this do not hinder teaching and learning activities because students of any type now enjoy technology and are familiar with it (kristianti, 2016). overall, students feel that they do not mind learning using edmodo. even students feel happy because the learning process in edmodo is easy to access. so, it requires careful preparation when using the edmodo application. the use of edmodo requires training for students starting from how to use and create accounts, class codes, how to access materials, and abilities that students must have when using the edmodo application (sari, 2015). the fifth stage is to develop relevant research instruments. at this stage, the instrument developed is a question of mathematical literacy ability consisting of 6 questions in the form of a description, each question made based on indicators and levels of mathematical literacy ability. karmila (2018) state that indicators of mathematical literacy ability are level 1 students that can use knowledge to solve routine problems and problems with a general context. level 2 students can interpret problems and solve them with formulas. level 3 students can carry out procedures well in solving problems and can choose problem-solving strategies. level 4 students can work effectively with models, choose and integrate different reprints, and connect with the real world. level 5 students can work with models for complex situations as well as be able to solve complex problems. level 6 students can use their reasoning to solve mathematical problems and generalize, formulate, and communicate their findings. the question was compiled by the researcher and assessed by the supervisor, and has been tested. the results of the question trial based on statistical processing and mathematical literacy ability are feasible for field trials. the sixth stage is the components of this e-learning-based teaching material which consists of twelve components. the essential components of e-learning-based teaching materials are an attractive appearance, a display filled with many interesting images, colors, and animations, and explicit material content and contents that make it easier for students to volume 11, no 2, september 2022, pp. 237-254 247 understand learning material. because with teaching materials that follow what is needed by students, learning will be enjoyable and can help achieve learning goals very well. the elearning system requires technological support in the digital era as a mechanism and exciting content (elyas, 2018). the use of videos and files uploaded to edmodo's account makes students more enthusiastic about learning (hanifah et al., 2019). the seventh stage is the established strategy and the abilities to be improved. based on the results of the validation of teaching materials, it produces teaching materials worthy of trial with an outstanding level of validity. going through the validation stages of teaching materials will make the developed teaching materials very good and get various criticisms and suggestions from validators that will help improve teaching materials. validity determines the quality of the teaching materials, and it will be seen what should be measured (azis, 2019). the eighth stage is the trial stage of e-learning-based teaching materials to students who are used as research samples. eight meetings have been held in the field trial stage; the learning was carried out online and through the edmodo application. the first meeting was with the giving of pretests to students. the second to the seventh meeting is giving materials and quizzes from each sub-topic, consisting of 3 sub-topic. moreover, the eighth meeting was the giving of the post-test. the teaching materials that are compiled must be excellent and correct because teaching materials are an essential part of learning. through teaching materials, students will be more helpful and easy to learn (magdalena et al., 2020). according to students, the results of interviews using google forms by learning to use edmodo-based e-learning teaching materials throughout the secular state that the appearance is attractive, reasonable, understandable, concise, and straightforward. then based on the material's content, teaching materials are straightforward to understand but problematic when faced with complex calculations. most students feel happy learning with this elearning-based teaching material because it is straightforward to understand and understand. e-learning-based learning can improve maximum learning outcomes (hartanto, 2016). the last stage is inputs on the improvement of teaching materials rather than validators that the teaching materials that have been prepared have several inputs. first, the teaching materials covered should be given authentic images of the material discussed in the media. the second is to include indicators in each discussion so that students know what should be achieved in the learning process. researchers have tried to improve the results of these inputs and produce suitable teaching materials. based on all the stages that have been carried out, the design of the developed teaching materials for statistics based on e-learning is assisted by edmodo. teaching materials are prepared by paying attention to the components of teaching materials suitable for e-learning-based. teaching materials can be accessed through the edmodo application easily. the teaching materials developed have gone through the validation stage by expert validators, media, and teachers so that teaching materials have good validity with excellent categories and can be used in the learning process for students. e-learning can make students access teaching materials or structured assignments independently without being limited by distance and time. however, in this case, there are also several obstacles faced in learning mathematics using e-learning media related to how difficult the material is to deliver because some materials are not easy to deliver, even face to face (hulukati et al., 2021). mathematical literacy ability refers to the pisa mathematical literacy ability level (karmila, 2018), which consists of 6 levels. at level 1, with mathematical literacy skills, students can use their knowledge to solve routine problems and problems with general contexts. most students can solve this problem well because, with their knowledge, students are asked to make data, and the data created by themselves must be processed so that the answers that arise from students will, of course, be different. a small part of there are supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 248 students who do not finish working on the problem, and some students suddenly make the final result in the form of a frequency distribution table. students can guess the answer, or students forget how to solve it. in addition, the lack of students in solving these problems is that students do not provide conclusions on the answers that have been made. this situation happens based on the results of the researchers' analysis; students may forget, or students are not accustomed to making conclusions outlined in written form. according to pisa, students' mathematical literacy achievement is concerning; 42.3% of students have not reached level 1 of the lowest proficiency (styawati & nursyahida, 2017). therefore, here the role of the teacher is needed to continue to train students so that their mathematical literacy skills develop to the maximum. level 2 literacy skills, namely interpreting problems and solving them and solving them with formulas, some students have been able to solve this problem, read the problem, and change it into a mathematical context well so that students can solve the problems faced. students can change the data presented as a bar chart into a frequency distribution table which is then searched for the mode value of the question posed. however, as in the case of mathematical literacy skills at level 1, students do not give conclusions at the end of their answers. at level 3 literacy skills, namely carrying out procedures well in solving problems and choosing problem-solving strategies, students are given questions about a table containing information about reports of visitors to a swimming pool. in this question, students' answers can vary depending on which side the student will answer. because what is being asked in this question are deciles, students can answer deciles 1, 2, 3, 4, or 5 so that the answer can be five alternatives. most of the students also have solved this problem well, can read the data, and complete what is asked in the question. however, it remains the same as the level 1 and 2 mathematical literacy skills. students do not provide reviews or conclusions from the results of their calculations. level 4 literacy skills can work effectively with models, choose and interpret different representations, and then relate them to the real world. mathematical literacy in learning mathematics in pisa develops competencies directly related to the real world (umbara & nuraeni, 2019). the oecd states that mathematical literacy can help someone understand the role or use of mathematics in everyday life. in addition, mathematical literacy emphasizes the ability of students to analyze, give a reason and communicate ideas effectively in solving mathematical problems they encounter (muzaki & masjudin, 2019). some students have solved the questions given, but at level 4, some are wrong in the calculations. in this matter, accuracy is needed, and knowing the formula and how to calculate it must be used. because not a few students answered the questions, not to completion, and did not give conclusions at the end of their answers. at level 5 literacy skills, namely being able to work with models for complex situations and solving complex problems, and level 6, using reasoning in solving mathematical problems, making generalizations, and formulating and communicating findings. for this level, most students cannot solve this problem well. most of them are wrong in answering the questions given. the results of the 2012 pisa research state that none of the indonesian students can answer the pisa level 5 and level 6 questions (putra et al., 2016). based on the question posed is about the variance or variance and standard deviation, where the variance and standard deviation are related. meanwhile, the calculations require calculations that students feel to be very difficult because there are calculations with unmistakable signs, powers, and formulas that are difficult to remember. thus, mathematical literacy skills at levels 5 and 6 still need to be improved to achieve everything more optimally. based on mathematical literacy skills from level 1 to level 6 seen from the results of student work, the abilities that have been achieved are level 1 to level 4, meaning that volume 11, no 2, september 2022, pp. 237-254 249 students can use their knowledge to solve, interpret, and interpret problems to solve them with formulas. in addition, students can carry out procedures well in solving problems and choose strategies. students can work effectively with models and relate them to the real world. because students have not been able to achieve optimally at levels 5 and 6, it means that students have not been able to work with models with complex situations and have not been able to solve complex problems. then students have not been able to use their reasoning in solving mathematical problems, formulating and communicating their findings. mathematical literacy plays a vital role as a life skill. therefore, teaching mathematics in schools should aim to develop mathematical literacy and improve the ability of each student to use and apply mathematical knowledge to solve real-life problems or situations (sumirattana et al., 2017). mathematical literacy skills are essential because mathematical literacy is an individual's capacity to formulate, use, and interpret mathematics in various contexts. these skills include mathematical reasoning and using mathematical concepts, procedures, facts, and tools to describe, explain, and predict phenomena. these skills lead individuals to recognize mathematics's role in life and make sound judgments and decisions that are needed by constructive and reflective citizens (sari, 2015). mathematical literacy ability does not automatically grow in every child; it takes effort to grow these abilities. for example, literacy-based learning is one of the efforts that can be done. in addition to improving students' mathematical literacy skills, appropriate teaching materials are needed so that students are trained to solve mathematical literacy problems at a higher level by frequently practicing problem-solving (masfufah & afriansyah, 2021). 4. conclusion based on the processing results and findings in this study, the development of edmodo-assisted e-learning-based teaching materials is very valid and can be used in learning. the method of developing e-learning-based teaching materials uses the dick and carey model with the following stages: (1) identify the instruction goal, the existing teaching materials are not accompanied by animation and the e-learning learning media is not varied; (2) conduct instructional analysis, relevant knowledge is statistical material, because its application plays a vital role in everyday life; (3) identify entry behaviours, students have not been able to manage and regulate themselves in thoughts, feelings and behavior; (4) write performance objectives, formulate the ability to use edmodo application learning media; (5) develop criterion referenced test, it is necessary to develop relevant items that improve mathematical literacy skills; (6) develop an instructional strategy, teaching materials are arranged according to the e-learning component; (7) develop and select instructional materials, display of teaching materials that are arranged based on the provisions of developing teaching materials; (8) design and conduct formative evaluation of instructional, statistical teaching materials have been designed equipped with contextual problems, animations and exciting pictures; (9) revision instruction, improvement of teaching materials based on criticism and suggestions, and (10) design and conduct summative evaluation, making improvements to teaching materials to increase the effectiveness of teaching materials. statistics teaching materials based on e-learning assisted by edmodo give pretty good results on mathematical literacy skills, most students have achieved the minimum completeness criteria scores, and there is an increase in mathematical literacy skills in the medium category. these results cause by students that can use their knowledge to solve routine problems with general contexts, students can interpret problems with formulas, students can complete procedures well, and students can work effectively and integrate with supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 250 real problems. so that students feel more helpful in understanding the material and can follow the learning process well even though it is done online. acknowledgements the authors would like to thank the postgraduate of universitas pasundan and smk negeri 2 baleendah, who have given the 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(2020). statistika pendidikan (konsep data dan peluang). jakad media publishing. yaniawati, r. p. (2012). pengaruh e-learning untuk meningkatkan daya matematik mahasiswa [the effect of e-learning to improve students' mathematical power]. jurnal cakrawala pendidikan, 31(3), 381-393. https://doi.org/10.21831/cp.v0i3.1137 yudiawati, n., trisaputri, f., & sari, n. m. (2021). analisis kemampuan literasi matematik dan kemampuan pemecahan masalah siswa ditinjau berdasarkan gender melalui pembelajaran reciprocal teaching [analysis of students' mathematical literacy skills and problem-solving abilities were reviewed based on gender through reciprocal teaching learning]. pasundan journal of mathematics education jurnal pendidikan matematika, 11(1), 65-77. https://doi.org/10.23969/pjme.v11i1.3691 yunianta, t. n. h., putri, a., & kusuma, d. (2019). development and comparison of mathematic mobile learning by using exelearning 2.0 program and mit inventor 2. infinity journal, 8(1), 43-56. https://doi.org/10.22460/infinity.v8i1.p43-56 https://doi.org/10.23969/symmetry.v6i1.3999 https://doi.org/10.22460/infinity.v8i2.p167-178 https://doi.org/10.33365/ji-mr.v1i1.252 https://doi.org/10.24256/jpmipa.v7i1.467 https://doi.org/10.21831/cp.v0i3.1137 https://doi.org/10.23969/pjme.v11i1.3691 https://doi.org/10.22460/infinity.v8i1.p43-56 supianti, yaniawati, osman, al-tamar, & lestari, development of teaching materials … 254 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 1, february 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i1.p145-162 145 the process of conceptualization in solving geometric-function problems masta hutajulu1, krisna satrio perbowo2, fiki alghadari3, eva dwi minarti1, wahyu hidayat 1* 1institut keguruan dan ilmu pendidikan siliwangi, indonesia 2university of warwick, england, uk 3stkip kusumanegara, indonesia article info abstract article history: received oct 27, 2021 revised jan 23, 2022 accepted jan 27, 2022 functional analysis has been of interest and the thinking of students should be prepared. analyzing the process of conceptualizing geometric function problem solving based on the dimensions of cognitive processes and knowledge was the purpose of this study. the subjects of this study were three students selected purposively from one of the secondary schools in indonesia. exploration of these studies with constant comparative techniques. the results of the data analysis show that the cognitive processes that operate and the knowledge applied by students focus on the conceptualization of algebraic representations. based on the conceptualization process, students' conceptual systems are still fragmented because of the problem of the relationship between the concept and the basis of its relationship. as a result, procedural knowledge is more dominant. keywords: bloom taxonomy, cognitive process, conceptualization, geometric-function, knowledge dimension this is an open access article under the cc by-sa license. corresponding author: wahyu hidayat, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi jl. terusan jenderal sudirman no. 3, cimahi, west java 40526, indonesia email: wahyu@ikipsiliwangi.ac.id how to cite: hutajulu, m., perbowo, k. s., alghadari, f., minarti, e. d., & hidayat, w. (2022). the process of conceptualization in solving geometric-function problems. infinity, 11(1), 145-162. 1. introduction in general mathematics education research, problems have often been involved in the construction or reconstruction of concept schemes, also sometimes intended to be meaningful learning experiences for students (mumu et al., 2017). for example, one approach that involves problems is problem-based learning (bartholomew & strimel, 2018; widodo et al., 2019), and besides, there are several other approaches, such as in hendriana and fadhillah (2019) which also uses the function of mathematical problems to solve problems. one of them is through a constructivist approach, problems are solved by students, used as learning resources to obtain a new mathematical concept (hendriana & fadhillah, 2019). in this case, solving mathematical problems is required to achieve learning goals. aside from being a learning resource that aims, the problem that is solved is also subjectively will increase the capacity of each student's ability concerning their mathematical conceptual https://doi.org/10.22460/infinity.v11i1.p145-162 https://creativecommons.org/licenses/by-sa/4.0/ hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 146 knowledge (hendriana & fadhillah, 2019; hendriana et al., 2018). for this purpose, the problem is functioned as a conditional source of learning and is said to be so because of the increase in the ability of students on the condition that the solutions they have been able to achieve so that their intellectual capacity increases. however, anything its function, it is subjective for each student that ideally, the solution to the problem is not easy to achieve immediately, and that is the main reason why the problem is not a routine and an intellectual challenge (alghadari & kusuma, 2018; dossey, 2017; polya, 1981). from the information above, for the two modes that function on a problem, the goal is that students can solve by passing conceptual constraints (dossey, 2017). there will be several cognitive processes that produce a bridging concept to arrive at that goal (alghadari & kusuma, 2018). hendriana et al. (2018) and nagle et al. (2013) for instance, said that reasoning is a process of problem-solving. furthermore, clancey (2001) says that the conceptualization process occurs in the problem-solving phase. the reasoning is aimed at conceptualizing the solution. on the other hand, polya (1981) has described cognitive processes and is a productive activity that can be used as a reference to overcome obstacles so that solutions are found. that is the concept of problem-solving steps. we believe that the existence of these steps to help anyone who is confused starts the settlement step. however, what is conceptualized is to complete, and any processes occur for each problem so that a solution appears, it is not a simple sequence (alghadari et al., 2019; clancey, 2001; glaser, 2002). if until now the problem-solving is considered important, both for learning purposes and improving mathematical abilities, then the conceptualization of problem-solving is part of that assumption. moreover, the problem solved requires the involvement of various dimensions with several hierarchies of mathematical concepts and different ways of solving that might be applied. this corresponds to the statement of clancey (2001) that conceptualization includes object and operating differences, and all about the experience, memory, and thought. because it is already known, there is more than one dimension involved, the reference to problem-solving processes may only be a procedure that is thoroughly seen between procedural and conceptual knowledge. glaser (2002) states that there is a conceptual theoretical burden when many concepts are gathered in one concept and because the concept of one event is not a careful generation of a pattern so that it becomes very unusual when there is no analysis. therefore, the occurrence of processes in a particular activity from the problem-solving phase, so that it becomes one goal for further analysis studies that are examined from several dimensional involvement. 1.1. for conceptualizing: the problem and solving process regarding a term from a problem, which is related to mathematics, it is said to be a mathematical problem, the existence of a particular procedure is not directly accepted as one of its characteristics, so there are conditions that can derail the definition of a problem. therefore, it needs to be translated more deeply into the meaning of the procedure, because it concerns the specifications of the problem definition. rittle-johnson and schneider (2015), and bartholomew and strimel (2018) suggest that the procedure is not only in the form of algorithms but also actions that may have to be sorted precisely to solve a problem. concerning the quotation, alghadari and kusuma (2018) reveal that procedures can be built by referring to the process when seeking a solution, then the collection of procedures found empirically is generalized as a problem-solving procedure, and it is not a solution to the intended procedure as an algorithm. furthermore, this is the fact that occurs in historical records concerning the special connection between matter and geometry. so, the procedure referred to and which is not the nature of the problem according to dossey (2017) and polya volume 11, no 1, february 2022, pp. 145-162 147 (1981) is about the existence of an algorithm. however, for certain cases there are conditions relating to the purpose of the last procedure, which is not an algorithm, and it develops through problem-solving practices (rittle-johnson & schneider, 2015). then, at the time of the practice, students are involved in active cognitive processes as well as building meaning (mumu et al., 2017; radmehr & drake, 2017, 2018). thus, there is an important meaning of the construction of student knowledge that involves the function of a problem (alghadari et al., 2019). this study uses open-ended conceptual problems, as claims, which fulfill the nature and criteria of mathematical problems according to dossey (2017), which is a problem in the problem itself. rittle-johnson and schneider (2015) have defined conceptual problems as problems that are relatively unfamiliar to students, so they must obtain from their conceptual knowledge compared to procedures they know. these two pieces of literature confirm information about the nature and criteria of the problem. in other words, no algorithm serves to determine completion as a symbolic procedure so that a solution is immediately found (alghadari & kusuma, 2018; dossey, 2017; polya, 1981). in the previous section, it was stated that in the phase of problem-solving, there is a series of cognitive processes that are not simple to produce a concept that becomes a bridge to a solution. for the concept in question, it is a process that is not simple and with the hierarchy of mathematical concepts applied, so that it relates to the statement rittle-johnson and schneider (2015), and wagner and sharp (2017) that mathematical knowledge requires repeated concepts and layered procedures. therefore, solving problems solved by students through conceptualization in which various cognitive processes occur in abstract mathematical concepts. here, there is a function of one reasoning process as mentioned, but the process does not take and rearrange the concept, but only activates the existing one, then connects it to form a general sequence and composition, and that conceptualizes the process of coordinating those relationships, which changes what the concept and the work they do (clancey, 2001). furthermore, regarding conceptualization, oberle et al. (2004) define it as the idea of ontology in the context of formal explicit specifications. or, the design of a process to produce a concept that is timeless and handles all that is needed so that it is a method applied to view events (glaser, 2002; nagle et al., 2013). furthermore, although there is a process that is not simple, conceptualization is a simplified abstract view of a representation for a purpose (oberle et al., 2004). so, conceptualization is a structured, mental prescribing process and refers to a series of procedures for developing a conceptual representation. 1.2. the dimension of cognitive process and knowledge with the conditions of the conceptual problem definition according to rittle-johnson and schneider (2015), then in the process of completion, there are cognitive and metacognitive roles so that both contribute to the development of conceptual schemes (iskandar, 2016). the role said by kilpatrick et al. (2001) related to strategic competence and adaptive reasoning. while creative reasoning is related to conceptual knowledge as a network where links separate pieces of information (bartholomew & strimel, 2018; hendriana et al., 2018). furthermore, in iskandar's literature (2016) it has been said that metacognitive plays a role in connecting between conceptual understanding and procedural experience. at that time, several processes occur such as analyzing the complexity of the problem, selecting important information used, and thinking about the thought process during problem-solving. these skills are in the context of monitoring self-understanding and problem-solving activities so that it is no different from adaptive expertise in cognitive studies by kilpatrick et al. (2001). up to this point, there are some different terms but there hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 148 is a similarity of words used. such as knowledge or conceptual understanding, and indeed these two terms describe different contexts based on a point of view in solving problems so that both need to be further detailed according to each dimension. furthermore, so that students are proficient in mathematics and solve problems, actually in kilpatrick et al. (2001) and hendriana et al. (2018) indeed it has been mentioned that there are 5 independent strands and represent different aspects of the complex whole, two of which have already been mentioned, strategic competencies and adaptive reasoning, while the other three are conceptual understanding, fluency of procedures, and productive dispositions. however, rittle-johnson and schneider (2015), and sahin et al. (2015) say that mathematical competence rests on the development of two knowledge, conceptual and procedural. however, without conceptual knowledge, procedural knowledge is of limited value and can be so limited that the consequences must be based on understanding concepts (österman & bråting, 2019; radmehr & drake, 2017, 2018; sahin et al., 2015). that is because organizational knowledge plays an important role in facilitating the use of knowledge (alghadari & kusuma, 2018). meanwhile, on the other hand, there have been difficulties to study the relationship between conceptual and procedural knowledge because test items or problems predominantly measure one type of two knowledge (rittle-johnson & schneider, 2015). so from that open-ended problem is played to see the interaction of the two knowledge, because problems with these criteria can invite students' thinking processes that are different in ways or approaches to problem-solving (bartholomew & strimel, 2018). the importance of this two differentiated knowledge is to understand the interaction between the two, and in the research study that it is fundamental because it is a source of information about the construction or reconstruction of knowledge (rittle-johnson & schneider, 2015). the intended interaction is about how students represent and connect pieces of knowledge so that the organization increases retention, improves fluency, and facilitates learning related material, which is a key factor whether they will understand it indepth and can use it in problem-solving (kilpatrick et al., 2001). the illustration above has led to the literature on radmehr & drake (2017, 2018) regarding bloom's revised taxonomy with a two-dimensional framework and has separated the dimensions of cognitive processes and knowledge. furthermore, the use of one or more frameworks can help understand the processes and phenomena observed when exploring mathematical concepts, and bloom's potential taxonomy is the largest, detailed, comprehensive, and flexible to explore a student's assessment of mathematical content. 1.3. knowledge in function and geometry to solve in mathematics, there are concepts of function and geometry. the two concepts are studied along the journey of students' education which is adjusted to their level and ability to think. the higher level of education causes the learning of the two concepts will increasingly show and utilize the relationship between them. such as in high school, which involves the concept of function and geometry is in the material sequence numbers or calculus. here, we focus only on both of them, because calculus is included in the indonesian mathematics education curriculum in high school and the conceptual problems that have been designed also for the material. the basic concept of calculus has used functions and geometry so that the level to study it is no longer recommended at the stage of concrete thinking. in the borji et al. (2018) it has been said that some of the special difficulties students were the graphical representations of derivatives, those were about the basic concepts of calculus. regarding that, wagner and sharp (2017), and sahin et al. (2015) said, calculus had a reputation as a concept with terms of procedures that caused students to be trapped in trying to memorize algorithms and rules that did not involve the development of volume 11, no 1, february 2022, pp. 145-162 149 related conceptual knowledge. strong conceptual knowledge serves to accelerate skills and procedures in calling information to solve problems (setyawan et al., 2017). therefore, the cognitive process for solving problems will involve all related potentials, so that conceptual and procedural knowledge that is no longer seen as two separate things, but combined in the complexity of abstract thinking processes. regarding mathematical concepts involving geometry, in van hiele's theory, it has also been mentioned in usman (2017) that there is a level of abstraction in which objects think of properties of a particular form intended to produce relationships between properties of that form. this becomes a reference that the process of thinking abstraction also exists for geometry even though it is not identical with concrete visualization. the theory is not for the framework of this study, but wagner and sharp (2017) have stated that it will be relevant if the conceptual problem is geometric. visual representation of a geometric shape is a clue to a pattern or characteristic that contains a concept, resulting in successful problem-solving will depend on the ability of the individual to identify how the components exist and complement each other (usman, 2017). unlike the concept of function, because kop et al. (2015) suggest that it cannot be accessed directly as a physical object but access to mathematical concepts can be obtained through representation. this study presents a mathematical conceptual problem of students that determine the graph of derivative functions, but the knowledge given as a source of information was showing the geometry of a function only, without the details for an algebraic function. therefore, some knowledge will be involved such as algebraic formulas, function domains, and meaningful relationships between representations (kop et al., 2015). choi and hong (2014) mention the ability to read the information provided only in graphical form requires thinking about the nature of complexity. according to the results of the study tokgoz and gualpa (2015), there is a tendency for students to solve problems with these characteristics by determining the algebraic representation. the problem is the concept of calculus which combines with the concepts of function and geometric shape, therefore we define it as a geometric-function problem. furthermore, tobin (2012) stated that a problem was seen as a geometry problem when the geometry elements of the curve can be communicated with the concept of negative or positive gradients at a certain point so that the problem was solved without involving an algebraic function formula. 1.4. the present study from previous studies, borji et al. (2018) reported that most students had problems developing mental constructs to calculate derivatives so they did not pay attention to the relationship between functions and their derivatives in an interval. the report has become a point of view for conducting relevant studies. kop et al. (2015) in his study of composing effective and efficient strategies for making graphs without the help of graphical tools, has shown that two-dimensional frameworks, recognition, and heuristics, can be used to describe graph formulation strategies. however, the dimensions of the study are different from this study. the dimensions of this study are cognitive processes and knowledge. the dimensions of cognitive processes are distinguished by the dimension of knowledge so that the details of the analysis with this framework approach are more comprehensive than just one dimension. radmehr and drake (2018) have made these differences based on the use of verbs for cognitive processes and nouns to show knowledge. furthermore, procedural knowledge refers to the use of symbols from a system of rules or procedures for solving mathematical problems, while conceptual knowledge is the knowledge that is rich in conceptual relationships (österman & bråting, 2019; radmehr & drake, 2017, 2018). the quotations above become the theory in this study to analyze the process of solving geometric hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 150 problems in student functions based on the dimensions of cognitive processes and their knowledge. 2. method 2.1. student's participant explorative studies with this simple technique of constant comparative were initially carried out purposively to seven students of class xi in the 2017/2018 school year in a science program at one of the secondary schools in the west jakarta area. the selection of them aims at the needs of this study, which shows the conceptualization process so that the students involved are those who have the potential to solve the problem. therefore, these students are recommendations from their mathematics subject teachers. then, all of them solved the geometry-function problem that we provided. in addition to solving problems with paper and ink, students also interpret what they write on the same sheet of paper so that it shows their flow of thinking in progress. 2.2. data collection this study involves problems related to function and geometry in calculus. of the seven students who have solved geometric-function problems, the results of the solutions are obtained along with their interpretations of what they have made. the problem they solved was adapted from tobin (2012), was to determine the sketch of the graph model of the derivative function of each graph in figure 1. figure 1. sketch of the function graph figure 1 contains three different problems, (a), (b), and (c). henceforth, sometimes we say figure 1 (a) as a problem 1 (a), and so for others. the three problems were solved by the students. we encourage them to complete sequentially, but not oblige because conceptual ideas of completion are purely the result of their thoughts written. each student's paper that reads is the primary data to be analyzed. each student will be sampled if they show a solution accompanied by the interpretation they describe in writing. 2.3. data analysis not all results of student completion become part of data analysis because not all of the analysis objects provide completeness as needed. after being examined from collected data, there were only three of the seven students selected to be the sample analysis in this study, and each of them was marked with the initials of their names, an, ni, and pr. from solving problems, there are several foci analyzes that become our point of view, namely mathematical concepts used for completion, cognitive processes involved and metacognitive functions that control, as well as conceptual and procedural knowledge that operates. the volume 11, no 1, february 2022, pp. 145-162 151 analysis refers to the radmehr & drake (2017, 2018) literature on the revised bloom taxonomy to compile a generalization of the conceptualization process for solving geometryfunction problems. 3. results and discussion 3.1. results it has been stated that there is a factor of subjectivity based on his knowledge which directs students to choose a particular way of solving a problem. therefore, in this study that of the three students. there was no one answer that is the same as the others, but the basis of the process of conceptualizing the solution is that it tends to be the same, for example for problem 1(b). there is a conceptualization process by a student who is completely different from the other two students, for problem 1(c) shown by pr, while for problem 1(a) is indicated by ni. it is clear that when different conceptualization processes will find different solutions. this is following the problem category as an open-ended type. although, the three students have answered the problem, each of them has missed a basic relevant mathematical concept to conceptualize the solution they made, namely the domain and range of functions. for more information about students' responses to problems, here are each of their completion processes. 3.1.1. an’s solving process the focus in figure 1 (a), an recalling his knowledge of functions and graphs. because it recognizes graphs, an sets x e as the most common classification for that. however, the graph of the exponent function did not mean the same as the image so there is another process to equalize the form, and that is called the "reversed" process. the process as an application of the concept of reflection in geometric transformation, although the detailed application of the concept was not written thoroughly, it was written that the formula of the function of the transformation results is x ey − −= . up to this point, an has not finished using the concept of transformation, then it was intended to make a curve until it was similar to the model image 1 (a), with translation  10 so that it passes through the base of the coordinates, and defines the graph asymptote with 11 lim =+− − → x x e . by its exemplifying process, there is no elemental detail in the problem that shows information about the asymptote boundary, so an represented the graph in figure 1 (a) in the algebraic function formula aey x +−= − , for a real number element. the algebraic formula after being derived was x ey − =' and sketched as shown 2(a). the graph model of the derived function that has been drawn by an is as follows. (a) (b) (c) figure 2. the completion problem by an hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 152 figure 2 was a problem-solving by an of the three graphical forms in figure 1. figure 2 (a) is a problem-solving for figure 1 (a), also the letters are adjusted for the other two. following is a description of the process for solving problem 1 (b). an outline of the substance of the conceptual idea of problem-solving was no different. in other words, the first step in the process of solving problem 1 (a) was not an exception so that it affected the settlement activity for the next two numbers. however, because there was an element of graphical representation that was loaded problem 1 (b) so with its conceptual knowledge on coordinates ( )4,0 and ( )0,5 , then an concluded that it was an ellipse. the pair of coordinates in the graph and the terminology of the forms already exist in the an knowledge dimension, so that the recalling process was carried out in the cognitive dimension. in further information, the results of checking and differentiating among the coordinate axes in the graphic image, so an considers that the graphic form was more "rectangular", and if the more square the higher the degree of power of the ellipse equation. finally, an produced the algebraic formula for the graph. 1 256625 44 =+ yx based on the implicit algebraic equation above, an implemented an explicit function and then used implicit differentiation, so that the derivative results obtained from the formula is as follows: ( ) 4 1224 4 256160000 251024 ' x x yxf −  == for the above function formula, an illustrated the graph of the derived function for figure 1 (b) as shown in figure 2 (b), so the problem is solved. furthermore, for the series of problem-solving processes 1 (c), several conceptions had been interpreted by an in the answer paper. such as, graphs that started from ( )a,0 then up and down so that it loaded functions xcos , but the model was getting down so it also loaded x− , then, the coefficients had begun to be adjusted resulting in the shape of the function approaching the figure 1(c). the process for solving this problem, an did not write many details, but an produced algebraic formulas in explicit function as follows. 4 2cos2 x xy −+= dan 4 1 2sin2' −−= xy the derivative of y for x was y '. the function y was an algebraic form of graphic figure 1(c). whereas function y was an algebraic form for figure 2 (c) representation. the concepts were solved by an, on the three problems illustrated the similarity of processes. 3.1.2. ni’s solving process all students in solving problems will certainly be recalling their relevant knowledge. ni had also done it to solve these three graph problems. he wasn’t different from the previous subject because the solution was conceptualized by determining the algebraic formula from the graph in figure 1, but there was something different from the algebraic formula, through his conceptual knowledge, ni had conceptualized that the classification for figure 1(a) led to the form 1− −= xy . however, factual knowledge assumed that the graph of the algebraic formula did not match the picture on the problem, and its least upper bound equally zero for real positive number function domain, then ni with its metacognitive knowledge made adjustments by adding a as an implementation of the translation concept so that it became the upper or minimum limit. here, we found that the algebraic formula was volume 11, no 1, february 2022, pp. 145-162 153 constructed by ni produced an explicit form axy +−= −1 and a a real number, but the answer was less precise, because a was not defined in detail as a positive real number, and the graph of the function also did not pass through the origin, it would have an impact on the representation for the derived function. (a) (b) (c) figure 3. problem answers by ni however, through the algebraic formula, the derivative functions and graphs, 2 /1' xy = immediately known and illustrated by ni, as shown in figure 3 (a), so that a did not become a problem that arose due to the derivative function of x. the following are three pictures from the graph of the derivative function as a result of the conceptualization problem-solving that had been constructed by ni. figure 3 is the three graphic images that ni had been made for each number of problems presented. each graph was drawn from the function constructed, then differentiable and represented by a sketch in the coordinate plane. the following is part of the description for the construction of functions so that figure 3 (b) is obtained. with his factual knowledge, ni stated that figure 1 (b) as an arc that intersects the x-axis was more curved than an arc that intersects the y-axis, so there was a difference in rank for the variables x and y. the more curved arc was interpreted as a variable of rank 2 while the others of the rank of 4, and ni produced the following equation accompanied by the equation for its derivative. 1 45 24 =      +      yx and 32 1 4 55 18'                     −−= − xx y based on the two algebraic representations above, the explicit form y' was derived from another representation. here it had indirectly seen that it was obtained from an initial algebraic representation (in the implicit form), which was first converted to an explicit form involving procedural and conceptual knowledge. derived algebraic functions were obtained and images made in the coordinate plane so that the result was figure 3 (b). in the figure, ni had been attributed to the domain for the representation of the derived result that the interval was 55 − x . however, we did not agree to the geometric shape of figure 3(b) at intervals 05 − x because the concept of the domain that the function had been implemented, was not in accordance that’s showed in figure 1 (b), and it was clear that at that interval, it should not be a function descending from the problem, but rather a zero equal gradient. thus the process of solving problem 1(b) had worked on by ni, continued with problem 1 (c). the first impression after seeing the problem figure 1(c) that had been made by ni was about trigonometric functions. this was the conception that the waves were classified as such x2 cos , and there are coefficients along with variables that were ax /− outside of its trigonometry due to a declining graphical model. then, he did not write down many other hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 154 interpretations, thus describing his procedural knowledge so that the resulting algebraic formula of the problem and its derivative functions is as follows: ( ) 2 2 2cos +−= x xy dan ( ) 2 1 2sin' −−= xy the formula above is the algebraic function formula y'. y' is the derivative of y for x. the two algebraic formulas are representations for each corresponding graph. the function formula y is for figure 1(c) and y’ is for image 3 (c). at this point, the completion model of two students has been the result of this study and as material for comparative analysis. the explanation above, along with all the ni’s descriptions were the process of problem-solving which conceptualized in constructing the function formula. 3.1.3. pr’s solving process there has been a different recalling mathematical concept that was done by pr because the two others used the trigonometry concept. it was polynomial, maximum, and inflection points for solving 1(c)’s problem. before describing it, it would begin with the process of problem solving 1(a). involving conceptual and procedural knowledge, there were several processes of exploration of the functions to obtain appropriate images. for example for problem 1(a) with the exponent function x ay = , exemplify a is equal to 2 and it turned out it was not by the shape of the graph so, pr reflected it on the x-axis to be x y 2−= . reflection continued on the y-axis, but in the end, he wrote x y − −= 2 , and to fit the picture on the given problem, with the concept of translation, he added its function with 1, it became x y − −= 21 . from the findings of the algebraic formula, the pr then generalized it with the number e. following successive representations for the functions and their derivatives that he described were x ey − −= 1 x xexy 2/'= and for 1(a). figure 4 is a graph of the derivative function for each of the graphs in figure 1. (a) (b) (c) figure 4. problem answers by pr figure 4 contains three parts, namely (a), (b), and (c). pr describes figure 4 (a), figure 4 (b), and figure 4 (c) respectively for the derivative functions of problems 1(a), 1(b), and 1(c). the completion process 1(a) has been described, and now is for the process of problem 1(b). for this problem, pr determines the initial form as follows: 1 4 4 4 4 =+ b y a x pr did not provide many detailed interpretations, but we assumed that his works were related to factual and conceptual knowledge about the concept of circles and graphical shapes. where a is the intersection point of the graph on the x-axis, and b is the intersection volume 11, no 1, february 2022, pp. 145-162 155 point with the y-axis, so that by substituting the values of a and b, we got an implicit form of algebraic representation for the graph model. furthermore, using the algebraic representation, pr also procedurally determined the representation for derivatives, and was written as below: 4 3 6251600008 625 ' x x y − −= based on the picture above, y’ is an algebraic representation of the function derivative in the graph model in figure 4 (b). pr described the graph of the derived function as the end of the settlement activity for problem 1(b). proceed with the settlement process for problem 1 (c) as a supplement for comparative materials for other solutions. pr’s knowledge of the problem, made him conceptualize that the function of the graph was a polynomial with the degree of four because there was a turning point and three stationary points, 0=x ax = bx = and. through these concepts, pr created a functional formula as follows: ( ) abxxbaxxbxaxxf +++−=−−−= 23))(()(' the formula above is two algebraic representation models for the same curve. it was written in pr’s work paper. pr constructed these representations, where the similarity of the two models was a symbol of his skills in the context of procedural knowledge. 3.2. discussion 3.2.1. student perspective in solving process the completion process, which was done by pr, had complimented our assumptions about the settlement concept of two of his friends who started. based on the completion process that has been described above, the three students had viewed that the problem is about constructing algebraic function formulas and had shown a series of cognitive processes that were not simple to the bridge of the solutions. therefore, the cognitive processes that operate and the knowledge applied by the three students to solve the problem, focus on the conceptualization of algebraic representations, to determine the derivative of functions and draw graphics. it could be said, they were not an intellectual challenge for them, and it had been shown by all three. this finding displays the same conditions as in the quote tokgoz and gualpa (2015) that students tend to find algebraic formulas from a function graph, as a bridge sketches a graph of its derivative functions. in quotations from several studies, it is stated that constructing a derivative function graph is to identify the gradient value along the curve, where the gradient value is relatively positive or negative, and increase or decrease (borji et al., 2018; hong & thomas, 2015; tobin, 2012), but the case here was identifying a graphical model from the algebraic function formula. as a result, the findings of this study are different from the discussion of tobin (2012), the problem is seen as a geometry problem where the geometrical elements of the curve are communicated with negative or positive gradient concepts at a certain point to produce the differential function graph. thus, generally, there are at least three processes of conceptual representation, in this case, that is constructing algebraic function formulas, determining derivative functions, and sketching graphs. the three processes in question are one example of a procedure but not an algorithmic or symbolic procedure (alghadari & kusuma, 2018), which was developed through problem-solving (rittle-johnson & schneider, 2015). hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 156 thus, for the findings in this study that the problem of students was a matter of representation of algebraic functions. in setyawan et al. (2017), and hong and thomas (2015), an item was said to be a matter of function if the concept can be seen in the representation of images and symbols. while in nagle et al. (2013) said that the mathematical concept related to the gradient of functions was reported that it presents challenges for students. so, it is fixed as the problem for students. however, what made students so conceptualized that it was a matter of function. setyawan et al. (2017) and dossey (2017) stated that students use a representation of a mathematical topic because it is significant to their conceptual knowledge, and the representation chosen at completion is the result of a decision of a phenomenon that has become its experience. then, it has been mentioned in the literature of setyawan et al. (2017) that conditions from the point of view as individual preferences based on information obtained because the perspective of an object is related to the appearance of the shape or size received by the senses. meanwhile according to borji et al. (2018), that it is because students are too late to realize the relationship between derivative functions and their main functions due to lack of respect for the relationship between the two in the graphical representation of the learning process. whereas translating between representations is a powerful communication tool for mathematical thinking (setyawan et al., 2017). in this case, we use the word main function to refer to the function before it is differentiable. then, choi and hong (2014) states that students' perspectives are inclinations that depend on algebraic thinking styles rather than geometric thinking styles. however, there were consequences for problems that were resolved through the process, students transform the graph so that it was represented in algebraic formulas. therefore, the process required the ability to read expressions and make rough estimates of the patterns that emerge in representations (kop et al., 2015). this was one reason why the process takes longer. exploration for a suitable graphical model has been demonstrated by all three students. it was also found in this study that the three students had skipped about the concept of the domain of function on the results of the derivative graph construction. we agree with the statement of borji et al. (2018) based on the findings of this study that students do not pay attention to the relationship between functions and their derivatives in one interval. this finding fits the statement of hong and thomas (2015) that using algebraic thinking performs poorly in exams. students usually show difficulty in using the function property (kop et al., 2015; tokgoz & gualpa, 2015), there was a factor that is less flexible in thinking about the concept of function where it is needed in mathematics and problemsolving (hong & thomas, 2015; kop et al., 2015) because it includes complex properties to think about when reading information provided only in graphic (choi & hong, 2014). it can be stated that perhaps the students' differential learning process places more emphasis on algebraic representations, so they will neglect a little graphical representation (borji et al., 2018), causing students to depend on these performance processes (hong & thomas, 2015), consequently, they are skilled in algebraic algorithm but difficulty in understanding concepts (choi & hong, 2014). 3.2.2. concept-image in conceptualizing with the findings of this study, the nature and criteria of the conceptual problem are indeed open-ended because there are conceptual constraints that students encounter thus inviting students to construct different processes and solutions. conceptual geometricfunction problems in this study are calculus problems related to functions and gradients. there are several functional concepts for deep understanding, such as domains and codes (setyawan et al., 2017), and that requires knowledge of boundaries, derivatives, asymptotes, properties, and intervals (hong & thomas, 2015; tokgoz & gualpa, 2015), as well as volume 11, no 1, february 2022, pp. 145-162 157 increase and decrease intervals, extreme values are elements and functional properties represented graphically (kop et al., 2015). however, the conceptualization process for problem-solving, and has been explained above, as stated (widodo et al., 2019), that it cannot be separated from students' viewpoints on problems, or how students perceive problems to be solved, but still relies on conceptual knowledge as well. nagle et al. (2013) have mentioned that learning calculus does require significant mathematical understanding to form a complete and connected concept image. therefore, knowledge and understanding of mathematical concepts is a capital that also supports the completion process to conceptualize the algebraic function formula. here, the three students explore the algebraic formulas of their functions and geometric shapes, until they are found to fit the model with the graph figure. the three completion processes of each student all illustrate a graph model of different derivative functions. this is because concepts are operated at different stages, or the choice of concepts for the construction of representations is also different. accordingly, dossey (2017) has said that students think differently because of some mathematical concepts applied. the difference is the acquisition of each individual's concept image as a total cognitive structure related to the concept (nagle et al., 2013), and student representation of a concept is influenced by the level of student development in forming mental models (setyawan et al., 2017). solving problems with the conceptualization process, as shown by the three students, which also involves covariational reasoning even though with a rough pattern, namely connecting between two variables, there is a graphical model that cannot be declared wrong for a certain interval. however, it is said in sahin et al. (2015) that although students are able and correct to solve derivative problems by implying procedural understanding, they do not understand the meaning of the concept of derivatives conceptually. furthermore, it is a result of the deficiency of conceptual understanding of important concepts in mathematics from the relationship between the concept and the underlying relationship so that students fill it with procedural and computational understanding as essential with little conceptual understanding. the results of this study are no different from the study report nagle et al. (2013) about concept images and internal conceptual systems of students who are still fragmented. 3.2.3. toward cognitive process and knowledge one of the findings of this study is that students explore more examples of algebraic’s expressions, it was a sign that students' procedural knowledge is more dominant and they believe in implementing it. students' confidence in their knowledge, because they are more familiar with the ideas that underlie the method of solving (hong & thomas, 2015). rightly stated sahin et al. (2015), and wagner and sharp (2017), and with this finding, it is still intended that calculus reputation is implied by students memorizing algorithms and rules. these cases were no different from the findings of sahin et al. (2015) which states that none of the respondents experienced difficulties in connecting mathematical concepts embedded in graphical models and exploration activities in models, even though the understanding was not relational, or was rather instrumental. furthermore, this has to do with the role of memorizing algorithms in learning derivatives so that it ignores important mathematical basic concepts, and graphs of derived functions rarely become visual objects that directly refer to the derivative function. then when referring to student learning textbooks, the interpretation of the geometry of the derivative function is the result of a sketch of the algebraic representation and rarely the reverse. therefore, related to these conditions, sahin et al. (2015) suggested that it might be better to emphasize derivatives as a function and hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 158 construct its graph through associating the slopes of different tangent lines of the curve representing the different slopes. glaser (2002) has mentioned that researchers may not be able to handle the conceptualization boredom with constant comparisons while coding, gathering with theoretical sampling and analyzing, writing theoretical memos and saturating concepts. however, studies of the conceptualization process of solving geometric-function problems have shown that the three responses of students describe their performance in the context that they depend procedurally and involve little covariational reasoning. while sahin et al. (2015) have mentioned that such conditions imply knowledge without understanding the basic meaning of the concept and how it is interrelated in the context of derivatives. furthermore, another important finding in this study is that students are not yet aware of the relationship between the function and it's derivative graphically without having to involve the construction of algebraic formula functions. in other words, they cannot connect between the graph of functions and the negative or positive of the gradient of a function at certain intervals. it is the basic concept of calculus about the relationship between the gradient of the tangent line at the point on the curve and the slopes of the secant lines. although in this study that students can reason about the role of concepts involved in the problem-solving process, but in sahin et al. (2015) mentioned that the relational understanding of derivatives must include all the underlying big ideas. this finding is appropriate as in the literature of borji et al. (2018) about the basic concepts of calculus which are of particular difficulty for students in the graphical representation of derivatives, and are the same as the report in the study of sahin et al. (2015) that students tend to have difficulty in understanding the basic concepts of calculus. from the results of this study to emphasize student learning in the next concept section adapted to the conceptualization that is most prevalent to mediate students in a didactic atmosphere from a strong knowledge base to build more advanced derivative function conceptualizations. this is important because it is part of the investigation suggestions for the concept image developed by students regarding how they understand and relate concepts (sahin et al., 2015), and school academic culture (nagle et al., 2013). 4. conclusion through solving conceptual geometric-function problems, students are required to recall their knowledge to be processed in cognitive space. such knowledge is from the point of view that students have seen it as a problem of algebraic function representation, and they have shown at least three processes of conceptual representation in this study, that are constructing algebraic function formulas, determining function derivatives, and sketching graphs. however, this conceptualization process has consequences, that is, students transform the graph so that it is represented in algebraic formulas, and that has resulted in the concept of the domain of function being overlooked. in the process, students explore the algebraic formulas of their functions and geometric shapes until they are found to fit the model with the graph figure. here, there is no similar construction process because mathematical concepts are operated at different stages, and the difference is the acquisition of each individual's concept image as the total cognitive structure associated with the concept which is influenced by the level of student development in forming mental models. in this case, some students are able and correct to solve the problem, but the work implies procedural understanding which does not understand the meaning of the concept of derivatives conceptually. this is a result of the lack of understanding of the concept of the relationship between the concept and the underlying relationship so that the internal conceptual system is still fragmented. this is a sign that students' procedural knowledge is volume 11, no 1, february 2022, pp. 145-162 159 more dominant and they believe in implementing it, not yet aware of the relationship between the function and it's derivative graphically, and this relates to the role of memorizing algorithms so that it ignores important mathematical basic concepts. this is a special part of the conceptualization process in problem-solving activities, namely precisely regarding the involvement of the relationship between functions and their geometric shapes. the process serves to emphasize student learning to construct conceptualizations of more advanced derivative functions that are adapted to the conceptualization that is most prevalent due to factors of school academic culture. therefore, further studies must be carried out for broader purposes. references alghadari, f., & kusuma, a. p. 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(2019). formal student thinking in mathematical problem-solving. journal of physics: conference series, 1188(1), 012087. https://doi.org/10.1088/1742-6596/1188/1/012087 https://doi.org/10.1088/1742-6596/943/1/012004 https://doi.org/10.18260/p.24733 https://doi.org/10.30935/scimath/9502 https://doi.org/10.5951/mathteacher.110.8.0618 https://doi.org/10.1088/1742-6596/1188/1/012087 hutajulu, perbowo, alghadari, minarti, & hidayat, the process of conceptualization … 162 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p193-210 193 effectiveness of flipped classroom model through multimedia technology in improving students’ performance in directed numbers haji muhamad hafizuddin haji mohamad ali, daniel asamoah*, masitah shahrill sultan hassanal bolkiah institute of education, universiti brunei darussalam, brunei darussalam article info abstract article history: received mar 20, 2022 revised apr 27, 2022 accepted may 04, 2022 the recent globalisation and the emergency of covid-19 require a teaching and learning environment that encourages the use of technology. through a mixed-method design and an action research approach, this study investigated the effectiveness of a flipped classroom model through multimedia technology in improving students’ performance in directed numbers, given the difficulty and misconceptions of students in this mathematical concept. a total of 30 year 9 students conveniently sampled from one of the secondary schools in brunei darussalam served as participants. the action taken involved a pretest, intervention, posttest, and interviews. the results of the paired sample t-test revealed that students’ performance in directed numbers significantly improved after the flipped classroom intervention. students had positive perceptions of the flipped classroom model as it encouraged their readiness, participation, and motivation. challenges such as time constraints and distractions when studying from home were reported. these results imply that amid covid-19, a flipped classroom through multimedia technology can be an effective and alternative way of teaching and learning directed numbers. it has the potential of encouraging student-centred learning and creativity, which are vital in teaching and learning mathematics. keywords: academic performance, brunei darussalam, directed numbers, flipped classroom model, multimedia technology this is an open access article under the cc by-sa license. corresponding author: daniel asamoah, sultan hassanal bolkiah institute of education, universiti brunei darussalam universiti brunei darussalam, bandar seri begawan, brunei darussalam. email: 20h9000@ubd.edu.bn how to cite: ali, h. m. h. h. m., asamoah, d., & shahrill, m. (2022). effectiveness of flipped classroom model through multimedia technology in improving student performance in directed numbers. infinity, 11(2), 193-210. 1. introduction education in the 21st century has witnessed several changes. approaches that encourage more student-centred compared to teacher-centred instruction continue to be developed and integrated into teaching and learning. globalisation, the development of technology, and the emergency of covid-19 have shown that traditional approaches to teaching and learning may not meet current educational needs and goals (jamil et al., 2022; https://doi.org/10.22460/infinity.v11i2.p193-210 https://creativecommons.org/licenses/by-sa/4.0/ ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 194 shahrill, noorashid, et al., 2021; shahrill, petra, et al., 2021). this has called for a need to constantly create instructional environment that encourages the use of technology (bishop & verleger, 2013). in responding to the changing needs of education, one of the approaches that have attracted recent research attention is the flipped classroom model. baker (2000) introduced this approach to instruction as a component of blended and inquiry-based instruction. it is an approach in which homework and classroom activities are interchanged ash (2012), such that students study instructional materials at home with their peers and teachers, followed by the teacher giving feedback and engaging in directed conversations during class time (lin & chen, 2016; say & yildirim, 2020). it exposes the students to study and understand instructional materials out-of-classroom, and in the classroom, do activities that support learning with the help of the teacher (moravec et al., 2010). students mainly study and master instructional materials through multimedia technology by watching videos, listening to podcasts, and reading e-books (balu, 2020; dominic-ugwu & nonyelum, 2019). using a flipped classroom model is associated with high students’ performance, motivation and retention (busebaia & john, 2020; say & yildirim, 2020; sirakaya & ozdemir, 2018), and this is irrespective of subject areas and educational level (strelan et al., 2020). it promotes student-centred learning, leading to student engagement, active participation, and self-directed learning (pierce & fox, 2012; qader & yalcin arslan, 2019). students can learn instructional materials at their own pace, ensures effective use of classroom time because most of the instructional tasks are done by the students at home, and the classroom time is used to clarify the misconceptions of students (fulton, 2012; hew & lo, 2018; matzin et al., 2013). given that a flipped classroom model encourages collaboration among their peers, high-ability students can assist low-ability students in constructing their knowledge on a given topic (ferreri & o’connor, 2013; nielsen et al., 2018). using a flipped classroom model encourages students to think creatively within and outside the classroom (herreid & schiller, 2013). however, the approach to teaching and learning has some disadvantages. teachers are unable to know if students learned instructional concepts at home. the lack of internet and other devices such as mobile phones, tablets and computers that make learning possible are other notable challenges (jenkins, 2012). other demerits of the flipped model include the high cost that comes with technology and the failure to create an environment where students can ask questions when learning at home (jenkins, 2012). despite these demerits, a flipped classroom model inverts the traditional model of instruction. it has the possibility of increasing student readiness as the learning content is introduced to students before the physical class. this allows teachers more instructional time to guide students through practical, active, and innovative ways of learning. previous studies have emphasised the need to use multimedia technologies in flipped classrooms and provide the technical support needed for such multimedia (oliveira, 2018). integrating multimedia in flipped learning has gained considerable relevance in teaching and learning. therefore, previous research attempts have focused on technological and conceptual improvement in using a flipped classroom model, especially on monitoring student learning while they are at home (jovanović et al., 2017). a flipped classroom model does not necessarily need to be implemented in an online platform, as it can also be done using multimedia technologies where text, visuals, animation, video, and sound can be merged to improve teaching and learning (abdurasulovich et al., 2020). since multimedia triggers multiple senses of audiences at a time, using various media in a flipped classroom model can produce conclusive results in teaching and learning (rajendra & sudana, 2018; yohannes et al., 2016). using multimedia in a flipped classroom serves as a varied source volume 11, no 2, september 2022, pp. 193-210 195 of information for students, making learning content accessible to students based on their preferences (aprianto et al., 2020; cevikbas & kaiser, 2020). how multimedia is used in a flipped classroom model to meet student needs and how students handle multimedia resources are also essential to determine the success of the flipped classroom methodology. oliveira (2018), for example, indicated that flipping the classroom is not a sufficient condition for improvement in student learning and engagement. however, how instructional resources in different formats reach the students and how well students can understand such resources are relevant. although student perceptions about flipped learning model are generally positive, students who have positive perceptions are those who can access instructional materials on time and without difficulty. at the same time, they prefer traditional teaching methods that are more interactive and learner-centred compared to a flipped classroom model (oliveira, 2018; yohannes et al., 2016). to this end, this study investigates the effectiveness of using multimedia technology in a flipped classroom model in improving students’ performance in directed numbers. directed numbers or integers have both direction and size, with one direction being positive and the other negative. for example, in 4-(-3), both 4 and (-3) are directed numbers with the 4 and (-3) being positive and negative, respectively. having a sufficient understanding of these numbers are applicable in everyday life (fuadiah & suryadi, 2017). they are helpful in reading temperature. a temperature of (-10) degrees means that it is 10 degrees away from and less than 0, suggesting a high level of coldness compared to a temperature reading of 10 degrees. understanding directed numbers is also helpful in profit and loss. it helps to know how less or more can be done to achieve satisfactory results in all aspects of life. profit is denoted by positive numbers in business transactions, while negative numbers indicate losses. understanding directed numbers helps to check account balance (makonye & fakude, 2016). if money is added to one’s account and there are adequate funds in the account, it is denoted by a positive number. in contrast, money withdrawn from one’s account is represented by a negative number. in the health sector, directed number is also applicable. normal blood pressure will record a positive number, while blood pressure below normalcy can be negative. this suggests that directed numbers are significant in every life, and the failure of students to have a good understanding of such numbers may cause serious challenges when applying such numbers in their daily activities. students have inadequate understanding and several misconceptions, despite the relevance of directed numbers (bofferding, 2014; lamb et al., 2012; vlassis, 2008). mostly, they are unable to differentiate between a negative sign used as an operation and one used as a symbol (lamb et al., 2012). this creates difficulties for them when performing operations involving directed numbers. student difficulty in directed numbers and other mathematical concepts exists in brunei darussalam (hereafter referred to as brunei). previous research has consistently reported that students have difficulty understanding algebraic expressions, fractions, simultaneous, linear, and quadratic equations (chong et al., 2022; hamid et al., 2013; japar et al., 2021; johari & shahrill, 2020; rosli et al., 2020; sarwadi & shahrill, 2014; shahrill, 2018). in most studies, the predominant cause of low performance was attributed to factors such as over-reliance on past examination questions, anxiety and stress, and teaching approaches (latif, 2021; salam & shahrill, 2014; shahrill, 2018; zakaria et al., 2013) and inadequate understanding and misconceptions about directed numbers (levison, 2016). despite these gaps, interventions to improve students’ understanding and performance in directed numbers are scarce. therefore, this study that provides a flipped classroom intervention to evaluate its effectiveness in improving students’ performance in directed numbers is timely. for the studies that focused on secondary schools in brunei, emphasis was placed on geography ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 196 (nawi et al., 2015) and history (latif et al., 2017). the paucity of studies that focused on secondary school mathematics did not consider directed numbers (manjanai & shahrill, 2016; toh et al., 2017). therefore, more studies are necessary to develop a more in-depth analysis of the effects of the flipped classroom model in the teaching and learning of secondary school mathematics. this present study provides a flipped classroom intervention to assess its efficacy in improving student performance in directed numbers using secondary school students in brunei. the following research questions were answered: (1) what is the effectiveness of a flipped classroom model through multimedia technology in improving students’ performance in directed numbers? (2) how do students perceive a flipped classroom model through multimedia technology as an instructional option? 2. method this study adopted a mixed-method design with an action research approach where both quantitative and qualitative data were collected sequentially (creswell & creswell, 2017). the quantitative data were collected through achievement tests before the qualitative data, which was collected through interviews. since this study sought to provide a flipped classroom intervention to address student difficulty in directed numbers (avison et al., 1999; mertler, 2013), the action included a pretest, an intervention, and a posttest. through a conveince sampling, 30 year 9 students were selected. they were from one of the government secondary schools in the belait district of brunei. the selected school had two year 9 classes. class a consisited of 14 students (five males and nine famales), while class b consisted of 16 students (seven males and nine feamles). the students in both classes served as participants for this study. they were used because directed numbers are taught at that grade level, and for this reason, providing intervention on the concept was feasible. both classes consisted of students with mixed abilities. students in both classes were combined to provide the intervention at the same time. there were two instruments used to collect data: achievement tests and a structured interview guide. the achievement tests involved a pretest and a posttest on directed numbers. the pretest was used to assess the entry behaviour of students before the intervention was provided, while the posttest was used to evaluate students’ performance after the intervention. both tests consisted of similar questions with the same level of difficulty. they involved four categories of questions that required students to find the magnitude (2questions), order (2-questions) add and subtract (2 questions), and solve temperature related questions (2-questions) involving directed numbers. the duration for each achievement test was 30 minutes. the total score for each of the achievnment test was eight marks, with one mark for each question. the structured interview guide asked participants their perceptions of the intervention provided. the main question that was asked in the interview is “how do you perceive the teaching and learning of directed numbers in the flipped classroom through multimedia technology? a total of six participants availed themselves for the interviews. six participants were suitable for the interview because the interview data was saturated afer the sixth participant. that is, we noicted that from the seventh participant who was interviewed, no new information was discovered on our study variables (saunders et al., 2016). to improve the content validity of the tests, specific lesson objectives were set for each of the areas in the test. a table of test specification was developed to ensure that all eight questions evenly covered all the areas of the lessons. the questions for the interview were given to four mathematics teachers who had more than 12 years of teaching experience. these experts volume 11, no 2, september 2022, pp. 193-210 197 evaluated the measurement quality of the questions. all suggestions from these experts were inculcated before administration. ethical clearance was obtained from the faculty’s ethics committee of the university. permissions were also obtained from school leaders through formal approval letters. students and parents also completed informed consent forms to indicate their willingness to serve as participants and allow their children to participate in this study, respectively. the information and identities of the students have been kept confidential and anonymous. the pretest was conducted before the main intervention. the test was organised online through a whatsapp group chat, where the pretest was posted. the students were given 30 minutes to answer all eight questions. they posted all their answers in a word file on the same platform, which we downloaded and scored. second, the lesson intervention, which involved a flipped classroom teaching and learning of directed numbers, was implemented. participants received a pre-class video lesson sent to them through the whatsapp group chat to learn at home. two youtube videos on directed numbers created by math antics (2014a, 2014b), were edited and used in the intervention. the duration of the videos was 13 minutes. the video lessons covered definitions, positive, negative, addition, subtraction, and representing directed numbers on a number line. all instructions regarding the video lesson were also provided in the group chat. students were asked to form groups of three to four members to prepare group presentations based on specific areas. they were given a week to learn the video lesson and prepare for the in-class presentations. after that, we had a collaborative and activity-based face-to-face classroom session. each group of students had a 15-minutes presentation to share what they had learned based on specific areas on directed numbers and were given a manila card to write their presentation answers. after each presentation, there was a question-and-answer section. the first author addressed and clarified all student misconceptions and learning gaps based on their presentations and the lesson task they learned at home. the first author facilitated all presentations. figure 1 illustrates the scenes of the activity-based in-class session of the flipped classroom intervention. figure 1. scenes of the activity-based in-class session of the flipped classroom intervention the students were actively involved in the activity-based instruction during the inclass intervention (see figure 1). they satisfactorily prepared their presentations using manila cards and presented their answers to the assigned tasks in front of the whole class. they were able to respond to each of their peers’ questions based on their presentations. the facilitator gave minimal guidance and comments since the students had completed the tasks individually and in groups at home before the in-class presentations and interaction. after the intervention, the posttest was conducted on the same day, and all 30 students were combined to take the test. there was a tranquil testing environment, and instructions regarding the posttest were given. the posttest lasted for 30 minutes, after which all scripts were collected, marked, and recorded. the pretest and posttest scores were entered into ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 198 spss, which aided the analysis. data cleaning was performed to address all missing values and outliers. a paired sample t-test was performed to compare the pretest and posttest scores to check and test for a mean difference. since the paired sample t-test is a dependent test, the dependent variable was the pretest and posttest scores. this statistical tool was used because pretest and posttest scores (interval data) from the same group of students were analysed before and after the intervention to examine performance (coman et al., 2013). additionally, the data for pretest (p=0.677>0.05) and posttest (p=0.561>0.05) were approximately normally distributed (fisher & marshall, 2009). therefore, the paired sample t-test was suitable to determine the effectiveness of the flipped classroom intervention. the transcribed interview data were analysed thematically. in the thematic analysis, we farmiliarised ourselves with the data before generating initial codes. potential themes were searched, reviewed and defined before producing the interview report (braun & clarke, 2012). 3. result and discussion 3.1. effectiveness of flipped classroom model in improving students’ performance in directed numbers the overall results showing the efficacy of the flipped classroom intervention in improving students’ performance in directed numbers are presented in table 1. table 1. paired sampled t-test between pretest and posttest scores descriptive statistics 95% confidence interval of mean difference mean sd mean difference sd std. error lower upper t df sig. (2-tailed) cohen d pretest 5.67 1.90 0.90 2.01 0.366 0.051 1.550 2.18 29 0.037 0.4 posttest 6.57 1.31 n = 30, sd = standard deviation; mean difference is significant if sig<0.05 the results of the paired sample t-test show that students’ performance in the posttest (mean = 6.57, sd = 1.31) is significantly higher than in the pretest (mean = 5.67, sd = 1.90) with t(29) = 2.18, p = 0.037<0.05 (see table 1). this suggests that performance in directed numbers significantly improved after the flipped classroom intervention. from the cohen’s d value of 0.4, it is observed that approximately 40% of students’ performance in directed numbers is accounted for by the intervention provided, which signifies a medium effect. since we are particularly interested in how students improved in answering each of the questions correctly, figure 2 illustrates the number of participants who answered correctly in both tests. figure 2. number of correct responses of participants volume 11, no 2, september 2022, pp. 193-210 199 of the eight questions, the participants improved their correct responses for six questions, from questions 3 to 8 (see figure 2). the number of participants who improved their pretest scores at the posttest level ranges from 1 to 12, with the most significant change of 12 participants for question 5. participants did not improve their pretest and posttest scores (27 marks for each) for question 2. for question 1, the pretest score (27 marks) is slightly higher than the posttest score (26 marks). we present specific errors and misconceptions based on the eight questions starting from figure 3. figure 3. common errors in questions 1 and 2 in figure 3, it is observed that participants scored both questions in the pretest but missed one of the questions in the posttest. they make a common error in the posttest by ignoring the negative sign of the negative integers. figure 4. common errors in questions 3 and 4 in question 3, the participants are confused and read positive integers as negative integers. they ignore the negative sign when arranging the numbers in ascending order and fail to arrange positive numbers first, followed by negative numbers (see figure 4). in question 4, participants arrange the numbers in ascending rather than descending order. they cannot differentiate the magnitude of positive and negative numbers but mix the numbers up (see figure 4). according to our observation, they also made the error of misplacing ‘0’ either at the beginning or at the end of the number line. this means that some of them were unable to know the place of zero. figure 5. common errors in questions 5 and 6 ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 200 the participants ignore the negative sign when performing the operation towards the given integers (see figure 5). they add two negative integers by disregarding the negative sign after adding. they also fail to differentiate the negative sign as an operation and a symbol. they perform correct calculations of two negative integers but forget to add the negative sign to the final answer (see figure 5). they still make calculation errors when adding and subtracting directed numbers. figure 6. common errors in questions 7 and 8 the participants think that monday at 0°c is the coldest night. they have a misconception of assuming that zero is the smallest integer regardless of the negative and positive integers. they similarly answered wrongly by neglecting the negative signs and only focusing on the values. this leads them to choose may with 6°c as the smallest value without considering the negative sign for the negative integers (see figure 6). this indicates that the participants still had misconceptions despite the significant improvement in performance after the intervention. 3.2. perceptions of students about the flipped classroom model intervention three themes emerged from the interview responses: the usefulness of the flipped classroom, the preferences of the flipped classroom to the face-to-face classroom, and the challenges associated with learning from a flipped classroom model. generally, the participants had positive perceptions about the flipped classroom model. despite their positive perceptions, they reported some challenges. 3.2.1. the usefulness of the flipped classroom given that traditional teaching and learning may not ensure the best use of instructional time, flipped classroom frees instructional time and provide the room for actvitites that can improve higher order student thinking. teachers may not waste time to provide instructional information to students when students can access that information from the internet or in textbooks (matzin et al., 2013; mazur, 2009). it affords students the opportunities to learn at their peace and take responsibility for their learning (fulton, 2012). flipped approaches to teaching and learning help students to learn compared to encountering instructional mateirals in the classrooom. generally, the students found the flipped classroom through multimedia technology useful. they saw the flipped classroom as helpful in improving their performance in directed numbers as it encouraged collaboration through group work and in-class presentations. this collaboration allowed them to share their knowledge of what they had learned individually and in groups. s19: i think learning through the video and having a follow-up physical class is very helpful. i understood what directed numbers are, and how to add and subtract such numbers as well. volume 11, no 2, september 2022, pp. 193-210 201 s5: engaging in a group work that allowed my peers to share their findings brought up different opinions and thinking among us. in case of any mistakes, my peers were able to correct them, which teaches us the correct way of learning. s23: … i was able to present in front of my peers, which made me very happy. there was teamwork before arriving at answers. there was knowledge sharing as well. the students voiced that watching the pre-class video lesson improved their readiness for the face-to-face class. since flipped classroom comes with flexibility, students are able to control how they learn instructional concepts at home before physical classes. it is worth noting that implementing a flipped classroom tranfers most of the classroom work to students. going through series of a carefully planned tasks provided by the teacher helps the students to be ready for physical classes. s23: i watched the video at home. in the class, we had presentations, questions, and answers. i was prepared for the physical class. in temperature, i have now understood that a negative number means too much coldness. a positive number is bigger than a negative number, and zero separates all of them… 3.2.2. preferences of the flipped classroom to the face-to-face classroom the participants found the flipped lesson as helpful and convenient to learn directed numbers. learning through pre-class video at home improved their understanding. at the same time, the face-to-face classroom helped clarified their misconceptions and received further explanations from their teacher. generally, they preferred to learn through the preclass video lessons and face-to-face classes. s5: it is easy to understand directed numbers when you watch the video at home and have the group work in a physical classroom. s19: i think i will prefer both. but it is always good to learn through the video at home before the actual class. flipped classroom is effective as face-to-face learning (stratton et al., 2020). this suggests that effective use of both instructional approaches in mathematics classrooms have the potential to improve student learning and performance. the latter promotes group work and helps clarify student doubts compared to learning through a pre-class video lesson alone. students exhibit high level of confidence to perform instructional tasks and positive percpetions when they are exposed to different instructional methods including hybrid, traditional and flipped classrooms (kaleem et al., 2016). given this background, it is not surprising that our participants preferred both the flipped classroom and face-to-face instruction. they are able to learn through pre-class video lessons by themselves at home and clarify their doubts during normal classes. s1: can we choose both? because both are helpful for students. the flipped classroom helps us understand the video and learn by ourselves. the traditional classroom helps us to get further explanations from the teacher. s26: both…by watching the video, i hear everything and can rewind it severally. in the normal lessons, i write any important information which i make reference to and ask the teacher the questions i want. ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 202 3.2.3. challenges associated with learning from a flipped classroom model although the participants found the flipped classroom model helpful in improving their performance in directed numbers and expressed positive perceptions of its use, they reported some challenges. they were faced with limited time for group discussions which affected their preparation for the in-class presentations. negative attitudes of group members also affected their meeting time. the participants voiced that the flipped classroom where pre-class videos are watched at home limits how they are can ask questions. s20: i did not have much time to discuss with my group, which affected my understanding and contributions. s1: there were also different opinions which sometimes disturbed the class. s29: …i am not able to ask the teacher questions when i learn from the pre-class video at home. i can only ask questions during the face-to-class. this made me to forget most of questions i wanted to ask. 3.3. discussion the results of this study indicate that the implementation of a flipped classroom through multimedia technology has the potential to improve students’ performance in directed numbers. this is expected because students are exposed to instructional tasks at home, preparing them for face-to-face instruction. the teacher facilitates classroom interaction through group work and activity-based learning. this encourages and motivate students to collaborate and share knowledge. the teacher also clarifies any misconceptions and misunderstandings that students may have. these are the characteristics of a flipped classroom model reported in the literature (ash, 2012; lin & chen, 2016; say & yildirim, 2020), which encourages student-centred learning (jamaludin & osman, 2014). therefore, students are expected to improve their performance. the results of this study indicating that a flipped classroom model has the potential of improving students’ performance in directed numbers confirm existing studies that support the efficacy of the flipped model in teaching and learning (busebaia & john, 2020; sirakaya & ozdemir, 2018; strelan et al., 2020). the use of multimedia and, in particular, video lessons enhance students’ performance. video lessons have the potential to improve students’ engagement and, at the same time, provide explanations and help them remember learning content compared to audio or other multimedia (hew & lo, 2018). students have the flexibility to rewind, pause, and skip the video lesson based on their preference, which may positively affect their understanding and performance (fulton, 2012; matzin et al., 2013). given this analysis, this might have helped students enhance their understanding when provided with the pre-class video lesson on directed numbers. this confirms the relevance of multimedia technology such as video lessons in improving students’ performance in a flipped classroom model (botha-ravyse & reitsma, 2015; oliveira, 2018), especially in directed numbers. therefore, it was not surprising that the students reported that they could access instructional content conveniently when given the pre-class video lesson, which confirmed the literature (aprianto et al., 2020). it is also interesting to reiterate that the students used in this present study generally perceived a flipped classroom environment positively. this suggests that the students benefitted from the filipped model. considering the zeal, participation, motivation, and understanding gained by the students, their positive views were expected (bofferding, 2014; oliveira, 2018). however, the students still had some misconceptions and volume 11, no 2, september 2022, pp. 193-210 203 misunderstandings and made some errors in directed numbers. this reminds us of student difficulty on directed numbers reported in the literature (bofferding, 2014; lamb et al., 2012; vlassis, 2008) and the errors students make, especially on negative integers acknowledged by previous studies (fuadiah & suryadi, 2017; lamb et al., 2012; levison, 2016). we agree with oliveira (2018) that a flipped class is not a sufficient condition for improvement in student learning. in this study, most students reported that they had limited time to present and discuss instructional tasks. they were not able to ask questions while studying at home, and were distracted when studying at home. these can contribute to create common errors in delivering their answers. therefore, it was not surprising that despite the efficacy of the flipped classroom intervention, students still preferred both flipped and traditional face-toface learning in directed numbers. given the improvement that is witnessed in the performance of students after the intervention provided, there are certain pedogicial approaches we undertook that might have contributed to students’ performance. we believe that these pedagogical approaches are worth practicing by other mathematics educators. we used whatsapp as a learning platform during the pretest. students were pleased and were able to respond to the learning tasks through whatsapp before the intervention. this suggests that during covid-19, social media plateforms can be carefully untlised to promote student learning in mathematics. our intervention was based on edited videos that illustrated series of concrete mathematical actions based on the concepts taught. the students experienced pictorial representations of what they were taught compared to teaching them in abstract terms. teachers should be able to develop and integrate the technology that enhance student learning to improve their performance in directed numbers. generally, the intervention involved an activity-based lesson through group work and collaboration, and allowed students to present their findings before addressing any questions they had. we planned the intervention in a such way that there were a lot of activities. this makes us to believe that most of the benefits of flipped classroom model depend on what happens in the classroom compared the instructional tasks students go through at home. teachers should plan carefully and ensure student-centred teaching and learning to improve performance in directed numbers. these attempts that characterised our flipped classroom through multimedia technology have the possibility to improve the performance of students as we have reported. 4. conclusion this study examined the effectiveness of a flipped classroom model through multimedia technology in improving students’ performance in directed numbers. the results showed that a flipped classroom model could improve students’ performance in directed numbers. generally, the students had positive perceptions of the flipped classroom model. the model improved student readiness, preparation, participation, and motivation in the teaching and learning process. using a flipped classroom also comes with challenges. students may lack group work and discussion time, and they may be disctracted by home activities. therefore, they prefered flipped classroom lessons and traditional face-to-face teaching and learning of directed numbers. our results imply that during the covid-19 pandemic, a flipped classroom through multimedia technology, such as video lessons, can be an effective and alternative way of teaching and learning directed numbers in secondary schools. it can encourage student-centred learning and improve students’ creative ability, which is vital in teaching and learning mathematics. this study draws the attention of mathematics teachers and schools to consider a flipped classroom as an alternative way to teach and learn directed numbers and other mathematics concepts. key stakeholders such as schools and parents may also consider ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 204 providing the necessary devices, materials, and environment to make a flipped classroom feasible. this study reminds mathematics educators that students have a great deal of misconception about directed numbers despite the intervention provided. they were confused about where to place “0” among positive and negative intergers, failed to differentiate between a negative sign as a symbol and as an operation, inability to determine the maginitute of positive and negative numbers and ignoring negative signs after calculations. these suggest areas where flipped classroom through multimedia technology can be used to improve the teaching and learning of directed numbers. this study covered a relatively small sample size to assess the effectiveness of the intervention provided. the effectiveness of our intervention was also confirmed in the teaching and learning of directed numbers. therefore, generalising our results to other mathematical concepts, subjects, and in different educational contexts should be done with care. given these limitations, we suggest that future studies consider replicating our study in other school and educational contexts, using a relatively large sample, to arrive at more robust conclusions on our study variables. future studies may consider extending the efficacy of a flipped classroom model to other mathematical concepts and subject areas across educational contexts. comparing other multimedia technologies in a flipped classroom to assess their effectiveness in improving students’ performance may also be an important focus of research. acknowledgements we are grateful to the students, teachers, and school leaders at the research site for their willingness to participate and contribute their time and resources to this study. references abdurasulovich, k. j., abdurasulovich, k., yangiboevich, k., anvarovich, a., & xolmurodovich, g. 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(2013). effect of cooperative learning on secondary school students’ mathematics achievement. creative education, 4(2), 98100. https://doi.org/10.4236/ce.2013.42014 https://doi.org/10.30935/cet.646888 https://doi.org/10.1016/j.edurev.2020.100314 https://doi.org/10.17501/icedu.2017.3120 https://doi.org/10.1080/09515080802285552 https://doi.org/10.14445/22315373/ijmtt-v39p510 https://doi.org/10.4236/ce.2013.42014 ali, asamoah, & shahrill, effectiveness of flipped classroom model through multimedia … 210 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p15-24 15 association among mathematical critical thinking skill, communication, and curiosity attitude as the impact of problem-based learning and cognitive conflict strategy (pblccs) in number theory course zetriuslita 1 , wahyudin 2 , jarnawi afgani dahlan 3 1 universitas islam riau, jl. kaharuddin nasution 113 pekanbaru riau, indonesia 2,3 universitas pendidikan indonesia, jl. setiabudi no.229, bandung, west java, indonesia 1 zetriuslita@edu.uir.ac.id, 2 wahyudin_mat@yahoo.com, 3 jarnawi@upi.edu.com received: november 17, 2017 ; accepted: january 13, 2018 abstract this study aims to find out the association among mathematical critical thinking ability, mathematical communication, and mathematical curiosity attitude as the impact of applying problem-based learning cognitive conflict strategy in number theory course. the research method is a correlative study. the instruments used include a test for mathematical critical thinking skill and communication, and questionnaire to obtain the scores of mathematical curiosity attitude. the findings showed that: 1) there was no association between critical thinking skill and mathematical curiosity attitude as the impact of applying problem-based learning cognitive conflict strategy, 2) there was no association between mathematical communication and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy, and 3) the impact of applying problem-based learning cognitive conflict strategy was more influential in developing critical thinking skill than communication skill and curiosity. keywords: cognitive conflict strategy, communication, critical thinking, curiosity attitude, problem-based learning. abstrak penelitian ini bertujuan untuk mengetahui asosiasi kemampuan berpikir kritis, komunikasi dan sikap rasa ingin tahu (curiosity) matematis sebagai efek penerapan pembelajaran berbasis masalah berstrategi konflik kognitif dalam mata kuliah teori bilangan. metode penelitian yang digunakan adalah penelitian korelasional. instrumennya berupa tes untuk menjaring kemampuan berpikir kritis dan komunikasi matematis serta angket untuk memperoleh skor sikap curiosity matematis. hasil analisis menunjukkan: (1) tidak terdapat asosiasi antara kemampuan berpikir kritis dan sikap curiosity matematis sebagai dampak penerapan pembelajaran berbasis masalah berstrategi konflik kognitif; (2) tidak terdapat asosiasi antara kemampuan komunikasi dan sikap curiosity matematis sebagai dampak penerapan pembelajaran berbasis masalah berstrategi konflik kognitif; (3) penerapan pembelajaran berbasis masalah berstrategi konflik kognitif didominasi dalam pengembangan kemampuan berpikir kritis sehingga kurang berdampak terhadap kemampuan komunikasi dan sikap curiosity matematis. kata kunci: berpikir kritis, curiosity, komunikasi, pembelajaran berbasis masalah, strategi konflik kognitif. how to cite: zetriuslita, wahyudin, & dahlan, j. a. (2018). association among mathematical critical thinking skill, communication, and curiosity attitude as the impact of problem-based learning and cognitive conflict strategy (pblccs) in number theory course. infinity, 7 (1), 15-24 doi:10.22460/infinity.v7i1.p15-24 mailto:zetriuslita@edu.uir.ac.id mailto:wahyudin_mat@yahoo.com mailto:jarnawi@upi.edu.com zetriuslita, wahyudin, & dahlan, association among mathematical critical … 16 introduction in education, especially mathematics education, critical thinking is a higher-order thinking skill. in bloom’s taxonomy, it consists of three levels: analysis, synthesis, and evaluation (ennis, 1993; duron, limbach, & waugh, 2006). johnson (2007) also stated that critical thinking is a well-directed and clear process used in mental activities such as problemsolving, decision making, persuasive activities,assumption analysis, and the ability to conduct scientific research. critical thinking skill can be perceived differently by many people and highly dependent on their thinking (thompson, 2011; hidayat, 2012). for instance, students must be able to interpret the given a problem into the mathematical sentence, solve it, evaluate the problemsolving and test or re-examine the accuracy of the answer to the given problem. chukwuyenum (2013) stated that critical thinking is a complex concept involving cognitive skills and affective dispositions that influence the way teachers present the concepts to students. falcione & falcione (1994) reinforces this idea in aktaş, & ünlü (2013) that the disposition of critical thinking can help to predict the critical thinking skill. critical thinking it also supports the intellectual curiosity. the thinking process generates one’s curiosity towards the problem. moreover, in communication, in nctm (national council of teachers of mathematics) (2000) stated that communication is an essential part of mathematics and mathematics education. students must learn mathematical communication, in addition to reasoning, and problem solving in primary and secondary education. kadir (2013) suggested the importance of communication in mathematics learning. a person is considered to understand something if he can explain it well, either orally or in writing. furthermore, the importance of mathematical communication is to the goal of education for students includes in the nctm standards document (nctm, 2000). in nctm, one of the goal of education is that students can communicate well mathematically. in the learning process, especially in mathematics learning, communication has a crucial role. if students cannot communicate, they cannot explain the meaning in mathematics if they do not understand mathematical language. according to rahmi, nadia, hasibah, & hidayat (2017), there are three kinds of mathematical communication: communication about mathematics, communication in mathematics, and communication with mathematics. communication about mathematics refers to how students are able to understand the concepts of mathematics, communication in mathematics means writing symbols that can be understood from mathematics, and communication with mathematics is a verbal communication, explain what is perceived about the concept of mathematics it self. in regards to curiosity attitude, curiosity is one of the scientific attitudes. binson (2009) defines curiosity as the disposition to inquire, investigate, and seek after acquiring knowledge. the tendency to inquire, investigate, and seek is a thinking framework of curiosity attitude about something more deeply. the high desire for learning something or looking for answers to specific questions is the catalyst for developing someone’s scientific abilities. reio jr, petrosko, wiswell, & thongsukmag (2006) said that curiosity is the desire for acquiring new knowledge and new sensory experience that motivates exploratory behavior. santoso (2011) argues that curiosity or a desire to know something is the basic nature of humans who keep asking whatever they see and find before asking why or how something happens. these volume 7, no. 1, february 2018 pp 15-24 17 questions then continue and develop into more advanced ones by asking why a problem occurs, how something happens and how to find a solution to this problem. such critical questions are typical to human beings and identified from the very beginning they can talk and express their feelings to other humans. in learning, curiosity is defined as a strong motivational force that can produce someone’s behavior for understanding or facing certain materials or problems. according to santoso (2011), curiosity doesn’t occur suddenly but needs to be cultivated and trained to develop. one of the ways for fostering curiosity is to provide challenging and complex problems that can create a variety of questions for students. their curiosity can increase by making them interested in learning through demonstration at the beginning of the lessons, asking questions that are challenging for their thinking ability, or giving complicated problems hopefully that they are curious to find the answers. suhadak (2014) stated that curiosity is a phenomenon characterized by an effort to seek and find something which leads to enthusiasm for learning, investigating, and observing. considering its significant role in learning, the development of curiosity should be the main focus of learning. the more curious learners feel about something, the closer they get to their learning environment, including their work groups (binson, 2009). mathematical curiosity is a strong motivation that learners haveto understand the materials or mathematical problems. curiosity about mathematical problems leads students to get the answer to the question. one way to get the answer is asking questions or inquiry. inquiry helps students to construct their understanding independently. understanding of concepts is obtained through constructing a better understanding compared with the acquired one. as a result, it is necessary to develop students’ curiosity because it motivates them to acquire new knowledge. mathematical curiosity means curiosity about a mathematical problem or conflict in mathematics learning. the relationship between critical thinking and mathematical curiosity can be seen from the characteristics. in critical thinking, questions typically occur like “what if?”, “what is wrong?”, and “what will we do?”regarding curiosity, the questions will be: “why can it be like this?”, “how to solve it?” one can be assumed that curiosity contributes to someone’s critical thinking about something. according to santoso (2011), the application of problembased learning can develop creative thinking skill and curiosity. both abilities should be well-trained because they don’t occur instantly. curiosity heavily influences mathematical critical thinking and communication about a particular problem or situation. facing the challenges requires critical thinking skill, mathematical communication and in-depth curiosity. especially for mathematics in higher education, mathematical critical thinking, communication and curiosity are the primary assets for students in dealing with the problems, both in learning and everyday social life. in this study, problem-based learning cognitive conflict strategy (pblccs) has been applied to develop mathematical critical thinking skill, communication, and curiosity in number of theory courses. the findings showed that the application of pblcss could significantly increase students’ critical thinking skill and has not been able to increase communication skill and mathematical curiosity attitude (zetriuslita, wahyudin, & jarnawi, 2017). zetriuslita, wahyudin, & dahlan, association among mathematical critical … 18 method the method of the research used correlative research. the instruments include thetest to examine students’ critical thinking ability and mathematical communication and the questionnaire to obtain the scoresof curiosity attitude. the chi-square test is used to process the data by using software spss version 21.00 version. the association is measured with the pearson correlation coefficient (rxy) with the interpretation as seen in table 1 below : table 1. interpretation to corellation value (pearson correlation coefficient (rxy)) rxyvalue interpretation 0.00 – 0.199 very weak or no correlation 0.20 – 0.399 weak 0.40 – 0.599 medium 0.60 – 0.799 strong or high 0.80 – 1.000 very strong or very high in order to find out whether the association is significant or not, the chi-square test is used with the following requirement: the test results for mathematical critical thinking skill and mathematical curiosity if sig ≥ α, with (α = 0.05), then accepted h0, meaning that there is no significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. if sig < α, (α = 0.05), then rejected h0, meaning that there is a significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem based learning and cognitive conflict strategy. the test results for mathematical communication skill and curiosity if sig ≥ α, (sig α = 0.05), then accepted h0, meaning that there is no significant association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. 2. if sig < α, (α = 0.05), then rejected h0, meaning that there is a significant association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. results and discussion results association between mathematical critical thinking skill and curiosity before discussing the association between mathematical critical thinking skill and curiosity, the hypotheses will be presented first. h0 : there is no significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. volume 7, no. 1, february 2018 pp 15-24 19 h1 : there is a significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. the association between mathematical critical thinking skill and curiosity was calculated using the chi-square test as seen in table 2 below. table 2. association of mathematical critical thinking ability and curiosity value df asymp. sig. (2-sided) pearson chi-square 182.813 a 200 0.803 likelihood ratio 91.629 200 1.000 linear-by-linear association 0.204 1 0.652 n of valid cases 25 as table 2 above shows asymp.sig. (2-sided) = 0.803 ≥ α, it means that there is no association between mathematical critical thinking skill and curiosity as the impact of applying problem-based learning and cognitive conflict strategy. in other words, students with high critical thinking ability don’t automatically have high mathematical curiosity either. it can be seen from the correlation coefficient in table 2. table 3. the correlation coefficient of mathematical critical thinking skill and curiosity critical thinking curiosity critical thinking pearson correlation 1 0.092 sig. (2-tailed) 0.661 n 25 25 curiosity pearson correlation 0.092 1 sig. (2-tailed) 0.661 n 25 25 table 3 shows that the pearson correlation coefficient (rxy) = 0.092, which means the association between mathematical critical thinking skill and curiosity is very weak, they tend to have no relationship. it can be seen in the value of contingency coefficient in table 4 below. tabel 4.significance value of mathematical critical thinking skill and curiosity value approx. sig. nominal by nominal contingency coefficient 0.938 0.803 n of valid cases 25 a. not assuming the null hypothesis. b. using the asymptotic standard error assuming the null hypothesis. table 4 shows that the absence of association can be seen from the coefficient of determination (0.938) 2 = 0.880 = 88%. it means that the absence of the association between mathematical critical thinking skill and curiosity is 88%. for further description, see figure 1 below. zetriuslita, wahyudin, & dahlan, association among mathematical critical … 20 figure 1. association between mathematical critical thinking skill and curiosity figure 1 above clarifies that there is no association between mathematical critical thinking skill and curiosity. the points that illustrate the kcm and cm scores scatter randomly without forming patterns, either linear orsquared pattern and so on. association between mathematical communication skill and curiosity hypothesis: h0: there is no significant association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. h1: there is a significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. similar to the association between mathematical critical thinking skill and curiosity, the chisquare tests are also used to find out the association between mathematical communication skill and curiosity attitude. table 5. association between mathematical critical thinking skill and curiosity value df asymp. sig. (2-sided) pearson chi-square 212.708 a 200 0.256 likelihood ratio 99.441 200 1.000 linear-by-linear association 2.618 1 0.106 n of valid cases 25 table 5 shows that asymp. sig. (2-sided) = 0.256 ≥ α. it means that there is no association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. in other words, students with high communication skill don’t automatically have high curiosity either (table 6). tabel 6. correlation between communication and curiosity ability communication curiosity communication pearson correlation 1 0.330 sig. (2-tailed) 0.107 n 25 25 curiosity pearson correlation 0.330 1 sig. (2-tailed) 0.107 n 25 25 volume 7, no. 1, february 2018 pp 15-24 21 table 6 shows that pearson correlation (rxy ) = 0.330, which means the association between mathematical communication skill and curiosity is weak. see table 7 for details. table7. the significance value of critical thinking skill and curiosity value approx. sig. nominal by nominal contingency coefficient 0.946 0.256 n of valid cases 25 25 a. not assuming the null hypothesis. b. using the asymptotic standard error assuming the null hypothesis. table 7 shows that the weak relationship can be seen from the coefficient of determination of (0.946) 2 = 0.895 = 89.5%. it means that the absence of the association between communication skill and curiosity is 89.5%. approx. sig = 0.256 ≥ α value means h0 is accepted. in conclusion, there is no significant association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy (figure 2). figure2. association between mathematical communication and curiosity attitude figure 2 shows that the points that illustrate the scores scatter randomly, which means that there is no clear pattern, whether it is linear or something else. discussion based on the analysis for both of the hypotheses above, there was no association or relationship between mathematical critical thinking skill and curiosity, and between mathematical communication skill and curiosity as the impact of problem-based learning and cognitive conflict strategy. white (2009) commented on the adoption of problem-based learning that curiosity was not teachable and rarely tested but needs to be cultivated and exercised. consequently, zion & sadeh (2010) also stated that open inquiry learning can improve curiosity. in inquiry learning, students are stimulated to find out the materials, one of which is to directly ask them towork with the phenomena that exist around them. however, accordingto the findings of this study, pblccs was not able to develop students’ curiosity. based on the statistical data, one of the reasons is that their curiosity is quite high. subsequently, lecturers were less focused on stimulating the students’ curiosity. although applying problem-based learning and cognitive conflict strategy can increase critical thinking skill, it has not improved mathematical communication skill and curiosity (zetriuslita, et al., 2017). it is an evident if critical thinking skill in creases while curiosity zetriuslita, wahyudin, & dahlan, association among mathematical critical … 22 does not; it influences the association of both variables. that is why there was no association between them, as well as the association between mathematical communication skill and curiosity. the lack of effort by educators also influences the development of students’ communication in learning. another problem arose when college students were not accustomed to communicate with, in, and about mathematics, either orally or in writing. in the implementation of learning, mathematical communication skill and curiosity were considered less important. it can be seen during the learning process. the activities were more focused on developing critical thinking skill instead of mathematical communication and curiosity. the other issue was the lack of learning instruments (li) such as student’s worksheet that focuses on developing communication skill. the worksheet heavily emphasized on developing critical thinking skill. in addition, lecturers did not work harder in facilitating communication in learning. however, according to lang & david (2006), an educator should be a good communicator. besides, the limitation time in this research depicted that students were still unfamiliar with the given material. they even have to get adjusted to the learning type. apparently, college students did not acquire the “pblccs” spirit in a relatively short time. maharani & laelasari (2017) argued that problem-based learning cannot be implemented if the situation is not possible, so there must be an alternative that suits the environment in which learning takes place. according to wang, li, pang, liang, & su (2016), to successfully of using pbl, students should be responsive to the learning process. then rohana (2015) state that affective areas were indirect objects of mathematics, while cognitive areas are direct mathematical objects. therefore, it requires a plenty of time and appears to be relatively slow. it affects the absence of association among the abilities as mentioned aboveafter applying problem-based learning and cognitive conflict strategy. however, if the variables were equally increasing, the relationship will automatically exist at the same time. conclusion in the light of the preceding discussions and summary of findings, the following conclusions i.e : there was no significant association between mathematical critical thinking skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy and there was no significant association between mathematical communication skill and curiosity attitude as the impact of applying problem-based learning and cognitive conflict strategy. acknowledgments i thank to my promoters who always gives guidance in writing this article. i also thank to kemristekdikti who funded this research which is part of doctoral dissertation research. references aktaş, g. s., & ünlü, m. 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(2007). curiosity and open inquiry learning. journal of biological education, 41(4), 162-169. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p133-150 133 how do students' decision-making ability in solving open-ended problems? wasilatul murtafiah1*, nurcholif diah sri lestari2, faridah hanim yahya3, davi apriandi1, edy suprapto1 1universitas pgri madiun, indonesia 2universitas jember, indonesia 3universitas pendidikan sultan idris, malaysia article info abstract article history: received jan 13, 2023 revised feb 2, 2023 accepted feb 24, 2023 an open-ended problem in learning mathematics is a problem with more than one answer or method of solving. in solving open-ended problems in learning mathematics, one of the abilities students must use is decision-making ability. each student has a variety of capacities, so this study aims to determine students' decision-making abilities in solving open-ended problems in learning mathematics. the type of research used is descriptive qualitative research. the subjects of this study were four students with different numbers of correct answers in working on open-ended problems. data collection was carried out using tests and interviews. the results showed that (1) the decision-making ability of the subject who answers correctly for both problems is complete because they fulfill all indicators, the subject can identify goals, make decisions, evaluate the results of determination, and present and remember between problems with things known to the problem and related to decisions that have been taken correctly; (2) decision-making ability for subjects with wrong answers on one number or two numbers is incomplete because they only fulfilled two indicators, the subject can identify goals, make decisions, is less able to evaluate decision results and present and remember between problems with things known to the problem, and related to decisions that have been taken with correct. mathematics teachers should often reinforce students to practice operating integers because it is a prerequisite for learning mathematics at the middle and high school levels. keywords: decision making, mathematical problem, open-ended, problem solving this is an open access article under the cc by-sa license. corresponding author: wasilatul murtafiah, department of mathematics education, universitas pgri madiun jl. setia budi no.85, madiun city, east java 63118, indonesia. email: wasila.mathedu@unipma.ac.id how to cite: murtafiah, w., lestari, n. d. s., yahya, f. h., apriandi, d., & suprapto, e. (2023). how do students' decision-making ability in solving open-ended problems? infinity, 12(1), 133-150. 1. introduction in everyday life, mathematics has many benefits, especially for solving problems. mathematics is a basic science that is essential in developing science and technology. for this reason, students must master mathematics early to improve human resources quality https://doi.org/10.22460/infinity.v12i1.p133-150 https://creativecommons.org/licenses/by-sa/4.0/ murtafiah et al., how do students' decision-making ability in solving open-ended … 134 (hendriana et al., 2022; hidayat et al., 2022). this follows the regulation of the minister of national education of the republic of indonesia no. 64 of 2013 concerning content standards, which explains that learning mathematics has goals. namely, students must have competencies such as a critical attitude, logical, analytical, responsible, creative, responsive, careful, thorough, and not easily give up on solving problems (afdareza et al., 2020; widodo et al., 2018). thus the important thing that must be taught to students through mathematics is the ability to decide on problem-solving strategies (septian et al., 2022; verawati et al., 2022; wijayanti et al., 2022). problem-solving in learning mathematics is the core of basic skills in the learning process (widodo, 2017; widodo et al., 2020). to hone students' skills in solving problems, teachers must be able to bring up students' creative ideas by using supporting facilities such as giving concerns about open-ended problems (hidayat & sariningsih, 2018). according to becker and shimada, an open ended problem is a problem that has many or several solutions and ways to get the correct answer (fatah et al., 2016; viseu & oliveira, 2012). therefore, giving problems regarding open-ended issues to students requires students to solve problems by looking for several alternative solutions (puspaningtyas, 2019). an open-ended problem has great potential to accommodate concern-solving, and students must have problemsolving skills to support them in finding solutions to problems (hernandez-serrano & jonassen, 2003; özreçberoğlu & çağanağa, 2018). the benefits of solving open-ended problems are appreciating students' mathematical understanding, involving students' roles in the learning process by using their skills and knowledge to determine alternative solutions to issues, and allowing teachers and students to hold discussions to discuss some of the ways used to solve problems (mariam et al., 2019). to solve this open-ended problem, students are also required to think creatively because problems can have more than one answer or solution strategy (sa’dijah et al., 2017; wijaya, 2018). this open-ended problem is a higher order thinking skills (hots) problem because it requires students' high-level thinking processes to solve it (ibrahim & widodo, 2020). thus, the open-ended problem supports students' readiness for the minimum competency assessment (ernawati, 2016). facts show that students' minimum competency assessment results are still low. students often experience difficulties formulating problems in everyday life into mathematical models, interpreting the context of real situations into mathematical models, and understanding the structure of mathematics, which consists of order, relationships, and problem patterns (syawahid, 2019). in addition to difficulty modeling mathematics, students have difficulty interpreting and choosing which mathematical concepts can be used to solve problems. students also experience difficulties determining or selecting the appropriate strategy to solve problems (sinatra et al., 2015; widodo et al., 2022), such as the ability to solve problems for the minimum competency assessment problems, students must think at a high level, namely thinking creatively in determining solutions (setianingsih et al., 2022). thinking creatively is part of a person's ability to make decisions. students can determine several ways to solve problems by using decision-making abilities. decisionmaking, commonly called decision-making, benefits students' thinking processes in solving problems (winarso, 2014) with several alternative solutions and differences in students' ability to solve open-ended problems in learning mathematics. therefore, students' decisionmaking abilities in solving open ended problems in mathematics learning need to be further researched to find out students' decision-making abilities in solving open ended problems in mathematics learning. the results of previous research indicate that there are several studies on decisionmaking in the field of mathematics education. research related to the values underlying the decision-making process of turkish and german mathematics teachers in group studies volume 12, no 1, february 2023, pp. 133-150 135 (dede, 2013). research on perception, interpretation, and decision-making in developing novice teacher competencies (santagata & yeh, 2016). decision-making research on student winners of student creativity programs in designing ict-based learning media (murtafiah et al., 2019). research on junior and senior teacher decision-making in developing math problems (murtafiah et al., 2020). an exploration of the decision-making of prospective teacher students in solving literacy problems shows differences in student decision-making abilities based on gender (murtafiah et al., 2021). research related to the decision-making of students winning microteaching competitions in designing plans and implementing mathematics learning (murtafiah et al., 2022). some of these studies indicate that there is still no research on decision-making in school students. in addition, previous research on decision-making is still dominated by teachers and prospective mathematics teachers. thus, it is necessary to research the analysis of students' decision-making abilities in solving open ended problems in learning mathematics. through this research, teachers can discover students' decision-making abilities in solving open-ended problems to be used as a basis for improving mathematics learning. in addition, this research can provide opportunities for teachers to innovate learning to enhance students' decision-making skills in solving problems in learning mathematics. 2. method 2.1. research design the research used in this research is descriptive with a qualitative approach. qualitative descriptive is a research method based on the philosophy of postpositivism, which is used to examine the condition of natural objects where the researcher is the critical instrument (creswell, 2012; creswell & creswell, 2017). qualitative descriptive research aims to describe, describe, explain, explain, and answer in more detail the problems to be studied by studying as much as possible an individual, a group, or an event (johnson & christensen, 2019; lambert & lambert, 2012). this study describes students' decisionmaking abilities in solving open ended problems. this research was carried out in stages in the even semester of the 2021/2022 academic year. 2.2. participant the subjects of this study were class viii students of junior high schools in madiun city. the selection of research subjects was based on truth in working on open ended problems, which were given to 36 students. the research subjects used four students in the grouping of research subjects: one student with two correct answers, one student with only one correct solution, one student with only two correct answers, and one student with all wrong answers. this is because researchers want to reveal how the decision-making abilities of each student's characteristics in solving open ended problems are based on the correctness of student answers. the selection of subjects in this study was also based on students' communication skills based on the considerations of the mathematics teacher. 2.3. research instrument the test instrument used in this study is an open-ended problem. open-ended problems have more than one correct answer or more than one correct solution method (bragg & nicol, 2008; sa’dijah et al., 2017; viseu & oliveira, 2012). the open-ended problem the researcher designed was tested on three validators: two mathematics teachers and one mathematics education lecturer. based on the expert validation test results, the open murtafiah et al., how do students' decision-making ability in solving open-ended … 136 ended problem used in this study is a problem that has more than one way of solving according to the characteristics of the students at the junior high school where the study was conducted. the open-ended problem used is as follows. 1. the solution to the system of equations 2𝑎 + 𝑏 = 3 and 3𝑎 + 4𝑏 = 7 is … 2. the solution to the following system of equations is … { 3𝑥 − 4𝑦 = 16 𝑥 − 2𝑦 = 20 in addition to the test instrument in the form of an open-ended problem to uncover research data, it is also strengthened by interview data with the interview guide instrument in table 1. table 1. interview guidelines decision-making ability indicators question it identifies the purpose of deciding a given problem and matters relating to it. a. do you know what the problem is asking? b. what are the steps for the solution? able to make decisions a. what solution method did you use? b. how many solutions do you know? c. why did you use this solution method? evaluate the results of decisions a. after you work on the problem, is it by what you learned at school? b. is the method you used the most effective way to solve the problem? able to present and remember the relationship between existing problems and things that are known in the problem and related to decisions that have been taken correctly a. after you decided to use the solution method, what materials/concepts did you use? b. why do you decide to solve the existing problem with this solution method? interview guidelines in this study were designed based on decision-making indicators (wang & ruhe, 2007). as with the open-ended problem test instrument, two math teachers and one math lecturer validated the interview guide. the results of the validation show that the interview guide can be used because it can reveal decision-making abilities and is by student characteristics. 2.4. data collection and analysis data collection techniques in this study used tests and interviews, so the research instruments used were also of two types: difficulties in the form of open-ended problems and interview guidelines. there are three types of data analysis techniques in this study, including (1) the reduction stage, where the researcher groups data and selects data according to the need to answer the problem formulation; (2) the data presentation stage, where the researcher presents the research results in the form of words adapted to indicators of decision-making ability; (3) the conclusion stage is by the research objectives (bogdao & biklen, 2003; kirk & miller, 1986; miles et al., 2014). the following in table 2 indicates the decision-making ability used to analyze research (wang & ruhe, 2007). volume 12, no 1, february 2023, pp. 133-150 137 table 2. decision-making ability indicators no decision making ability indicator 1. it identifies the purpose of deciding a given problem and matters relating to it. 2. able to make decisions. 3. evaluate the results of decisions 4. able to present and remember the relationship between existing problems and things that are known about the problem and related to decisions that have been taken correctly. at this stage of data collection and analysis, a triangulation method was used to check the validity of the data (carter et al., 2014; guion et al., 2011; natow, 2020; renz et al., 2018). triangulation of the technique in this study was carried out by comparing the data on the results of open-ended problem tests and interviews. 3. result and discussion 3.1. results four groups were obtained based on the results of selecting research subjects. namely, students with correct answers to 2 problems, students with accurate answers to only number 1, students with correct answers to only number 2, and students with wrong answers can all be presented in table 3. table 3. grouping research subjects answer group the number of students selected subject code it's all true 19 i1 number 1 is correct, and number 2 is wrong 7 i2 number 1 is wrong, and number 2 is right 7 i3 numbers 1 and 2 are wrong 3 i4 the following describes students' decision-making abilities in solving open ended problems. subject i1 the following in figure 1, results from subject i1's work in solving open ended problems. subject i1 can complete with correct answers on problem numbers 1 and 2. figure 1. subject test results from i1 murtafiah et al., how do students' decision-making ability in solving open-ended … 138 based on the indicators of decision-making ability in identifying the purpose of decision-making, subject i1 knows what is known and asked in the problem so that subject i1 can determine what method to use to solve the problem. this was conveyed by subject i1, "i use the elimination of one of the variables, then use substitution to obtain the other variable". subject i1 also wrote down the steps for solving the problem according to his chosen method. subject i1 solves problems number 1 and 2 by eliminating one of the variables asked to determine the value of one of the variables. then subject i1 substitutes the variable's value found into one of the known equations to determine the value of the other variable. the work of subject i1 is shown in figure 1, where for numbers 1 and 2, the subject uses the same method. this is supported by the subject's statement during the interview, "in my opinion, questions 1 and 2 can be solved in the same way, namely using a mixed method". subject i1 was decided by determining the method to be used, namely the elimination and substitution method. subject i1 then evaluates the results of his decision by making corrections before the subject collects the results of his work. for the results of solving number 1, the values obtained are a=1 and b=1, and for number 2, the values obtained are x= –24 and y= –22. subject i1 can present and remember the relationship between existing problems and things that are known about the problem and make decisions that are taken correctly. subject i2 the following in figure 2, results from subject i2's work in solving open ended problems. subject i2 can complete the correct answer on problem number 1 and the wrong answer on problem number 2. figure 2. subject test results from i2 based on indicators of decision-making ability in identifying the purpose of decisionmaking, subject i2 knows what is known and asked in the problem so that subject i2 can determine what method to use to solve the problem. this is supported by the interview results where subject i2 stated, "i chose the combined method of elimination first, then substitution". subject i2 also wrote down steps to solve the problem. subject i2 solves problems number 1 and 2 using elimination and substitution. subject i2 writes 2a+ b=3 as the first equation and 3a+4b= 7 as the second equation. then subject i2 eliminates one of the variables, variable a, by multiplying the first equation by 3 and the second by 2. so the first equation becomes 6a+3b=9, and the second equation becomes 6a+8b=14. so the subject gets the result b = 1. after the subject receives the b value, the subject substitutes the b value into the first equation, namely 2a+b=3, the result is a=1, and the answer the subject gets is correct. volume 12, no 1, february 2023, pp. 133-150 139 the same applies to problem number 2, and subject i2 writes 3x–4y=16 as the first equation and x–2y=20 as the second equation. then subject i2 eliminates one of the variables, namely the variable, namely variable x, by multiplying the first equation by 1 and multiplying the second equation by 3. so that the first equation becomes 3x–4y=16 and the second equation becomes 3x–6y=60, then subject i2 gets the result y=22. the results obtained by the subject are wrong because the answer received should be y= –22, so finding the value of the variable x is also wrong. subject i2 was less thorough in operating 16 minus 60, which should be –22. through interviews, subject i2 stated that "oh yes, i was not careful, ma'am, the results should have been negative". subject i2 can make decisions by determining the method used, namely the method of elimination and substitution, so that in evaluating the results of the decision, subject i2 corrects it first before the subject collects the results of his work. for solving number 1, the values a=1 and b=1 are obtained, but for problem number, the outcome of the solution is still not quite right. subject i2 is less able to present and remember the relationship between existing problems and things that are known about the problem and make decisions that are taken correctly. subject i3 the following in figure 3, results from subject i3's work in solving open ended problems. subject i3 finished with the wrong answer on problem number 1 and the correct answer on problem number 2. figure 3. subject test results from i3 based on indicators of decision-making ability in identifying the purpose of decisionmaking, subject i3 knows what is known and asked in the problem so that subject i3 can determine what method to use to solve the problem. the statement of subject i3 supports this during the interview, "i use the method of elimination and substitution ma'am." subject i3 also wrote down the steps to solving the problem. subject i3 used the elimination and substitution methods to solve problem number 1. subject i3 wrote 2a+b=3 as the first equation and 3a+4b=7 as the second equation. then subject i3 eliminates one of the variables, variable a, by multiplying the first equation by 4 and multiplying the second equation by 1. so that the first equation becomes 8a+4b=12 and the second equation becomes 3a+4b=7 then subject i3 gets the result a=5. the results obtained by subject i3 are wrong because the answer received should be a=1, so finding the value of variable b is also wrong. subject i3 was not careful in operating 5a=5, namely 5 divided by 5, which should be a=1. for number 2, subject i3 uses the same method as number 1, namely the elimination and substitution method, to solve the problem. subject i3 writes 3x–4y=16 as the first equation and x–2y=20 as the second. then subject i3 eliminates one of the variables, namely murtafiah et al., how do students' decision-making ability in solving open-ended … 140 the variable, namely variable x, by multiplying the first equation by 1 and multiplying the second equation by 3. so the first equation becomes 3x–4y=16, and the second becomes 3x– 6y=60. then subject i3 obtains the result y= –22. after the subject brings the y value, the issue substitutes y into the second equation, the result x= –24, and the answers you get are correct. subject i3 can make decisions by determining the method used, namely the method of elimination and substitution, to evaluate the decision results. subject i3 corrects it first before the subject collects the results of his work. for solution number 1, the values a=5 and b= –7 are wrong, but for problem number 2, the results are correct, namely x= –24 and y= – 22. subject i3 is less able to present and remember the relationship between existing problems and things that are known about the problem and make decisions that are taken correctly. subject i4 figure 4 shows the result of subject i4's work in solving open ended problems. subject i4 finished with the wrong answer on problem number 1 and the correct answer on problem number 2. figure 4. subject test results from i4 based on indicators of decision-making ability in identifying the purpose of decisionmaking, subject i4 knows what is known and asked in the problem so that subject i4 can determine what method to use to solve the problem. this is supported by the results of an interview with subject i4 "i chose elimination first, then substitution because it's easy". subject i4 also wrote down steps to solve the problem. subject i4 used the elimination and substitution method to solve the problem for number 1. subject i4 wrote 2a+b=3 as the first equation and 3a+4b=7 as the second equation. then subject i4 eliminates one of the variables, variable a, by multiplying the first equation by 4 and multiplying the second equation by 1. so the first equation becomes 8a+4b=12, and the second equation becomes 3a+4b=7. then subject i4 obtained the result a=5. the results obtained by subject i4 are wrong because the answer got should be a=1, so finding the value of variable b is also wrong. the results of the interviews show that the subject feels the answer is correct. this is by the statement of subject i4 that "in my opinion, my answer is correct". for number 2, subject i4 uses the same method as number 1, namely the elimination and substitution method, to solve the existing problem. subject i4 writes 3x–4y=16 as the first equation and x–2y=20 as the second. then subject i4 eliminates one of the variables, namely x, by multiplying the first equation by 1 and the second by 3. so the first equation becomes 3x–4y=16, and the second becomes 3x–6y=60. so subject i4 gets the result y= –22. after the subject obtains the y value, the issue substitutes y into the second equation. the volume 12, no 1, february 2023, pp. 133-150 141 result is x=64. subject i4 was careless in operating, so the wrong answer was obtained. the subject's acknowledgment supports this through interviews, "oh yes, ma'am, i made a mistake in counting, negative meets negative should be positive". subject i4 can make decisions by determining the method to be used, namely the method of elimination and substitution. subject i4 did not evaluate the results of the decision or make corrections before the subject collected the results of his work. subject i4 was careless in calculating to produce inaccurate solving results for both problems. subject i4 is less able to present and remember the relationship between existing issues and things that are known in the problem and make decisions that are taken correctly. 3.2. discussion the results showed that each subject had different decision-making abilities. this difference can be seen in the skills of subjects i1, i2, i3, and i4 in each indicator of decision making, as shown in figure 5. figure 5. the subject's decision-making ability in each indicator at the stage of identifying the purpose of making decisions from a given problem and matters relating to the situation, subjects i1, i2, i3, and i4 were able to determine what was known and what was asked in the questions even though all subjects did not write explicitly on the answer sheets. from the interviews with the four subjects, they said that they were used to working on a system of two-variable linear equations without writing down what was known and what was asked. as stated by one of the subjects, i1, ”i did not write down what was known and asked because we usually wrote this down in the form of story questions”. even though all subjects did not write down what was known and asked in the questions, based on the results of all interviews, all subjects could identify the purpose of decision making from the given problem. this result is in line with research by widodo, istiqomah, et al. (2019), which also found that although the subject did not write down what was known and asked, the students could the process of solving the problem in the next stage. problem identification is part of the student's understanding of the problem (felmer et al., 2016). how to identify these students is different from the results of research conducted by lee (2016), that at the stage of understanding the problem, students write down what is known and asked. in the murtafiah et al., how do students' decision-making ability in solving open-ended … 142 identification stage of this goal, all topics are looking for a set of solutions to a system of two-variable linear equations. at this stage, the subject can understand the problem even though students do not write down what is known and asked in their work. identifying the purpose of deciding on this problem requires a person's ability to understand the situation, in this case, the issue (dauer et al., 2017; schoenfeld, 2015; wang & ruhe, 2007). identifying this goal is marked by determining what is asked of the problem (hutajulu et al., 2019; widodo, turmudi, et al., 2019). at decision-making stage, all subjects i1, i2, i3, and i4 can choose a solution method. all subjects chose the same techniques, namely solving the problem using a mixed elimination and substitution method, as shown in figure 6. subject i1 subject i2 subject i3 subject i4 description: elimination method; subtitution method figure 6. all subjects chose the same method to solve the problem number 1 and 2 volume 12, no 1, february 2023, pp. 133-150 143 the results of interviews with the four subjects stated that subject i1 said, ”i chose this method because it was more concise and faster to solve this problem and as exemplified by the teacher in class”. subjects i2, i3, and i4 have almost the same reasons as submitted by subject i3 ”i use this method because it is as exemplified by the teacher”. at this stage, only subject i1 had the right reason that choosing the combined elimination-substitution method would make the solution to a system of two-variable linear equations more concise and faster. the reason for choosing this method is in line with previous research if students choose the elimination-substitution combination method because it is more effective for solving a system of two-variable linear equations (bariroh et al., 2023). when asked if there was another way to use it, subjects i1, i2, i3, and i4 stated that they knew if other methods had been taught and could be used to solve the problem. subject i1 said, "in my opinion, this problem can be solved by using the method of elimination, substitution, and graphics". subjects i2, i3, and i4 have the same statement: can this problem be solved using only the elimination method or substitution? subject i1 has better knowledge when compared to subjects i2, i3, and i4. subject i1 can mention if to solve the problem can also use the graphical method. subject i1 stated that he obtained information regarding visual methods from textbooks and the internet. subjects i2, i3, and i4 do not mention graphical methods. when asked if there was another method to solve the problem, they gave the same answer: none. they said the teacher taught only the elimination, substitution, and combined elimination-substitution methods. in this decision making stage, all subjects used one method, the combined elimination-substitution method. this result contradicts the theory that solving open questions differently should be used for questions 1 and 2 (siswono, 2008). the subject should be able to use different ways and methods of solving (baker et al., 2001). in addition to the reasons given by i1, the technique chosen is concise; he uses one way for all subjects because they are fixated on the example given by the teacher. this shows that students are not used to using other methods or practices that vary besides those exemplified by the teacher. at this stage, students' creative thinking skills are needed to determine various solutions that can be used to solve the problem correctly. students can use the use of other methods as a compare of the final answer from solving the problem. in making decisions, one must be creative in collecting various ways to solve problems (murtafiah et al., 2021; murtafiah et al., 2019). when evaluating the decision results, subject i1 can consider correcting the answers to obtain the correct answers for numbers 1 and 2. subjects i2 and i3 are less able to assess because they have one correct answer and one wrong. subjects i2 and i3 were less thorough in performing integer division operations. subject i4 could not evaluate because he had incorrect answers in numbers 1 and 2. subject i4 felt that the answer was correct in question number 1, even though he was not careful in operating the division of integers. some 7th grade students in malaysia also experienced subject errors in integer division operations, and they were weak in multiplication and division caused of poor basic knowledge of arithmetic operations (khalid & embong, 2019). the errors experienced by subjects i2, i3, and i4 differ from high-ability grade 6 students in indonesia who can correctly carry out the division operation (nur et al., 2022). in the evaluation of the result stage, there were other mistakes made by subject i4. in question number 2, subject i4 was not cautious in operating the subtraction of negative integers. this is supported by the interview results where subject i4 stated, ”i am often confused when using numbers with a negative sign”. the error in evaluating experienced by subject i1 is not surprising because understanding the concept of abstract negative numbers is an obstacle for students; 60.4% of the respondents had difficulty when they gave examples of contexts which are integers involving negative numbers (fuadiah et al., 2017). in line murtafiah et al., how do students' decision-making ability in solving open-ended … 144 with this, the results of previous research show that students with low abilities have difficulty operating negative integers. in contrast, students with high skills have no problem working with negative numbers (utomo, 2020). at the evaluation stage of the results of this decision, all subjects require fundamental knowledge, i.e., arithmetic operations. arithmetic operations, which include addition, subtraction, multiplication, and division of integers,s are prerequisite knowledge needed in advanced mathematics. in addition to basic skills related to arithmetic operations, critical thinking is also required. these critical thinking skills are necessary to evaluate and re-check work and answers to solving problems that have been resolved. previous research also supports that critical thinking skills are needed to produce decisions with correct problemsolving (murtafiah et al., 2020; swartz et al., 1998). at the stage of presenting and remembering the relationship between existing problems and things that are known in the problem and about decisions taken correctly, only subject i1 fulfills this. this is also supported by the results of an interview with i1, that i1 presents a solution to a problem that has been rechecked for the suitability between what was asked and the answer to the problem-solving that has been done. subjects i2, i3, and i4 have not been able to present conclusions accurately because they have not used the concept of integer operations correctly. this shows that there is still a lack of students' understanding regarding the relationship between mathematical concepts to solve problems. students need knowledge of the prerequisite material/concept in learning mathematics. this is in line with previous research that the subject's ability to relate mathematical concepts to problems, one of which is influenced by the basic skills possessed by students related to arithmetic operations (khalid & embong, 2019). moreover, in solving open ended problems, knowledge and understanding of prerequisite materials and ideas are needed by students to be able to provide more than one solution method/strategy (anggoro et al., 2021; pramuditya et al., 2022). thus, the decision making by the subject answers correctly for both problems (i1); the subject can do the problem precisely and fulfills all indicators of decision making ability. decision-making for the subject by answering correctly only one problem (i2 and i3), namely, the subject can work on the situation well, with errors operating problem-solving. the issue can fulfill several indicators of decision-making where the subject is not sure about the decisions taken, but the subject can work on some of the problems well. decision making for the subject by answering the two concerns incorrectly (i4), namely, the subject can work on the issue to completion. still, it cannot calculate correctly, and the subject fulfills several decision making indicators. the subject's decision-making abilities in solving the twovariable linear equation system problem can be presented in table 4. table 4. decision making ability subject indicators decision making ability identification of problems make decisions evaluate the result of the decision present and remember the relationship between existing problems i1 complete i2 incomplete i3 incomplete i4 incomplete description: fulfill; not fulfill volume 12, no 1, february 2023, pp. 133-150 145 the subject who answered correctly for both problems has complete decisionmaking abilities because they meet all the indicators that have been set. subjects who responded to only one problem precisely had incomplete decision making abilities because they only fulfilled two indicators. subjects who answered both questions incorrectly had incomplete decision making abilities because they only fulfilled two indicators. this shows that the basic skills of high school students need to be taken seriously. the results showed that the weak essential ability of students in performing integer operations had a significant influence on other students' skills, as in this case, it affected students' decision making abilities. this finding is an unanswered question, as many high school students still experience errors in operating integers (fuadiah et al., 2017; khalid & embong, 2019; utomo, 2020). is there anything wrong with learning mathematics in elementary school? this requires teachers to seriously innovate in teaching students essential skills such as integer operations. mathematics teachers should often reinforce students to practice operating integers because it is a prerequisite for learning mathematics at the middle and high school levels. in addition, this also provides an opportunity for future researchers to overcome student errors in performing integer operations. 4. conclusion this study concluded that based on decision-making the subject answers correctly for both problems. the subject can identify goals, make decisions, evaluate the results of determination, and present and remember between problems with things known to the problem and related to decisions that have been taken correctly. subjects who answered correctly for both questions have complete decision-making abilities because they meet all indicators. while decision-making for subjects with wrong answers on one number or two numbers, namely the subject can identify goals, make decisions, is less able to evaluate decision results and present and remember between problems with things known to the problem, and related to decisions that have been taken with correct. subjects who answered correctly only one problem or answered incorrectly for both problems had incomplete knowledge abilities because they only fulfilled two indicators. furthermore, mathematics teachers should often reinforce students to practice operating integers because it is a prerequisite for learning mathematics at the middle and high school levels. acknowledgements the authors would like to thank the universitas pgri madiun, universitas jember, smpn 7 madiun, and drtpm ditjen dikti, who have facilitated and provided funds for this research. references afdareza, m. y., yuanita, p., & maimunah, m. 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(2022). learning achievement of extroverted students in algebraic operations by tutorial learning: a single subject research. international journal of evaluation and research in education, 11(1), 99-107. https://doi.org/10.11591/ijere.v11i1.21747 widodo, s. a., turmudi, t., & dahlan, j. a. (2019). an error students in mathematical problems solves based on cognitive development. international journal of scientific & technology research, 8(07), 433-439. wijaya, a. (2018). how do open-ended problems promote mathematical creativity? a reflection of bare mathematics problem and contextual problem. journal of physics: conference series, 983(1), 012114. https://doi.org/10.1088/17426596/983/1/012114 wijayanti, a., widodo, s. a., pusporini, w., wijayanti, n., irfan, m., & trisniawati, t. (2022). optimization of mathematics learning with problem based learning and 3n (niteni, nirokke, nambahi) to improve mathematical problem solving skills. indomath: indonesia mathematics education, 5(2), 123-134. winarso, w. (2014). problem solving, creativity dan decision making dalam pembelajaran matematika [problem solving, creativity and decision making in learning mathematics]. eduma: mathematics education learning and teaching, 3(1), 1-16. https://doi.org/10.1088/1742-6596/948/1/012004 https://doi.org/10.1088/1742-6596/1657/1/012092 https://doi.org/10.1088/1742-6596/1188/1/012087 https://doi.org/10.11591/ijere.v11i1.21747 https://doi.org/10.1088/1742-6596/983/1/012114 https://doi.org/10.1088/1742-6596/983/1/012114 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p349-366 349 development of combined module using contextual scientific approach to enhance students' cognitive and affective muhamad yusup kurniansyah1, wahyu hidayat2*, euis eti rohaeti2 1stai darul falah cihampelas, indonesia 2institut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received aug 17, 2021 revised sep 21, 2022 accepted sep 29, 2022 this research on the development of experimental product test designs aims to develop learning products to solve problems that occur in the field and find out and analyze the mathematical problem-solving abilities and habits of mind of mts students whose learning uses a combination of scientific approaches and contextual approaches compared to mts students who use scientific approaches. the research subjects were conducted on students in indonesia with details of 1 class at mts azzahra batujajar for a limited trial, three classes from mts bhakti pertiwi cililin, mts albidayah, and mts al-mukhtariyah rajamandala for field trials, and two classes at mts nurul hidayah batujajar for product testing and used as the experimental class and the control class. the instrument in this research is a set of mathematical problem-solving abilities and a non-test to determine the students' habits of mind. the results of this study indicate that the process of developing teaching materials is carried out well, and the product is feasible to use. mathematical problemsolving abilities and habits of mind of students who receive learning using a combination of scientific and contextual approaches are better than those who receive knowledge using a scientific approach. keywords: development, habits of mind, problem solving this is an open access article under the cc by-sa license. corresponding author: wahyu hidayat, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi jl. terusan jenderal sudiman no. 3, cimahi city, west java 40526, indonesia. email: wahyu@ikipsiliwangi.ac.id how to cite: kurniansyah, m. y., hidayat, w., & rohaeti, e. e. (2022). development of combined module using contextual scientific approach to enhance students' cognitive and affective. infinity, 11(2), 349-366. 1. introduction mathematics is a subject that requires a high level of understanding and also has a very important role in solving a problem that we often encounter in life. this is contained in regulation of the minister of education and culture of the republic of indonesia number 58 of 2013 concerning the 2013 curriculum for junior high school/mts, which states that the objectives of learning mathematics in junior high schools are, among others, students are expected to be able to understand concepts, use patterns in solving problems (rosidin et al., 2019; suhirman et al., 2016). https://doi.org/10.22460/infinity.v11i2.p349-366 https://creativecommons.org/licenses/by-sa/4.0/ kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 350 problem solving is one of the important things contained in the purposes of learning mathematics. the students are able for problem solving in mathematics learning then they will be better in mathematics learning (siagan et al., 2019; simamora & saragih, 2019; sumirattana et al., 2017; surya et al., 2017). mathematical problem solving ability become one of the basic mathematical abilities that must be possessed by every student. because the problem-solving skills is very important, then this ability should be given to students from early age as a provision in preparing competition in the real life so the students could be familiar to the challenges that they faced (basri, 2019; cevikbas & argün, 2017; irwanto et al., 2018; tyas & naibaho, 2021). however, the scope of reality is not accordance to all of theories that described above, the problem which is difficult to be solved at school, it is including on the two variables linear equation’s theory. two variables linear equation is one of the subject matter in math which is a theory that is often considered difficult so that some students have adversity when working on modified math problems in the story question form. this is evident from previous research which found various errors made by students when working on linear equations of two variables (fatoni et al., 2021; putra et al., 2020; santoso et al., 2019; wicaksono, 2020). in addition, the results of research conducted by sukestiyarno et al. (2021) also concludes that the weak mathematical problem solving ability of students is caused by the lack of variety of teachers in optimizing teaching materials in the form of learning modules. the learning module used by the teacher in providing subject matter to students is considered inadequate. this affects students' ability to package concepts properly and correctly, so that it has an impact on low problem solving abilities. the weak ability of students to solve a problem can be sourced from students' thinking habits (pei et al., 2018; umar, 2017). students' thinking habits in dealing with and solving a problem will make students accustomed to finding new problems every time (hodiyanto & firdaus, 2020; papadopoulos, 2019). this shows that students' habits of mind have an important role in the learning process of individual development in helping to solve a problem. according to akdeniz and ekici (2019) that habits of mind or the habit of thinking as a step or method taken by intelligent people when facing a problem whose solution cannot be easily identified. this confirms what was conveyed by prasad (2020) that it is difficult to get students to think mathematically because the assessment scheme carried out by teachers in learning is considered to lack respect for students' intellectual courage and humility, so that students' habits of mind do not develop. this can have an impact on the students' habits of mind continue to be trained, will synergize the right brain and left brain which in the end students get used to finding solutions to even difficult problems. therefore, efforts must be made to improve students' mathematical problem solving abilities and habits of mind in particular to solve problems in two-variable linear equations story problems is to develop innovative teaching materials related to students' daily lives. the urgency of developing these teaching materials needs to be packaged in a learning product that influences the achievement of students' mathematical problem solving abilities and habits of mind. considering that developing mathematical problem solving skills and familiarizing students' thinking processes is not easy, it is necessary to have a learning approach integrated with teaching materials to encourage students to be active and find concepts independently, especially those related to problems in everyday life. one of them is the contextual teaching and learning approach, which is the most likely approach to help students find concepts oriented to everyday life. contextual teaching and learning (ctl) approach is learning that regulates students to be actively involved in the learning process so that they can find concepts learned through students' knowledge and experience (selvianiresa & prabawanto, 2017). contextual volume 11, no 2, september 2022, pp. 349-366 351 approach is applied by exposing students to contextual problems that will be solved by relating them to real situations. in addition, in this effort to develop teaching materials, it also needs to be associated with a scientific approach. kurniati et al. (2015) states that the development of mathematics learning tools with a contextual approach integrated with a scientific approach can produce valid, practical, and effective learning tools. based on these problems, this study aims to determine the effectiveness of developing learning modules that combine two approaches, namely a scientific approach and a contextual approach, in improving students' mathematical problem solving abilities and habits of mind. 2. method the methods in this research are development research (research & development). the development research is an endeavor to develop the effective product to be used in school, and it does not aim to verify a theory. the development research flows described on figure 1. figure 1. the development research flows figure 1 shows the stages of development research that the authors did to develop teaching materials from a combination of scientific approaches and contextual approaches. in the early stages, the authors conducted a preliminary study in the form of library research, school surveys and drafting. in the literature study, the researchers conducted a study of the theories related to the approach to be developed, besides that it also examined the characteristics of students, especially in their mathematical problem solving abilities. a school survey was conducted to see the spldv learning that had been carried out by mts teachers, and interviews were also conducted with these teachers to find out what learning deficiencies should be improved. then, the authors create the designs and drafts of module that will be developed for limited trials. after a limited trial, the authors seek input from the teacher to find out whether or not there is a revision of the developed teaching materials, if finding to be revised then the module must be revised, if it is not the teaching material must be continued to a broad trial. after extensively checked then product is validated by the kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 352 experts, if they found a revision then the teaching materials are revised for the second time, if they are not then the teaching materials are directly allowed to be tested. then after testing the product, the compiler has the final form of teaching material and is ready to be socialized. product testing activities are carried out with an experimental design involving two classes for the sample which are used as the experimental class and the control class. the subject for the research trials is the students of 8th grade in mts nurul hidayah, batujajar, which the selected sample based on considerations created by the researcher and according to certain criteria that is total students, class situation, and teacher’s suggestions. in this research, the selected sample as a research sample is students in grade of viii a which have a total 27 students for being a control class, and the grade of viii b which have a 28 students become an experimental class whereas all of students from both of class, totally 55 students, have been taught as the research the research location for the limited trial was carried out at mts azzahra batujajar with a total of 29 students, while for the field test it was carried out in three schools namely mts bhakti pertiwi cililin, mts terpadu albidayah, and mts al-ikhsan batujajar, all of them are located in kabupaten bandung barat. the compilers used questionnaires and interviews to determine the students' readability on the products developed in limited trial activities and field tests. data processing was carried out using ibm spss statistics version 22 to test the results of product tests and using percentages presented descriptively. 3. result and discussion 3.1. result 3.1.1. teaching material development process from the description of the research objectives, to develop a teaching material using a combination of two approaches, a validation is needed for the teaching material itself. validation is carried out on expert validators, namely lecturers, school supervisors, then subject teachers, and it is proven that the level of readability is proved to the level students concerned in the research. lecturers provide validation for teaching materials whether or not it is appropriate for the developed teaching materials to be used with revisions or without revisions. the subject teacher also validates whether the content is in accordance with the proposed level or not and can be used with or without revision. while students are allowed to read carefully then interviewed whether they are able to achieve a high level of readability or not. the following is an initial view of the teaching materials before the limited trial is carried out, presented on figure 2. volume 11, no 2, september 2022, pp. 349-366 353 figure 2. initial appearance of teaching materials before trial figure 2 shows the design or initial draft of the teaching materials that the authors developed prior to the trial. the validation results show that the average percentage of the validation aspects on the validation sheet of the teaching materials is still at 32%, so the expert validators conclude that the teaching materials can be tested with revisions. the next step of this development research that the authors worked is a limited trial. the limited trial was carried out at mts azzahra batujajar with 29 students present when the authors made observations in the field or called a limited trial. this limited trial was conducted to measure the students' readability of the teaching materials that the compilers developed. the technique that the authors used to obtain data from the trial is the form of interviews and giving questionnaires about the teaching materials that have read (see table 1). table 1. limited trial questionnaire results no questions response agree do not agree 1. do you understand the problems given in the teaching materials? 12 17 41.40% 58.60% 2. do you understand the commands from the tables in the teaching materials? 8 21 27.60% 72.40% 3. do you understand the language contained in the teaching materials? 24 5 82.80% 17.20% 4. is the language in the teaching materials effective/uncomplicated? 26 3 89.70% 10.30% 5. are you able to capture the information contained in the teaching materials? 14 15 48.30% 51.70% 6. are the teaching materials more practical than textbooks? 16 13 55.20% 44.80% overall average 57.50% 42.50% kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 354 to strengthen the data readability of students during a limited trial at mts azzahra batujajar, in addition to using a questionnaire, the compilers also conducted interviews with several students to find out what parts the students cannot understand in the modules that the compilers developed (see table 1). based on the results of interviews in general for aspects of writing with good and correct indonesian rules and the use of effective and efficient language, students do not experience problems or find a language they do not understand. meanwhile, for the clarity of presentation, students experience problems in terms of observing and reading tables, which may be caused by the lack of mastery of prerequisite material. this shows that the connection between mathematical material needs to be understood by students to solve the problems they are experiencing (jannah & apriliya, 2017; kenedi et al., 2019; siregar & surya, 2017). the following is a documentation of the limited trial activities that the authors carried out at mts azzahra batujajar (see figure 3). figure 3. limited trial activities after conducting a limited trial, the teaching materials received input from the teacher and the original validator as well as from interviews and student questionnaires regarding the initial appearance of the teaching materials. the following is an initial view of the teaching materials after a limited trial was carried out and the first revision regarding the stages of learning such as praying should be included in the teaching materials to make them more interactive. the initial view is presented on figure 4. figure 4. display of teaching materials after limited trial volume 11, no 2, september 2022, pp. 349-366 355 then after doing a limited trial and getting a revision, the next stage is a field test. there is no difference in data collection techniques for field tests, such as limited trials, namely questionnaires and interviews. a field test was carried out in three schools in west bandung, namely mts bhakti pertiwi cililin with 19 students, mts albidayah cangkorah with 23 students, and mts al mukhtariyah rajamandala with 27 students, so the overall number of students in the test try wider is 69 students. the following is a table of results from a field test that the authors have carried out in three mts located in bandung barat (see table 2). table 2. field test questionnaire results no questions response agree do not agree 1. do you understand the problems given in the teaching materials? 31 38 44.90% 55.10% 2. do you understand the commands from the tables in the teaching materials? 23 46 33.30% 66.70% 3. do you understand the language contained in the teaching materials? 44 25 63.80% 36.20% 4. is the language in the teaching materials effective/uncomplicated? 37 32 53.60% 46.40% 5. are you able to capture the information contained in the teaching materials? 29 40 42.00% 58.00% 6. are the teaching materials more practical than textbooks? 35 34 50.70% 49.30% overall average 48.00% 52.00% according to the results of interviews for a written aspects in a good and correct indonesian rule and the use of effective and efficient language, there are some students who do not understand the language in the modules that the compilers developed. meanwhile, for the clarity of presentation, it is the same as during the limited trial, students experience problems in terms of observing contextual problems in teaching materials and reading tables, which may be caused by a lack of mastery of prerequisite material. the following is a documentation of the limited trial activities that the authors carried out at mts bhakti pertiwi cililin, mts al bidayah, and mts al mukhtarah rajamandala (see figure 5). figure 5. field test activities after conducting field test, the next stage is the final validation (expert validation) carried out to find out whether the teaching materials developed after the discovery of several shortcomings at the trial stage such as limited trials and field tests and getting revisions from expert validators along with teacher to proceed to the product test stage. the results of the validation showed that the average percentage of the validation aspects on the validation kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 356 sheet of the teaching materials was 80%, so the expert validators concluded that the teaching materials could be tested for the product. after conducting field test of teaching materials, they received feedback from the teacher and the original validator as well as from interviews and student's questionnaires regarding the content of the teaching materials. the following is a display of teaching materials after field test and getting a second revision regarding apperception activities must be in the teaching materials before the core activity takes place, so apperception activities are not only delivered orally by the teacher but are in teaching materials. the appearance of apperception on teaching materials after extensive testing and validation is presented on figure 6. figure 6. display of teaching materials after field test and validation the next step is after the teaching materials get validation from the expert validator then the teaching materials are tested. this product test activity was carried out at mts nurul hidayah batujajar in 2 classes of students’ grade 8th with an experimental class and a control class conducted for 8 meetings on the material of linear equations of two variables. in practice, there are pretest and posttest activities to measure mathematical problem-solving abilities before and after teaching materials are used on students. the following are product testing activities carried out by the compilers (see figure 7) figure 7. product test activities the last step of the product development stage carried out by the compilers is product socialization which is carried out during teacher meeting activities at mts nurul hidayah batujajar to provide information to teachers where the compilers have developed a teaching material compiled from a combination both of a scientific approach and a contextual volume 11, no 2, september 2022, pp. 349-366 357 approach as one of the solutions for learning, especially in mathematics. the following are product socialization activities that the authors carry out (see figure 8). figure 8. product socialization the following is the final version of the teaching materials when the product socialization activities are carried out, after the teaching materials have gone through the development flow that the compilers did (see figure 9). figure 9. final version of teaching material 3.1.2. analysis of students' problem solving ability and habits of mind to describe the problem solving skills and habits of mind of students in learning mathematics, data is needed in the form of values derived from pretest scores, posttest scores, and gain values. the pretest score is the student's initial test score before receiving a combined learning from a scientific approach and a contextual approach. post-test scores are student test scores obtained after receiving a combination of scientific approach and contextual approach learning. gain is the increase in student scores obtained from the difference between the pretest and posttest scores. this test was given to students of class viii a as a control class who received learning with a scientific approach and class viii b as an experimental class who received kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 358 learning using a combination of a scientific approach and a contextual approach. descriptive calculations to describe problem solving abilities can be briefly seen in table 3. table 3. description of problem solving ability scores and habits of mind mathematical ability statistics learning approaches combined scientific and contextual approach scientific approach pretest posttest gain pretest posttest gain problem solving x̅ 36 71 0.54 35 62 0.42 % 36.00% 71.00% 35.00% 62.00% s 11.80 6.30 8.60 12.10 ideal score 100 1 100 1 habits of mind x̅ 69.43 79.77 % 44.08% 48.7% s 8.20 6.87 ideal score 157.50 163.48 table 3 shows that the average pretest of the two classes is in the low category; this is indicated by the initial ability of the experimental class and control class students who are not much different. then after the treatment, it can be seen that the average post-test of the two classes has increased and are both in the medium category. for the average of n-gain, both of classes are also in the medium category. then for the standard deviation of mathematical problem solving abilities, it can be seen that the standard deviation of the experimental class pretest is greater in value than the standard deviation of the control class, which means that the values in the control class are more spread out than the experimental class, but after the treatment is given, the opposite applies to the control class posttest scores larger than the experimental class, which means that the posttest value of the experimental class is more spread out. then for the results of the habits of mind posttest after the msi calculation, it can be seen that the average of the two classes is in the medium category with the average value of the experimental class under the control class, while for the standard deviation of habits of mind, it can be seen that the control class is smaller than the experiment class which indicates that the data obtained from the control class is more spreading out. to clarify the description of the research results, data analysis was carried out on the results of the pretest experimental class and control class to find out the differences in students' pre-abilities in the experimental class and control class through statistical tests which included the normality tested, if the data is not normally distributed then the test continued by doing non-parametric test, namely the mann whitney test, if the data is normally distributed then it will be continued with the parametric test which begins with the homogeneity test, if the data is not homogeneous then it will be continued to do the t' test, if the data is homogeneous then it will be carried out to do the independent samples t test the first testing step is the normality test for the pretest value of the experimental class and control class, which is used to find out which direction the next testing should be, whether parametric or non-parametric tests. normality test was performed with the help of ibm spss statistics version 22 program. test criteria :if sig. > α=5%, then the sample data is normally distributed the results of the pretest normality test's calculation for the experimental and the control class for students' mathematical problem solving abilities are presented in table 4. volume 11, no 2, september 2022, pp. 349-366 359 table 4. normality test results pretest values problem solving ability kolmogorov-smirnova shapiro-wilk statistic df sig. statistic df sig. experiment 0.179 27 0.027 0.930 27 0.068 control 0.149 27 0.128 0.940 27 0.123 a. lilliefors significance correction the calculation of the kolmogorov-smirnov test results (see table 4), the significance value of the experimental class pretest < α is 0.027 while the control class pretest significance value > α is 0.128, which means that the experimental class pretest data is not normally distributed while the control class is normally distributed. because one of the classes is not normally distributed, the next step is the non-parametric test, namely the mann whitney test. the non-parametric test, namely the mann whitney test, is carried out if the data obtained are not normally distributed or the data is normally distributed but not homogeneous. the mann whitney test was carried out with the help of the ibm spss statistics version 22 program with the following hypotheses: h0: u1 = u2 h1: u1 ≠ u2 test criteria: if sig. > α=5%, then h0 been accepted the results of the mann whitney test's calculation of students' mathematical problem solving ability pretest scores are presented in table 5. table 5. mann whitney test results pretest of mathematical problem solving ability value mann-whitney u 365.000 wilcoxon w 743.000 z -.220 asymp. sig. (2-tailed) .826 monte carlo sig. (2-tailed) sig. .833b 95% confidence interval lower bound .826 upper bound .841 monte carlo sig. (1-tailed) sig. .417b 95% confidence interval lower bound .408 upper bound .427 a. grouping variable: group b. based on 10000 sampled tables with starting seed 2000000. table 5 show that the significance value is obtained 0.826. because it has value of sig. > α then h0 been accepted, so that it can be concluded that there is no difference in the mathematical problem solving abilities of mts students who receive learning using a combination of scientific approaches and contextual approaches with those who receive learning using a scientific approach. after known, the conclusion from the t-test for the pretest value of the experimental class and the control class are no difference in the preability of each class, then the author will re-analyze the posttest results for the experimental class and control class to find out the achievements made by each class. the first step of test is the normality test for the posttest scores of the experimental class and the control class, which is used to find out which way the next testing should be, whether parametric or non-parametric tests. normality test on α = 5% by using the ibm spss statistics version 22 program. kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 360 test criteria: if sig. > α, then the sample data is normally distributed the results of the posttest normality test calculation for the experimental and the control class for students' mathematical problem solving abilities are presented in table 6. table 6. normality test results posttest values problem solving ability kolmogorov-smirnova shapiro-wilk statistic df sig. statistic df sig. experiment 0.141 27 0.178 0.952 27 .234 control 0.125 27 0.200* 0.964 27 .461 *. this is a lower bound of the true significance. a. lilliefors significance correction table 6 show that the posttest significance value of the experimental class is 0.178 while the significance value of the control class is 0.200, which means that the posttest data of the two classes is normally distributed because it has a normal value sig. > α. the next step is parametric test. parametric test is performed if the obtained data is normally distributed. parametric tests were carried out by helping of the ibm spss statistics version 22 program with the following hypotheses: h0: 𝜇1 = 𝜇2 h1: 𝜇1 > 𝜇2 test criteria: if sig. > α, then h0 been accepted the results of the independent samples t parametric test's calculation, the post-test scores for students' mathematical problem solving abilities are presented in table 7. table 7. the posttest results of mathematical problem solving ability levene's test for equality of variances t-test for equality of means f sig. t df sig. (2tailed) mean difference std. error difference 95% confidence interval of the difference lower upper score equal variances assumed 12.36 0.001 3.37 53 0.001 8.750 2.592 3.551 13.94 equal variances not assumed 3.33 38.57 0.002 8.750 2.620 3.448 14.05 the levene's test column the value of sig. < α (see table 7), which means that it can be concluded that the two variances are not homogeneous. because the obtained data from the results' test are normally distributed and not homogeneous, the test is carried out with the t-test. the test carried out is the t test, then the data used is equal variances not assumed so that it can be seen that the value of sig. (2-tailed) is about 0,002. however, because the hypothesis testing uses a one-party statistical test (1-tailed) so for further ensure the correctness of the test above, we do it manually by comparing the calculated to the table. it can be seen from the test table that what we get is 3.33. meanwhile, for the table value for the one-sided (1-tailed) test at 5% significance level in the test table above, df = 38 is obtained. the value from the obtained table is 2.024 which can be seen in table t. then the volume 11, no 2, september 2022, pp. 349-366 361 next step, we will compare the count to the table which gives the result that count > table then h0 is rejected which means that at the 5% significance level can be concluded that students' mathematical problem solving ability who receive a learning using the combined scientific approach and the contextual approach are better than students who receive learning using a scientific approach. furthermore, the authors also tested to the affective behavior, namely the habits of mind of students, the following are the results of testing of students' habits of mind. test criteria: if sig. > α=5%, then the sample data is normally distributed the results of the calculation of the posttest data normality test of the students' attitude scale habits of mind are presented in table 8. table 8. normality test results of the habits of mind students kolmogorov-smirnova shapiro-wilk statistic df sig. statistic df sig. experiment 0.085 27 0.200* 0.983 27 0.931 control 0.097 27 0.200* 0.986 27 0.964 *. this is a lower bound of the true significance. a. lilliefors significance correction table 8 show the experimental class and the control class is equal to 0.200. so it can be concluded that the average posttest value on the habits of mind data is normally distributed. parametric test is performed if the obtained data is normally distributed. parametric tests were carried out with the help of the ibm spss statistics version 22 program with the following hypotheses: h0: 𝜇1 = 𝜇2 h1: 𝜇1 > 𝜇2 test criteria: if sig. > α=5%, then h0 been accepted the results of calculation of the parametric independent samples t-test of the students' habits of mind are presented in table 9. table 9. the results of the students’ habits of mind levene's test for equality of variances t-test for equality of means f sig. t df sig. (2tailed) mean difference std. error difference 95% confidence interval of the difference lower upper score equal variances assumed 0.611 0.438 -5.03 53 0.000 -10.342 2.054 -14.462 -6.223 equal variances not assumed -5.05 51.8 0.000 -10.342 2.047 -14.450 -6.235 the levene's test column gets the value of sig. > α (see table 9), which means concluded that the two variances are homogeneous. because the obtained data from the test results are normally distributed and homogeneous, then the test is carried out to the t-test. the test carried out is the t test, the used data is equal variances assumed so that can be seen that the value of sig. (2-tailed) of 0.000. however, because of hypothesis testing uses a one kurniansyah, hidayat, & rohaeti, development of combined module using contextual scientific … 362 sided (1-tailed) statistical test, so for further ensure the truth of the test, it be manually done by comparing the count to the table. it can be seen from the test (see table 9) that the count we get is -5.03. meanwhile, the table value for the one-sided (1-tailed) test at the 5% significance level is obtained dk = n1 + n2 – k whereas n is the number of data, while k is the number of variables. so dk = n1 + n2 – k = 28 + 27 – 2 = 53. for the value from the obtained table is about 2,005 which can be seen in table t. then the next step, we will compare the count to the table which gives the result that count > table so h0 is rejected, which means that the significance level at 5% can be concluded that the habits of mind of mts students who receive learning using a combination of scientific approaches and contextual approaches are better than those who receive learning using scientific approaches. 3.2. discussion this research aims to develop the learning materials, so they can be used as a solution to problems found in the field, regarding the lack of student problem solving abilities because the teaching used materials are less qualified. this is because the available teaching materials only focus on the material and practice questions, so learning becomes monotonous. teachers should improvise so that education is not dominated by textbooks which have an impact on student activities in learning (hasibuan et al., 2019; rahmi et al., 2019; sari & yaniawati, 2019; siagan et al., 2019; ulandari et al., 2019). related to the process of developing teaching materials, it has been running smoothly without any problems. the teaching materials produced have gone through the validation stage by experts, teachers, and students to be feasible and effective to be used as learning teaching materials. this shows that good product development of teaching materials is teaching materials that go through expert validation tests to get excellent and practical categories (andini & yunianta, 2018; putri et al., 2020; rahmi et al., 2019). in addition, one of the objectives of this research apart from developed teaching material products that are able to become solutions to problems regarding teaching materials in schools, this study also aims to examine the achievement of mathematical problem solving abilities and the habits of mind of mts students after teaching materials using the combination of the scientific approach and the contextual approach has been developed compared to those using the scientific approach in schools as usual without any development for teaching materials. the research was carried out at the product test stage in development research after going through the development stages which were carried out in accordance to the stated by budiarti and haryanto (2016) in conducting their development research, if the product has been assessed as good then it will be said to be feasible to use and tested using two classes. as a sample to be used as an experimental class and a control class. the average of pretest result is not found significant differences both the experimental class and the control class such as the research conducted by yanti (2017) with an initial test of the pretest value and obtained the results that the basic abilities of the two classes were the same before being given treatment. furthermore, the two groups were given different treatments where the experimental class was treated with teaching materials that had been developed while the control class was treated with a scientific approach that exists generally at school. then, based on the results of posttest data analysis in the experimental class and control class, the mathematical problem solving ability in the experimental class is higher than the control class. it can be concluded that the mathematical problem solving ability of students who receive learning using a combination of scientific approaches and contextual approaches is better than students who receive learning using a scientific approach. this is in accordance to the research conducted by mulhamah and putrawangsa volume 11, no 2, september 2022, pp. 349-366 363 (2016) who conducted research by applying a contextual approach in their learning to junior high school students equivalent to the results that there was an increased in students' mathematical problem solving abilities from cycle 1 to cycle 2, while it was also in accordance to the research conducted by nuralam and eliyana (2017) who used a scientific approach in their learning by getting the results that problem-solving abilities whose learning uses a scientific approach are higher than the realistic approach. the research conducted by rahmawati et al. (2014) found out that the gain value obtained by the upper-class and the lower class were both included in the moderate criteria. this thing is the same as this study whereas the gain value obtained by the experimental class and the control class, its increasing is included in the medium category. then, based on the results of analysis of the non-test of affective behavior data in the experimental class and control class, the students' habits of mind in the experimental class were higher than the control class. it can be concluded that the habits of mind of mts students who receive learning by using a combination of a scientific approach and a contextual approach are better than those who receive learning using a scientific approach. according to the analysis of research data's result, it is known that the mathematical problem solving ability and habits of mind of mts students who receive learning using a combination of scientific approaches and contextual approaches are better than those who receive learning using scientific approaches this is indicated by the difference in the average scores of pretest, posttest, and n-gain solving abilities and the average non-test scores of students' habits of mind in the experimental class and the control class. after being given treatment to experimental class students by using a combination of scientific approaches and contextual approaches and control classes with scientific approaches, the results of the analysis obtained support the hypothesis which states that mathematical problem solving abilities and habits of mind students who learn using a combination of scientific approaches and contextual approaches better than those who learn by using a scientific approach. 4. conclusion based on the results of research that has been carried out regarding the development of teaching materials to improve problem solving abilities and the habits of mind of mts students using a combination of scientific approaches and contextual approaches, it can be concluded: the process of developing teaching materials has been carried out well which is divided into 6 stages after the teaching materials are made, starting from (1) initial validation (expert judgment); (2) limited trial; (3) field test; (4) final validation (expert validation); (5) product test (experiment class sample and control class); and (6) product socialization to obtain a statement that the product is suitable for use. the mathematical problem solving ability of students who learn by using a combination of a scientific approach and a contextual approach is better than students who learn by using a scientific approach. the habits of mind of mts students who learn by using a combination of a scientific approach and a contextual approach are better than those who learn by using a scientific approach. references akdeniz, h., & ekici, g. 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(2017). penerapan model problem based learning (pbl) terhadap kemampuan komunikasi dan kemampuan pemecahan masalah matematika siswa sekolah menengah pertama lubuklinggau [application of problem based learning (pbl) model on communication skills and mathematical problem solving abilities of lubuklinggau junior high school students]. jurnal pendidikan matematika raflesia, 2(2), 118-129. https://doi.org/10.12973/iejme/3966 https://doi.org/10.5539/ijel.v6n5p170 https://doi.org/10.24815/jdm.v8i1.19898 https://doi.org/10.1016/j.kjss.2016.06.001 https://doi.org/10.22342/jme.8.1.3324.85-94 https://doi.org/10.29333/iejme/5721 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p45-54 45 the difference of students’ mathematical achievement by using guided-discovery and cooperative learning model jigsaw type anna angela sitinjak 1 , herman mawengkang 2 1,2 university of sumatera utara, medan, jl. bioteknologi no.1, medan, north sumatera, indonesia 1 annaangelasitinjak@gmail.com, 2 hmawengkang@yahoo.com received: october 25, 2017 ; accepted: january 24, 2018 abstract the type of this study is a quasi-experiment study with its purpose to know any difference in students’ achievement in mathematics which using the model of guided discovery learning with cooperative learning model jigsaw type. the population of this study is all students in sma n 3 p. siantar. the sampling technique applied was cluster random sampling. the experimental class i that chosen is x-1 consisted of 36 students, meanwhile, the experimental class ii that chosen is x-6 consisted of 36 students. the instrument used to measure the students’ mathematics achievement was an essay test. the normality test used was lilliefor’s test, get that data is normal and the homogeneity test by using fisher test, get that data is homogeny. the data analysis technique was t-test at the level of significance α = 5%.the study result showed that there is the difference of students’ achievement in mathematics which using the guided discovery learning model with cooperative learning model jigsaw type in grade x sma n 3 p. siantar where obtained = 2.504 at  = 0.05 and = 1.995, then . keywords: guided discovery, jigsaw, mathematics achievement. abstrak penelitian ini termasuk pada penelitian eksperimen tipe kuasi dengan bertujuan untuk mengetahui perbedaan prestasi siswa terhadap matematika yang menggunakan model pembelajaran penemuan terbimbing dengan model pembelajaran koperatif tipe jigsaw. populasi penelitian adalah siswasiswa sma n 3 p.siantar. dengan menggunakan “cluster random sampling”, dipilih sampel untuk kelas eksperimen i adalah x-1 dengan 36 siswa dan kelas eksperimen ii adalah x-6 dengan 36 siswa. instrumen yang digunakan untuk mengukur prestasi siswa berbentuk esai. data yang diteliti adalah normal dan homogen setelah diuji dengan liliefors dan fisher. teknik analisis data menggunakan ttest dengan  = 5%. hasil penelitian ini menunjukkan bahwa ada perbedaan prestasi siswa terhadap matematika yang menggunakan model pembelajaran penemuan terbimbing dengan model pembelajaran koperatif tipe jigsaw dimana diperoleh = 2.504 at  = 0.05 and = 1.995, maka . kata kunci: hasil belajar matematis, jigsaw, penemuan terbimbing. how to cite: sitinjak, a. a., & mawengkang, h. (2018). the difference of students’ mathematical achievement by using guided-discovery and cooperative learning model jigsaw type. infinity, 7 (1), 45-54. doi:10.22460/infinity.v7i1.p45-54. mailto:hmawengkang@yahoo.com sitinjak & mawengkang, the difference of students’ mathematical achievement … 46 introduction mathematics is the science which is very important to be mastered, because it is indispensable in daily life. in order to achieve the goals of learning mathematics, it is needed the role of various components such as: students, teachers, learning indicators, subject content, learning model, methods, media, and evaluation. teacher as one component of teaching and learning activity has a very important role in achieving the learning objectives and determine the success of the educational process. teacher must be able to motivate their students to engage in the teaching and learning process. unfortunately, it is not easy to motivate students, especially senior high school students to be active in teaching and learning activities because teachers often apply the traditional learning model. so most of the teachers directly provide mathematical formulas to the students and the students only see and memorize the formulas. so do not be surprised if they think math is very bored and dreaded. they think if they cannot memorize formulas they will get low score at math test and worse when they just memorize formulas without knowing the actual math concepts (how to get the formula). as the result, when teachers give a different question from the example, the students are confusedand fell difficult to solve it. so the math scores of students are still low. martunis, ikhsan & rizal (2014) says that to study mathematics is required a good understanding of the concept in which in order to form the new concept understanding, necessary understanding of the concept before. however, as revealed by hendriana, hidayat, & ristiana (2018) that many children after studying mathematics, simple part was much he did not understand, because many of the concepts are misunderstood, which means that the students' understanding of the concept of the low. the learning and teaching process which is implemented by teacher at the class is classically and only rely on textbooks with a teaching method that emphasizes the process of memorizing rather than understanding the concept. so that when students are given problems or test, will have difficulty in solving and can get low score. the low of students’ achievement for mathematics in indonesia is also proved from the results of international research (hidayat, 2017). this problem is also occurred in sman 3 pematangsiantar. the math teacher of sma n 3 pematangsiantar complained that the score average of daily math test is still around 60 whereas kkm (minimum passing criteria) in that school is 75. it indicates that it does not achieve the value of kkm for math. based on data obtained from dkn in grade x sma n 3 pematangsiantar, the average of students’ achievement for mathematics also has not been satisfactory that is just 61.7. one of the ways is refinement to the arranging and application of learning models used by the teacher. the learning model which is appropriate, effective and making the students closed to the teacher will make students enjoy to study and be more active, so students can improve their understanding of mathematics. from some existing learning models, one way to deal with the above problem is by guided-discovery learning model. discovery learning can be defined as the learning that takes place when the student is not presented with the subject matter in the final form, but rather is required to organize it himself (janssen, westbroek, & van driel, 2014; mayer, 2004; wagensveld, segers, kleemans, & verhoeven; 2015). discovery occurs when individuals are involved, especially in the use of their mental processes to discover some concepts and principles. for example, the teacher presents a volume 7, no. 1, february 2018 pp 45-54. 47 problem and students solve the problem until they find out the interrelationship. but because students in high school are not accustomed to find out own the solving of the problems presented, they still need the guidance of the teacher and the model is known as the guideddiscovery learning models. so that, the closeness between teachers and students keeps well (herman, 2016; rahmawati, fitriana, & setiawan; 2017). cooperative learning model also can make students to be active. cooperative learning enables skills in working as teams, skills that are in dire demand in the workplace. jigsaw is type of cooperative learning model in which each student becomes a member of two groups, namely the member of the home group and the member of the expert group so that students do not get bored because the discussions during the lesson they not only meet in one group. jigsaw cooperative learning model makes every student to be responsible and foster a desire / effort to understand the parts of the lessons to be learned and deliver the material to the other group members. so that students can develop the positive relationships among his friends who have different capabilities, to help friends who have difficulty in understanding mathematical concepts and improve self-esteem of student (huda, 2016; sari, 2017). based on the above description, the researcher is interested in conducting research with the title "the difference of students’ mathematical achievement by using guided-discovery and cooperative learning model jigsaw type” and the purpose of this research is to know any difference of students’ achievement in mathematics which using the model of guided discovery learning with cooperative learning model jigsaw type. method type of research used in this study is a quasi experiment because students who are the subject of research have been formed in a particular group (in a class) so that the student may not be randomly selected. as a result, this type of experiment cannot be used to control the external variables that affect the doing of the experiment. in this study, two groups are randomly selected and then given different treatment, namely experimental class 1 taught by guideddiscovery learning model and experimental class 2 taught by cooperative learning model jigsaw type. so the design of this experimental research is the post-test two experimental groups design (cohen, manion, & morrison, 2013). this research was conducted at sma n 3 pematangsiantar with the population in this study is all of students at sma n 3 p.siantar registered in even semester in the academic year 2014/2015 and selected sample randomly which taken by cluster random sampling technique is class x-1 (36 students) as an experimental class i taught the guided-discovery learning model and class x-6 (36 students) as an experimental class ii taught the cooperative learning model jigsaw type. the sample is taken from population by assumption that the characteristics from both of classed are equal. research instrument used is test which was tested the validity and reliability, to measure students’ achievement in the cognitive domain at level knowledge, comprehension, application. to analyze the experimental result, would be used the test score of both of group. sitinjak & mawengkang, the difference of students’ mathematical achievement … 48 results and discussion results based on the data of students’ mathematics achievement who taught by guided-discovery learning model, the minimum score is 76 and the maximum score is 98. based on the calculation, the mean score is 85.69, the variance is 46.33, the standard deviation is 6.81.the frequency of students’ mathematics achievement in experimental class i can be shown on table 1. table 1. the students’ mathematics achievement in experimental class i interval predicate criteria total precentage (%) 96-100 a very high 3 8.33 90-95 a high 8 22.22 86-89 b high 6 16.67 80-85 b low 9 25.00 75-79 c low 10 27.78 <75 c very low 0 0.00 total 36 100.00 the data of distribution table can be drawn in bar diagram (histogram) that can be seen on figure 1. figure 1. the histogram of students’ mathematics achievement in experimental class i from the above diagram can be seen that the students’ mathematics achievement taught by guided-discovery learning model is not lower than kkm value. but, this following is table 2 which showing the students’ achievement for each indicator: volume 7, no. 1, february 2018 pp 45-54. 49 table 2. the students’ mathematics achievement in experimental class i for each indicator no indicator ni si st sa pa pi 1. determine trigonometri ratios formulas of right triangle 8 6 216 178 82.41 82.41 2. determine reverse formula of trigonometry 2 10 360 357 99.17 99.17 3. use trigonometri identity to solve problems 4 8 288 262 90.97 88.31 4. 5 6 216 185 85.67 5. determine the value of trigonometric ratios for specific angle 1 8 288 264 91.67 93.89 6. 7 10 360 346 96.11 7. determine the value signs of trigonometry in all quadrants 6 10 360 210 58.33 58.33 8. use trigonometric ratios formula to solve problem in real life 11 12 432 336 77.78 85.84 9. 12 10 360 338 93.89 10. determine trigonometric ratios formula of related angles 3 8 288 204 70.83 80.25 11. 9 6 216 185 85.65 12. 10 6 216 182 84.26 note: ni = number of item si = score maximum for an item st = score total maximum of class for an item (36 x si) sa = score total which is achieved by students for an item pa = students’ achievement for an item pi = students’ achievement for each indicator from above, we can see that students’ achievement for indicator which is lower than 75 is determine the value signs of trigonometry in all quadrants with just achieving 58.33. this indicator belongs to sub matter of the value sign of trigonometric ratio. based on the data of students’ mathematics achievement who taught by cooperative learning model jigsaw type, the minimum score is 61 and the maximum score is 94. based on the calculation, the mean score is 81.03, the variance is 78.31, the standard deviation is 8.85.the frequency of students’ mathematics achievement in experimental class ii can be shown on table 3. sitinjak & mawengkang, the difference of students’ mathematical achievement … 50 table 3. the students’ mathematics achievement in experimental class ii indicator predicate criteria total precentage (%) 96-100 a very high 0 0 90-95 a high 6 16.67 86-89 b high 9 25.00 80-85 b low 8 22.22 75-79 c low 2 5.56 <75 c very low 11 30.55 total 36 100.00 the data of distribution table can be drawn in bar diagram (histogram) that can be seen on figure 2. figure 2. the histogram of students’ mathematics achievement in experimental class ii from the above diagram can be seen that the students’ mathematics achievement taught by cooperative learning model jigsaw type, few of students get the score <75 with the total of 11 students (30.56%). but, this following is table 4 which showing the students’ achievement for each indicator: table 4. the students’ mathematics achievement in experimental class ii for each indicator no. questions indicator ni si st sa pa pi 1. determine trigonometri ratios formulas of right triangle 8 6 216 188 87.04 87.04 2. determine reverse formula of trigonometry 2 10 360 343 95.28 95.28 3. use trigonometri identity to solve problems 4 8 288 188 65.28 62.97 4. 5 6 216 131 60.65 volume 7, no. 1, february 2018 pp 45-54. 51 no. questions indicator ni si st sa pa pi 5. determine the value of trigonometric ratios for specific angle 1 8 288 250 86.81 93.41 6. 7 10 360 360 100.00 7. determine the value signs of trigonometry in all quadrants 6 10 360 154 42.78 42.78 8. use trigonometric ratios formula to solve problem in real life 11 12 432 350 81.02 82.18 9. 12 10 360 300 83.33 10. determine trigonometric ratios formula of related angles 3 8 288 258 89.58 85.88 11. 9 6 216 171 79.17 12. 10 6 216 192 88.89 note: ni = number of item si = score maximum for an item st = score total maximum of class for an item (36 x si) sa = score total which is achieved by students for an item pa = students’ achievement for an item pi = students’ achievement for each indicator from above, we can see that students’ achievement for indicator which is lower than 75 is use trigonometric identity to solve problems with just achieving 62.97 and determine the value signs of trigonometry in all quadrants. this indicator belongs to sub matter of trigonometric identity and the value sign of trigonometric ratio. based on above data, it is done testing hypothesis by using t-test. after testing the requirements of data analysis, it is known that the data used normal distribution and homogeneous. therefore, testing the average difference of the two experimental classes using t-test with data normal and homogeneous.summary of the study hypothesis test calculations results are presented in the following table 5: tabel 5. summary of t-test result note experimental class i experimental class ii n 36 36 ̅ 85.69 81.03 46.33 78.31 1.995 conclusion ho is rejected from the table 5, it can be known that = at  = 0.05 and = 1.995, then thus ho is rejected. it means that there is the difference of students’ achievement in mathematics which using guided-discovery sitinjak & mawengkang, the difference of students’ mathematical achievement … 52 learning model with using cooperative learning model jigsaw type in grade x sma n 3 p. siantar. discussion mathematics achievement test is given to both of classes after taught two different learning model to determine how the results of student achievement in the two classes after being given treatment. from the research, the average obtained by experimental class i is 85.69 and the experimental class ii is 81.03. then testing the hypothesis by using t test and obtained = 2.504 at  = 0.05 and = 1.995, then (ho is rejected). it means that there is the difference of students’ achievement in mathematics which using the guided discovery learning model with cooperative learning model jigsaw type in class x sma n 3 p. siantar. this can be accepted because the guided discovery learning model enables students to discover, think self on the material being studied from the beginning to the end of the meeting. when student does not understand a subtopic and appears curiosity about it then he can ask a friend beside him or teacher, so that student will take seriously the explanation of a friend or teacher. as a result, student will have a more meaningful learning experience thus will take longer to remember the material they have learned. guided discovery learning model makes students are actively involved in the process of learning and the topics are usually intrinsically motivating, the activities used in discovery contexts are often more meaningful than the typical classroom exercises and textbook study, students acquire investigative and reflective skills that can be generalized and applied in other contexts and also builds on the students’ prior knowledge and experience (sabidin, ismail, tasir, nihra, & said, 2014; yuliana, tasari, & wijayanti, 2017). while the experimental class ii, there are students who are less able to explain to home group members about subtopic which he is mastered, so that other members of the group are lack understand it, so need more time. in addition, in jigsaw group, there is advanced student dominates in discussing subtopic given, so that other students just record the answers without understanding the subtopic. this is same with revealing of sugandi (2013) that the cooperative learning model jigsaw type needs the ample research resources to complete their project, need to spend more time helping the less advanced students in jigsaw group, and the dominant students might try to control the jigsaw group so, other students do not understand the subtopics it and less able to explain it to his friend in home group. however, in this study that there is weaknesses of researcher namely in the learning process, researcher can allocate time less well in both classes. because the experimental class i, there were students who did not understand a subtopic and asked the teacher but as a brief discussion, the teacher was not able to guide the students. in the experimental class ii, brief time discussing was conducted by a group of experts and they were not able to learn how to explain to the home group members about subtopic mastered because not all students have a great ability to give an explanation to his friends so easily understood. volume 7, no. 1, february 2018 pp 45-54. 53 conclusion based on the result of research, it can be concluded that: (1) there is difference of students’ achievement in mathematics which using the model of guided discovery learning with cooperative learning model jigsaw type in class x sma n 3 p. siantar; (2) students who are taught by guided discovery learning model have the higher score that by cooperative learning model jigsaw type. references cohen, l., manion, l., & morrison, k. (2013). research methods in education. london: routledge. hendriana, h., hidayat, w., & ristiana, m. g. (2018). student teachers’ mathematical questioning and courage in metaphorical thinking learning. in journal of physics: conference series (vol. 948, no. 1, p. 012019). iop publishing. herman, t. (2016). pengaruh penerapan model pembelajaran discovery learning terhadap peningkatan kemampuan berpikir kritis matematis dan self confidence siswa kelas v sekolah dasar. eduhumaniora: jurnal pendidikan dasar, 7(2), 140-151. hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. huda, m. k. (2016). penerapan pembelajaran kooperatif tipe investigasi kelompok untuk meningkatkan kemampuan pemecahan masalah matematika siswa pada materi persamaan garis lurus. infinity journal, 5(1), 15-24. janssen, f. j., westbroek, h. b., & van driel, j. h. (2014). how to make guided discovery learning practical for student teachers. instructional science, 42(1), 67-90. martunis, m., ikhsan, m., & rizal, s. (2014). meningkatkan kemampuan pemahaman dan komunikasi matematis siswa sekolah menengah atas melalui model pembelajaran generatif. jurnal didaktik matematika, 1(2), 75-84. mayer, r. e. (2004). should there be a three-strikes rule against pure discovery learning?. american psychologist, 59(1), 14. rahmawati, f., fitriana, l., & setiawan, r. (2017). eksperimentasi model pembelajaran kuantum dan discovery learning terhadap prestasi belajar ditinjau dari motivasi belajar siswa pada materi aturan sinus, kosinus, dan luas segitiga di sma negeri 5 surakarta. jurnal pendidikan matematika dan matematika solusi, 1(6), 82-91. sabidin, z., ismail, z., tasir, z., nihra, m., & said, h. m. (2014). a meta-analysis: pedagogical strategies for teaching mathematics among aboriginal students. in international education postgraduate seminar 2014 (p. 552). sari, d. m. (2017). analysis of students’mathematical communication ability by using cooperative learning talking stick type. infinity journal, 6(2), 183-194. sugandi, a. i. (2013). pengaruh pembelajaran berbasis masalah dengan setting kooperatif jigsaw terhadap kemandirian belajar siswa sma. infinity journal, 2(2), 144-155. wagensveld, b., segers, e., kleemans, t., & verhoeven, l. (2015). child predictors of learning to control variables via instruction or self-discovery. instructional science, 43(3), 365-379. sitinjak & mawengkang, the difference of students’ mathematical achievement … 54 yuliana, y., tasari, t., & wijayanti, s. (2017). the effectiveness of guided discovery learning to teach integral calculus for the mathematics students of mathematics education widya dharma university. infinity journal, 6(1), 01-10. infinity journal of mathematics education p–issn 2089-6867 volume 6, no. 2, september 2017 e–issn 2460-9285 doi 10.22460/infinity.v6i2.p157-168 157 analysis of didactical contracts on teaching mathematics: a design experiment on a lesson of negative integers operations nyiayu fahriza fuadiah 1 , didi suryadi 2 , turmudi 3 1 universitas pgri palembang, jl. jend a. yani lorong gotong royong no. 9/10, palembang, north sumatera, indonesia 2,3 universitas pendidikan indonesia, jl. setiabudi no.229, bandung, west java, indonesia 1 nyiayufahriza@univprgri-palembang.ac.id, 2 ddsuryadi1@gmail.com, 3 turmudi@upi.edu received: july 11, 2017 ; accepted: september 11, 2017 abstract this paper presents the analysis of teaching of math that took place in the classroom to identify the characteristics of didactical contracts that occur as part of a didactical situation. the design experiment was conducted on 31 seventh grade students and a mathematics teacher on negative integer operations lessons. the researchers analyzed the ongoing learning, how the didactical situation stages evolve in the teacher-student interaction that allows the formation of new knowledge or concepts in the student and how the teacher organizes the responsibility for achieving the learning objectives. the analysis showed that students, at an early stage, can perform negative integers operations by utilizing the basic concepts they get in primary school. this concept can bridge into a new knowledge that identifies the properties of integer count operations and builds power of mind on problem-solving. keywords: didactical contract, didactical situation, negative integers operations. abstrak artikel ini menganalisis pengajaran matematika yang berlangsung di dalam kelas dalam mengindentifikasi karakteristik kontrak didaktis yang terjadi sebagai bagian dari situasi didaktis. disain eksperimen dilakukan terhadap 31 siswa kelas 7 dan seorang guru matematika pada pelajaran operasi bilangan bulat negatif. peneliti melakukan analisis terhadap pembelajaran yang berlangsung, bagaimana tahap-tahap situasi didaktis berkembang dalam interaksi guru-siswa yang memungkinkan terbentuknya pengetahuan baru atau konsep pada siswa dan bagaimana guru mengorganisir tanggung jawab untuk mencapai tujuan pembelajaran. analisis menunjukkan bahwa siswa, pada tahap awal, dapat melakukan operasi hitung bilangan bulat negatif dengan memanfaatkan konsep dasar yang mereka dapatkan di sekolah dasar. konsep ini dapat menjembatani menuju pengetahuan yang baru yaitu mengidentifikasi sifat-sifat operasi hitung bilangan bulat dan membangun daya pikir pada pemecahan masalah. kata kunci: kontrak didaktis, situasi didaktis, operasi bilangan bulat negatif. how to cite: fuadiah, n. f., suryadi, d., & turmudi (2017). analysis of didactical contracts on teaching mathematics: a design experiment on a lesson of negative integers operations. infinity, 6 (2), 157-168. doi:10.22460/infinity.v6i2.p157-168 fuadiah, suryadi, & turmudi, analysis of didactical contracts on teaching … 158 introduction mathematics learning is a process of interaction between teachers and students that involve the development of thinking patterns and processing logic in a learning environment that deliberately created by teachers. good interaction between teacher and student describes the learning process can be done well. the interaction among teachers and students is a social activity that does not escape the constraints that can come from both. brousseau (2002) refers to the obstacles of the theories conveyed by bachelard in 1938 and piaget in 1975 on "errors", that mistakes and failures play a role that is not simple. this type of mistake is unpredictable, called obstacles. this error is a part of the acquisition of knowledge. it is also the underlying theory of the didactic situation, namely obstacles. the analysis of this constraining problem becomes important to be expressed in order to better optimize the construction of new knowledge processes and reduce the obstacles that occur. analysis of learning problems during this time is more emphasized on students while the analysis of how the teachers teach has not received more attention. both these analyzes can synergize in designing the learning in accordance with the conditions and needs of students. analysis of teaching can improve teacher strategy in future teaching, priority material emphasis, more reproduction of time for a concept, the type of task that must be given to students, anticipate the questions and responses that may arise, and teacher’s control to know the extent to which the learning objectives have been achieved. these things are teacher's responsibility to plan effective learning. we can say that the teacher's job is not easy. the greatest role of math teachers in the formation of new knowledge on the students is ‘to turn on’ mathematics through creating situations in which they are planning to teach (sarrazy & novotna, 2013). teaching means to create conditions in which something new may emerge. this creation is central to the teacher's work: to create problems and solving the problems. teachers and students have their respective responsibilities that cannot be exchanged. this is what didactics of mathematics calls the didactical contract. didactical contract the concept of the didactical contract is undoubtedly one of the best known concepts of didactics of mathematics (sarrazy & novotna, 2013). didactical contract was described by brousseau (2002) as a set of rules that define the responsibilities of teachers and students in an interaction of teaching and learning. further, didactical contract has meaning that 'teachers are required to teach and students must learn', teachers assign tasks and students perform these tasks. didactical contract is also defined as the rules of the game and strategy in a didactic situation. it is a justification for the situation presented by the teacher. situational changes allow for the modification of contracts in which new situations occur (brousseau, 2002). didactical contract is divided into two categories: devolution contract that teachers organize student’s mathematical activities whose respond to the activities, and institutionalization contract that students propose the outcomes they get and the teachers provide directions that match the knowledge reference (hersant & perrin-glorian, 2005; brousseau, sarrazy & novotna, 2014). the process of devolution and institutionalization then was introduced to relate the dimensions of acculturation and adaptation of educational endeavor even though they were under the responsibility of the teacher. according to artigue, haspekian & corblinlenfant (2014) the devolution process is the process of negotiating with teachers through a volume 6, no. 2, september 2017 pp 157-168 159 didactical contract which temporarily allows for the transfer of responsibilities regarding the teacher's knowledge goals to the students. through devolution, teachers make their students accept math responsibility to solve problems without ignoring didactic goals, and maintaining them, creating conditions that should be a means of learning through adaptation. through institutionalization, teachers help students to connect their contextual knowledge has been built in an adidactical situation in accordance with the targets of knowledge to be achieved and thus the teacher re-assigns decontextualization and transforms into "savoirs" (brousseau (2002) distinguished between "knowledge" and "knowings" knowledge is the "connaissances" of individual cognitive awakening, while knowings are "savoirs", the cognitive construct that appears in the social aspect) furthermore, the idea of didactical contract has been developed to differentiate some types of contracts. the level of didactic contract structure proposed by hersant and perrin-glorian (2005) were: macro-contract, meso-contract, and micro-contract. macro-contract primarily relate to the purpose of teaching, meso-contract with the realization of an activity, such as exercise resolution, while the micro-contract corresponds to an activity focused on the unit of mathematical content, eg concrete questions in practice. chevallard and barquero (arias & araya, 2009) introduced different interactions characterizing the practice of adidactical potential in a potential adidactical contract. this type of contract offers new responsibilities to the student, an important responsibility to identify. this form of accountabilities required students to explain their suggestions and contribute to rebuilding the cognitive path that leads them to the learning objectives. teachers do not need to answer all questions and they should contribute to ask questions to students. they must also develop the skills to argue, in properly communicating ideas and knowing about learning in other ways (metacognition). an important aspect set in the potential adidactical contract is that teachers (with students) make clear rules in applying these skills in other contexts. students' difficulties on negative number integer is the first number faced by students that require reasoning with numbers less than zero which cannot be modeled in real (stephan & akyuz, 2012). in mathematics education, students' difficulties in negative numbers are often found (vlassis, 2008; bellamy, 2015). some articles detailed outlines the struggle of mathematician with integers, especially with the meaning of number less than zero (hefendehl-hebeker, 1991; gallardo, 2002). bishop, lisa, philipp, whitacre, schappelle, & lewis (2014) conducted a series of studies to identify cognitive obstacles about students' perceptions in negative numbers. results of the research shown that students have problems in understanding negative numbers because of they assume that result of addition cannot be smaller and result of subtraction cannot be greater. this opinion is based on their perception that number is a real so that no number which is "less than nothing" or below zero (bishop, et. al, 2014). a study conducted by lamb, bishop, philipp, schappelle, whitacre, & lewis (2012) found that only about a quarter of high school students interviewed were correct in identifying – (-4) as greater than -4. furthermore, they did not realize that they have enough information to determine which is greater than -x or x. seng (2012) also revealed that students' difficulties in arithmetic operations involving negative numbers become one of the major obstacles in understanding the concept of algebra. the result of the error analysis that he studied on grade 7 students showed that students had difficulties in integer manipulating, which is the basic skill in solving algebraic expressions. fuadiah, suryadi, & turmudi, analysis of didactical contracts on teaching … 160 this error came about because students applied the rules in arithmetic incorrectly that they have learned, students more likely use procedural approaches in manipulating negative integers. research questions and objectives based on the described descriptions and theoretical framework, the researchers focused on the activities undertaken by the teacher, how the teacher managed the class by creating anticipation of the student's responses, and how the learning trajectory passed by the students. all of these questions will lead to how the characteristics of the didactical contract in the learning of the counting operation of negative numbers. this research was expected to answer the question how didactical contract phenomenon in a learning process. analysis of this teaching is expected to be used as a benchmark teacher in teaching to improve the quality of learning next. method participant arithmetic operations of negative integer in this research is a part of the hypothetical learning trajectory (hlt) that was designed by researchers in negative integers material. this study is an implementation of hlt in phase of design experiment after pilot experiment. implementation of the design was carried out in 31 students in one of the 7th grade in a junior high school in palembang (11-12 years old) and a mathematics teacher as a model teacher (5 years teaching experience) in the odd semester of the year academic 2016/2017. the first researcher took a role as an observer to observed the learning activities directly and record all the didactical activities within the classroom that can be observed. data collection and analysis three cameras were used to record all learning activities. students and teachers activities then were transferred into a continuing conversation transcript through the coding process to identify the characteristics of the didactical contract. all data were analyzed descriptivel y qualitatively to get a comprehensive picture of the learning process, how the teacher manages the learning, and how the student process finds new ways in addition and negative integer reduction operations. the learning process was described based on the stage of didactical situation. results and discussion the learning was done in two stages, 1) simple problem solving on integers, and 2) integer count operation. the whole learning was done in four learning activities (see table 1). the learning objective is the student can use the counting operation and determine the result exactly in solving the problem (macro-contract). table 1. learning scheme of operations of negative integers learning phase goal learning activity simple problem solving know that in the operation of a number can produce positive or negative numbers simple addition and subtraction of integers the used of addition and subtraction for simple problem solving. volume 6, no. 2, september 2017 pp 157-168 161 learning phase goal learning activity operations of integers can perfom the addition and subtraction involving negative integers the concept of operations with tools the concept of operations without tools activity 1: simple addition and subtraction of integers this section was the initial stage of learning in which the didactical contract is implicitly emphasized on the "call the concept back" on the student, especially the addition and subtraction that students have learned when they were in elementary school, including integer operations, and student knowledge that the results of the operation of numbers can be either positive or negative. looking back at the extent students mastered this concept will greatly help teachers create a didactical situation for the next learning at a higher level. this process is a devolution contract whereby the teacher assigned responsibility to the student to solve the problem as a learning tool through the adaptation process. one of the problems and answers of student 1 can be seen in figure 1. figure 1. the answer of student 1 translate: a ladder has 15 steps. amira is on the 3rd step, then up 9 steps up. because there was a pencil that fell, amira down 6 flights of stairs down. on what steps is amira now? the mistake at student 1 was the operation that is applied to answer the problem about the ladder. it should be amira positions before she was down is 9 + 3. the answers like this also happen to some other students. the purpose of the teacher that students can make appropriate counting operations for this simple problem is failed so that teacher needed to straighten out the concept of students. as an anticipation of these student errors, teacher directed students using vertical numbers to represent the staircase then provide some questions that lead the students. the use of number line as a model has known in general by the students. didactical contract is the process of negotiation that allows the transfer of knowledge goals from the teacher to the students (artigue, haspekian, & corblin-lenfant, 2014). activity 2: use of operation of integer to solve the problem. this activity offered the use of integer operations in solving the problems. the situation that will built was how students can apply the concept of counting operations in solving problems in everyday life with the construction of correct thinking. this conditioning is very important to improve students' thinking structures at a later stage with appropriate instructional design. here are the problems given to students: fuadiah, suryadi, & turmudi, analysis of didactical contracts on teaching … 162 here's the answer of one student (student 2), figure 2. the answer of student 2 translate: mr. andi's money rp. 250.000, mr. cahyo’s money rp. 150.000, rp. 250.000,+ rp. 100.000,= rp 350.000,(mr. andi must add rp. 100.000,to be rp. 350.000,-). he can borrow it to mr. cahyo. student 2's answer show that he did not take into account the loan as one of mr. andi's balance factors. this loan should be subtraction first from savings so that the remaining balance was final. the process of filing a problem to construct new knowledge (which will later be used to solve a problem) sometimes does not guarantee a didactical situation under a fully undertaken didactical contract (hersant & perrin-glorian, 2005). this is what will be the basis when the responses given by the student show that the student has not been able to perform the operation exactly, then the teacher action is create a new didactical situation in accordance with the needs of the students. in this case a meso-contract agreement was adopted through a new situation, ie the teacher provided help by elaborating the problem so that it is easily solved by the students. in the ordinary learning process, an adidactical situation is sometimes rare, but some situations have the potential to occur (adidactical potential). it is said to be 'potential' because teachers can get involved managing the situation, evaluating students' answers without waiting for students to react to feedback from the milieu. brousseau (2002) explained that in a didactical situation, if the teacher feels a failure in the learning process, the student does not meet the expected learning goals, so the teacher is said not to meet student expectations implicitly. students will be 'complain' because they cannot solve the problem given by the teacher. this situation leads to a conflict in the teacher, why this is happen. conflict experienced by teachers, negotiation, and the search for a new contract will continue the didactic relationship through a new didactical situation. in this case the teacher assumed that previous learning and new conditions bring the student to new learning possibilities. this is done by the teacher after, which is designing learning calculation operations so that students can understand the concept of counting operations and not just memorize the procedure. this learning design will carried out in the next learning activity that is the counting operation of negative integers. mr. andi and mr. cahyo have savings of rp. 250.000, and rp. 150.000, -. besides, they also have a loan, mr. andi rp. 175.000, and mr. cahyo rp. 50.000, -. how is the way for mr. andi's balance to be rp. 350.000,? volume 6, no. 2, september 2017 pp 157-168 163 activity 3: the concept of addition and subtraction involving negative number with tools in this learning activity, students were not faced with the usual problems but learning was directed entirely by the teacher at an early stage with the aim of establishing new knowledge about the operation of a negative integer (micro-contract). the concept of operation was built using the tools in the form of a jar containing green and red candies, each of which amounts to 25 pieces. students were divided into 4 groups based on the seating row. each group got a jar of candies. these candies illustrate the positive and negative numbers with the illustrations shown in figure 4. a. addition b. subtraction figure 3. operations of addition and subtraction with a jar and candies implicitly, students were introduced to the concept of inverse in addition through one red candy and one blue candy equal to zero. using the concept of summation that summing is a "batch" of two quantities, students will later realize that the sum can also be negative. the concept of subtraction that students understand as "take away" was still used here. in the construction of other concepts built on this activity, the teacher does not simply inform that "negative is negative as positive" but how students find the concept that a (-b) will be equal to a + b. students in group 2 perform a operation for 4 (-8) by including 4 blue candies (representing positive 4) and 8 red candies (representing negative 8), then issuing 8 red candies (subtraction). once calculated results obtained are 4 blue candies or equal to 4. response as already predicted by previous teachers. teacher : what are our previous agreements for reduction operations? students : mmm ... (weak voice) teacher : all right. have you put 4 positive candies into the jar? how about negative 8? surya : eight red candies, ma’am. teacher : that's it? is the same 4 with 4 minus 0? students : it is the same, ma’am teacher : is 8 blue candies with 8 red candies same with zero? akbar : same thing mom, meaning 8 + (-8) = 0 teacher : right. now, there are 4 with 4 blue candies then added 8 blue candies and 8 red candies? students : it’s the same, ma’am teacher : why? surya : because 8 blue candies and 8 red candies are equal to zero. teacher : so what are you going to do? students : (put 8 blue candies and 8 red candies into the jar then take out 8 red candies and count the rest) fuadiah, suryadi, & turmudi, analysis of didactical contracts on teaching … 164 didactical contract can change when students were unable to perform subtraction of negative numbers immediately so that teacher needed to guide students by demonstrating their procedures over and over through some examples (meso-contract). this needed to be done so that the mathematical concepts that are obtained not just memorize it. teacher had been monitoring student work in the group to ensure that learning objectives are well achieved. observations conducted by the researchers showed students did not experience obstacles in performing the operation in this way. the most important part of this stage is that students can perform count operations for larger numbers that may not be represented with concrete objects or number line. with the concept that has been obtained, students are expected to perform abstraction on operations without tools. activity 4: arithmetic operation without tools the concept of arithmetic operations that had been studied was very useful in determining the results of operations for multi-digit numbers. jar and candies were still used as abstract models at the beginning of the calculation, but then gradually no longer use the model directly even though the rules were still used. this method can be easily understood by students compared to using a number line or other tools. teacher assistances were still needed in every stage of mathematical abstraction so that students can actually calculate with the right concept. at the beginning of this activity students are asked to perform operations on 23 – 45. besides the students can count it, the teacher also wanted to implant the concept indirectly that 23 45 is the nature of 23 + (-45). seeing many students who have not given the right answer, the teacher took the initiative to explain the procedure used procedure of subtraction that they have learn (figure 4). so that, figure 4. the operation’s procedure for 23 – 45 to fill the theorems and strategies on the board is one way to show teachers' understanding of mathematical material (kinslenko, 2005). the teachers then gave some questions as assignments for the students. this stage was a validation situation, i.e. students convey their ideas and teachers play a role to bridge their knowledge to achieve the intended knowledge (brousseau, 2002). that was in line with what was stated by kinslenko (2005) that teachers start with what is known and end with the knowledge of mathematics through the construction process. students were required to solve the problem and they make clear and complete explanations of the theory and any way that has been used to solve the problem. almost all students experienced constraints in determining the operating results of 23 + (-32) and 15 (-33). repeated work is required to get the correct answer. teacher : how are you doing? where are you up to? indra : we're still having trouble in 15 (-33), ma’am. teacher : let’s try to remember the previous way, using the candies. how to subtract? fadhil : use candies, maam -45 23 23 -45 23 – 45 = 23 + (-45) 45 +23 -45 -23 +23 -22 -22 volume 6, no. 2, september 2017 pp 157-168 165 teacher : well with this. if less -33 what will we do? students : mmm ... (weak voice) teacher : you must write first 33 and what? bimo : negative 33 (write the numbers +33 and -33 above the number 15) teacher : then, how much should be taken away? students : negative 33. teacher : now what number is left? let’s try to count on that! the errors that occur precisely the starting point in the process of constructing knowledge. the teacher's responsibility in this process is to ensure that students use the right strategy. of course requires patience from both of, teacher and students. arias and araya (2009) described a different type of interaction between students and studentteacher whose characteristics illustrate the adidactical potential practice in a potential adidactical contract. according to arias and araya further, the teachers not answer all the questions and they have to contribute to questioning among students. also, they must develop skills as to argue, to communicate precisely the ideas, to know about other ways of learning (metacognition). in this case, it was necessary for teacher intervention to modify the milieu so that students can improve their understanding of a concept. this type of contract offers new responsibilities to the student, an important responsibility to identify. this form of responsibility requires students to explain their suggestions and contribute to rebuilding the cognitive path that leads them to the learning objectives through the questioning that builds the intended knowledge to the students. the properties of the counting operation is a concept that students get then after the concept of counting operations. knowledge of these traits should not be obtained by memorizing as it has been so far. observations made by the researcher showed that teacher tend to provide information or formulas of operations. consequently there was much misconception on the students because students simply apply the rote without knowing the concept (fuadiah, suryadi & turmudi, 2017). to embed this concept to students, students are given several operations with the same result then students compare between an operations with other operations. thus the student know the properties of the operation, for example, a – b will give the same result as a + (-b) or a (-b) will equal the result with a + b. noteworthy in this study is the student's emphasis on the concept of base-ten number and inverse addition. this concept is indispensable for students to count multiplex numbers, especially negative numbers. mastery in arithmetic operations will affect students in solving problems. the study showed that students who can determine the mathematical model in solving the problem can find obstacles in performing the arithmetic operation (hunges, 1986). at the next level of learning, such as algebra, students can algebraically model correctly but incorrectly in their countdown operations (seng, 2012). based on students' difficulties, vlassis (2004) suggest that the minus sign plays a major role in the development of understanding and using negative numbers. in theory of didactical situation, to improve students' autonomy, teachers should be more likely to have a lesson-learned intervention (brousseau, 2002; manno, 2006; perrin-glorian, 2008; ruthven, laborde, leach, & tiberghien, 2009; artigue, haspekian, & corblin-lenfant, 2014). this situation, argued by laborde and perrin-glorian (2005), adidactical situations are designed to minimize the involvement of the teacher in the learning process. however, in the practice of learning that occurs, teacher assistance is needed as a teacher anticipation of the fuadiah, suryadi, & turmudi, analysis of didactical contracts on teaching … 166 response of students who developed during the learning such as providing questions that can lead students to an understanding. student responses that indicate the learning objectives that have not been achieved will result the didactical contract was broken. therefore teachers anticipated by creating new situations that involve the role of teachers larger. those were expressed by hersant and perrin-glorian (2005) that the process of filing a problem to construct new knowledge (which would later be used to solve a problem) sometimes does not guarantee a didactic situation under a fully undertaken didactical contract. in this activity, as an act of student response, the teacher provided a re-explanation of the meaning of "take away" of the subtraction. teacher's action is to give an example of an easier operation than ever so that the students can understand more easily by recalling previously learned concepts. in a didactical situation where students are given the opportunity to solve problems without teacher intervention, teacher assistance is still needed as a part of didactical contract. conflict experienced by teachers, negotiation, and the search for a new contract will continue the didactic relationship through a new didactical situation (brousseau, 2002). this new situation is then built by the teacher when the student has not been able to perform the counting operation through the correct concept, so that the teacher needs to redirect the possibility of the operation algorithm that students can do. conclusion the analysis of the teaching and learning process on this study that has been going on there are some didactical situational changes in anticipation of the broken didactical contract, namely: 1) creating a new didactical situation to support students' ability in performing the arithmetic operation because the teacher's expectation that the student can perform the counting operation is constrained on negative number, 2) teacher assistance as one of the implementation of didactical contract can be done through the questions that lead can bridge the students to understanding the concept so that not only know the procedure only. action and feedback through a strategy will enable the building of a new knowledge. based on the identification of the characteristics of the didactical contract in this study, there are two types of dominant didactical contract, namely 1) mayeustic socratic contract, occurring in the early stages of the learning process of adaptation where the teacher does not fully dominate but helps the child's learning process through the provision of key questions that exploring the experience and early knowledge of the students to elicit relevance to the concepts to be studied; and 2) potential adidactical contract, in student learning describes the question and does not wait for the answer. also, students are called to answer other students' questions, so teachers are not the only ones who have the right answers. students should explain their suggestions and contribute to rebuilding the cognitive way that leads them to a result (arias & araya, 2009). this type of contract occurs in institutionalization where there is awareness among students about a concept that has become a self-sustaining student (savoirs). analysis of the implementation of the design of learning needs to be done continuously by analyzing the demands and needs of students are very diverse. the diversity needs to be accommodated in learning, because the act of uniformity of students with diverse reality is not wise and proportional. references arias, f., & araya, a. 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(2008). the role of mathematical simbols in the development of number conceptualization: the case of the minus sign. philosophical psychology, 21(4), 555– 570. http://dx.doi.org/10.1080/09515080802285552 https://www.unige.ch/math/ensmath/rome2008/wg5/papers/perrin.pdf https://doi.org/10.3102/0013189x09338513 http://dx.doi.org/10.1007/s11858-013-0496-4 http://dx.doi.org/10.1007/s11858-013-0496-4 http://dx.doi.org/10.5951/jresematheduc.43.4.0428 http://dx.doi.org/10.1016/j.learninstruc.2004.06.012 http://dx.doi.org/10.1080/09515080802285552 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 11, no. 2, september 2022 e–issn 2460-9285 https://doi.org/10.22460/infinity.v11i2.p325-348 325 “flipped classroom type peer instructionbased learning” based on a website to improve student's problem solving tuti azizah, ahmad fauzan, yulyanti harisman* universitas negeri padang, indonesia article info abstract article history: received aug 26, 2022 revised sep 28, 2022 accepted sep 29, 2022 this study aims to develop learning tools in the form of lesson plans, student worksheets, learning videos, and ppt, based on web for first-grade seventhschool material linear equations with flipped classroom-type peer instruction. the development of this web-based learning tool uses the plomp model. the steps for developing web-based learning tools include preliminary needs analysis, curriculum analysis, concept analysis, student analysis, and literature analysis. then product development and assessment. this research was carried out during the product development stage based on needs analysis, curriculum analysis, concept analysis, student analysis, and literature analysis and then self-evaluated. the preliminary analysis obtained information that teachers need learning tools that can help students improve problem-solving skills, students need engaging learning media, students are interested in using learning media using videos with a period of 10-15 minutes, and ppt, accessed in the web. then the product is compiled and evaluated on its own, related to the apparent error in using it. based on the preliminary analysis, learning lessons, student worksheets, learning videos, ppt based on web and selfevaluation have been produced, which can then be validated by experts. keywords: flipped classroom, peer instructions, problem-solving, website this is an open access article under the cc by-sa license. corresponding author: yulyanti harisman, department of mathematics education, universitas negeri padang jl. prof. dr. hamka, kota padang city, west sumatra 25171, indonesia. email: yulyanti_h@fmipa.unp.ac.id how to cite: azizah, t., fauzan, a., & harisman, y. (2022). “flipped classroom type peer instruction-based learning” based on a website to improve student's problem solving. infinity, 11(2), 325-348. 1. introduction mathematics as one of the basic sciences has an important role in the development of science and technology. the learning objectives of mathematics in the decree of the ministry of education and culture no 64 of 2013 are related to the learning objectives set by the nctm (palgunadi et al., 2021). one of the objectives is that students are required to have basic problem-solving skills. therefore, it is very important for students to have the ability to solve problems in order to achieve maximum learning outcomes in accordance with the objectives of mathematics learning (kasem et al., 2018; roheni et al., 2017). in fact, https://doi.org/10.22460/infinity.v11i2.p325-348 https://creativecommons.org/licenses/by-sa/4.0/ azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 326 the mathematical problem-solving ability of students in indonesia still requires improvement and special attention, as shown by the results of the 2018 pisa study; indonesia's mathematics score was 379 below the international average score of 489 and the 2015 timss results for indonesia reached 397 with an international score of 500 (guhn et al., 2014; oecd, 2019; palgunadi et al., 2021). the national council for mathematics teachers (nctm) also mentioned the importance of solving learning problems. according to nctm (palgunadi et al., 2021), the mathematical thinking process in mathematics education includes five basic standard competencies: problem-solving skills, reasoning skills, communication skills, and presentation skills. this low ability results in the low quality of human resources as indicated by low problem-solving abilities. this is because, until now, education has not provided opportunities for students to develop problem-solving skills. students can also integrate realworld problem-solving skills to solve real-world problems and competition (cahyani & setyawati, 2017). indicators of mathematical problem-solving ability in this research are based on the combination and modification of polya (sam & qohar, 2016; siahaan et al., 2019), which are: (1) understanding the problem by identifying the adequacy of the required data; (2) developing a problem-solving plan by presenting the problem in a mathematical model; (3) solving problems by selecting and implementing the strategy; and (4) inferring or interpreting the solution. the process standards, learning must be planned, assessed, and supervised (ramadhani, 2016). on the other hand, with the covid-19 pandemic affecting learning activities, teachers must innovate in learning, including using technology in learning (herliandry et al., 2020). based on observations conducted at one of the junior high schools in padang, it is known that the learning process is carried out 3 days face-to-face and 3 days at home by giving assignments, one hour of mathematics lessons at the school is only done for 25 minutes. assignments are developed from student worksheets which mostly contain routine questions, so students have not been able to develop problem-solving skills well. educators also find it difficult to catch up on materials not understood by students. based on these problems, it is necessary to design effective learning tools that can help students develop their problem-solving skills, according to the characteristics of students and the covid-19 pandemic situation. internet-based learning technology with the flipped classroom model is a solution for transitioning traditional learning to a virtual system. students feel familiar with the materials that will be discussed in class (gariou-papalexiou et al., 2017). in several types of the flipped classroom, peer instruction flipped learning models have the most potential to improve problem-solving skills. this is because when in class students are given learning that begins with contextual problems; students are trained to understand and formulate the problems given through the concept test which will be answered individually so that students will get used to answering the questions given independently. learning tools will be developed later in the form of lesson plans, student worksheets, learning videos, and power point media to put in the web. studies which use web as a means of providing material are still rare, even though web provides complete tools, thus helping students learn comprehensively. based on research conducted by nakamura (2011), the advantage of the web is that there is no limitation on the number of web pages and it is easy to insert new pages to accommodate the needs of students who need to improve or add material. research on the development of learning tools using flipped classroom and its effect on students' mathematical abilities has been carried out by researchers, such as the research conducted by prayitno and masduki (2016) focusing on the development of blended learning media with the flipped classroom model in mathematics education courses. in the research, volume 11, no 2, september 2022, pp. 325-348 327 learning media were produced in the form of learning videos, quizzes, e-modules, and learning videos suitable for students based on expert tests, and students could use these media as a supplement to lecture materials. the effectiveness of the flipped classroom on attitudes and skills in learning mathematics in vocational high schools showed an increase in aspects of students' attitudes and skills in applying the concepts of sequence and series as well as being more active in problem-solving activities (damayanti & sutama, 2016). the development of a flipped classroom-based mathematics learning model in class xi of smkn 1 gedangsari gunungkidul showed that the flipped classroom-based mathematics learning model can maximize learning with one-on-one interaction through learning videos uploaded online and offline (damayanti & sutama, 2016). research shafique and irwin-robinson (2015) on the study of the effectiveness of the flipped classroom in 9 mathematics classes showed that learning at the university becomes more effective when flipped learning is implemented. in addition, the flipped classroom will also create a good learning experience for students. furthermore, hayati (2018) focusing on the theoretical study of the flipped classroom in mathematics learning showed that the flipped classroom learning model can help students learn both inside and outside the classroom, resulting in students being directly involved in the learning process along with the development of information and technology which requires teachers to apply technology in teaching and learning based on the characteristics of mathematics learning. furthermore, research conducted by prayitno and masduki (2016) on a comparative study of the flipped traditional learning model with peer instruction flipped on problem-solving abilities showed that the average value of the problem-solving ability test results for class viii students who received learning using the peer instruction flipped learning model was higher than the average score of the test results from students who experienced learning using the traditional flipped learning model. in addition, research related to the use of flipped classroom model has been carried out by researchers, such as research conducted by thohir et al. (2021) and herliandry et al. (2020). based on the description above, this research is different in the products developed, namely learning tools in the form of lesson plans, student worksheets, learning videos, powerpoint slides, and web which is arranged using flipped classroom type of peer instruction to see the effectiveness of the problem-solving abilities of the first grade of junior high school students. 2. method this research employed the research and development design. product development was conducted through several stages adapted to the model chosen, namely the plomp (plomp, 2013) which has been simplified into three stages as follows (1) preliminary research phase. the analysis activities carried out include needs analysis. the activity carried out is to analyze what is needed and expected for development. the information collection was carried out by the method of interviews. the results of the needs analysis were considered in the design of learning tools in order to achieve learning objectives and meet the needs of students in improving problem-solving skills. in curriculum analysis, the activities carried out were identifying the topic or subject matter, and compiling them in the right order, aiming to study the scope of the material, learning objectives, and the selection of appropriate strategies. the method used is a documentation method using a checklist of learning tools in schools. analysis of learners, the activities carried out were collecting information on the characteristics of students, and adjusting to the preparation of learning materials and tools. the purpose of student analysis is to find out the product that students want and identify the students' understanding of ppt applications. the method used in the azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 328 student analysis stage is the provision of questionnaires. the next is concept analysis; the activities carried out were identifying the main concepts of the material, detailing, and compiling them with concept maps systematically. relevant teaching materials are to be taught based on curriculum analysis. in the literature analysis stage, an analysis was carried out on reading/book resources used at schools to see the suitability of books with the curriculum. the appropriate books will be used as a reference for the preparation of learning tools to improve the problem-solving abilities of the first grade of junior high school'. furthermore, relevant research journals that can be used as the basis for developing products were reviewed. the next was (2) development phase or prototyping phase; in this phase, tools were designed in the form of lesson plans, student worksheets, learning videos, ppt slides, and web-based flipped classroom type peer instruction based on the results of the analysis in the preliminary research phase. the result of the design at this stage is called prototype 1. each prototype was evaluated with reference to the formative evaluation of tessmer's development in figure 1. figure 1. formative evaluation of tessmer development in plomp and nieveen the formative evaluation steps used are outlined as follows: (a) self-evaluation were carried out to double-check the completeness of the device components. the method used was self-evaluation and discussion with colleagues. the result of the revision of the learning device is called prototype 2. (b) expert reviews; expert assessment aims to validate the device by providing assessments and advice according to the expert's field. the validators involved were five people consisting of three mathematics education experts, one linguist, and educational technologist. (c) one-to-one evaluation, the one-on-one evaluation was carried out by asking for suggestions from users of learning devices, namely an educator and three students. students were selected based on high, medium, and low ability levels. the purpose of this individual evaluation is to identify possible errors in the student worksheet, learning videos, and website in the form of material, implementation, and to see the technical volume 11, no 2, september 2022, pp. 325-348 329 quality and practicality of the learning tools compiled. (d) small group evaluation; the revised learning tools produced prototype 3, then an evaluation was carried out involving 6 students selected by mathematics educators. each of the two learners represents a high, medium and low ability group. in this evaluation, aspects of presentation, time allocation, andreadability of the device were assessed. the data collection methods used in small group evaluation are interviews and observations (e) field test; in this stage, a trial was carried out, called a field trial. the goal of this trial is to evaluate learning tools in actual classroom situations. after field tests were carried out, the students were given a practicality questionnaire. (3) assessment phase; the assessment phase was carried out to determine the level of effectiveness of the tools by looking at the process and test results of mathematical problem-solving questions of students who have learned using learning devices. tests were carried out before and after students used the learning tools. the test results were processed based on the rubric of scoring mathematical problem-solving ability. this research was carried out in the stage of preliminary research, product development, and self-evaluation, while expert validation and product effectiveness will be reported in a different article. 3. result and discussion in the preliminary research phase, identification or analysis was carried out to develop a peer instruction type flipped classroom-based learning tool and analyze the limitations of the subject matter to be developed. the purpose of this stage is to establish and define the conditions needed in the development of learning tools. this stage started in july 2022. 3.1. needs analysis results the needs analysis stage is carried out with the aim of producing a flipped classroombased learning device of the peer instruction type that can be adapted to the needs. a needs analysis was conducted by interviewing the seventh-grade mathematics educators. based on the results of the interview, the following conclusions were obtained; (a) learning activities carried out after post-covid are face-to-face learning; (b) the school has applied flipped classroom learning, but the implementation is fully virtual; so, the material given is not explained again in class. (c) educators need learning tools that can help students maximize learning time, due to less learning time in class. (d) the problem-solving ability of students is still low, shown by the way the students understand the problem, plan a solution, complete the settlement, and re-examine the answers. (e) students enjoy learning using technology and using a variety of media. based on the results of the analysis, learning tools are needed to adapt to the learning conditions at one of the junior high schools in padang. at the same time, the tools can help students in solving problems. one of the efforts to overcome the problems in the learning process is to develop a peer-instruction flipped classroom in the form of lesson plans, learning media (ppt, and learning videos), student worksheets, and web as pages for students to access materials designed in such a way with the aim of overcoming problems. 3.2. curriculum analysis results curriculum analysis was carried out by examining the curriculum used at one of the junior high schools in padang. based on the results of the curriculum analysis, it is known that the curriculum used in schools is an independent curriculum in the seven-grade class semester 1. the curriculum began to be implemented on july 17, 2022. curriculum analysis aims to find out what materials about linear equations are presented in the curriculum in azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 330 accordance with the expected competencies, whether the materials are adequate to achieve learning objectives, and whether the materials have been properly ordered. the results of this curriculum analysis are used as a basis for formulating learning objectives in developing linear forecasting topic learning tools based on flipped classroom type peer instruction classes for the first grade of junior high school students. curriculum analysis is focused on analyzing learning outcomes in order to obtain learning objectives that become a reference in the development of lesson plans, student worksheets, ppt slides, and learning videos (see table 1). table 1. curriculum analysis results element learning outcomes learning objectives linear equation at the end of phase d, learners can recognize, predict and generalize patterns in the form of an arrangement of objects and numbers. they can express a situation in an algebraic form. they can use the properties of operations (commutative, associative, and distributive) to produce equivalent algebraic forms. learners can understand relationships and functions (domain, codomain, range) and present them in the form of arrow charts, tables, sets of sequential pairs, and graphs. they can distinguish some nonlinear functions from linear functions graphically. they can solve one-variable linear equations and inequalities. they can present, analyze, and solve problems by using relationships, functions, and linear equations. they can solve a twovariable linear safekeeping system through several ways to problemsolving. 1. state the relationship between two magnitudes (<,>,=,≤,≥) in a problem. 2. understand the correctness of mathematical sentences of equations when letters are substituted with numbers in a problem. 3. determine the solution of an equation without substituting numbers into letters in a problem. 4. solve equations using the properties of equations in a problem. 5. solve equations using the idea of moving tribes in a problem. 6. solve equations in the form of decimals and fractions in a problem. 7. solve problems by using linear equations 8. understand ratio relationships by using linear equations in a problem. 9. solve problems related to ratios by using linear equations the elaboration of learning objectives is carried out so that the materials can be explained in an orderly manner. this aims to make the materials easy to understand by students. regarding inequality, the curriculum guidelines for junior high schools stipulate that "the relationship between the amounts is expressed using the inequality", while the nature and completion of the inequality are studied in the next class. then, the researcher also paid attention to the small school class time, which is 30 minutes for one class hour. therefore, it only studied how it was used between the two-class hour. this was also part of a discussion with educators at one of the junior high schools in padang. 3.3. concept analysis based on the curriculum analysis, there are 9 learning objectives. to achieve the learning goal, appropriate and relevant materials are needed. the results of concept analysis reveal that teaching on the topic of the system of linear equations so far has not developed volume 11, no 2, september 2022, pp. 325-348 331 the ability to build concepts from linear equations and solve these problems. teaching the topic of linear equations so far has been directed to the abstract form of the general form of linear equations without starting from concrete/ contextual problems that can be observed by learners. solving direct linear equations is explained by the steps in the book, such as the method of substitution. based on the results of concept analysis, the topic of this linear equation begins by stating the relationship between two magnitudes. presenting the relationship of two quantities in the form of equations and inequality is an initial concept that students must master before solving the form of linear equations, the use of calculating operations, properties, and the application of linear equations to the form of more complex problem situations. finally, the comparison material on linear equations is an additional concept that aims to provide students with an understanding of problems that uses a comparison ratio, including the ratio of the size of sugar to one cup of coffee. based on the results of the analysis, the main concepts are studied in the material of linear equations and systematically compiled according to the flow of their presentation. the materials and concepts needed in learning linear equations are arranged in the form of concept maps. the concept map can be seen in figure 2. figure 2. concepts maps 3.4. results of student characteristics analysis student analysis was carried out to determine the characteristics of students so that the design of the learning tools is in accordance with expectations in the mathematics learning process. the analysis of students was carried out by interviewing educators and distributing a questionnaire on the characteristics of first-grade seventh school of junior hight school. based on the results of the interview, students in the class have diverse academic abilities consisting of high, medium, and low abilities. the characteristics of the learners analyzed include academic ability, group work ability, experience background, charm for colors and images, and student attitudes. azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 332 based on the questionnaire given to students, some information about the students was collected. the results obtained using the questionnaire can be seen in table 2. table 2. results of student characteristic analysis aspect information conclusion constraints during the learning process learning activities carried out by students at school are going well of the 32 respondents, 20 voted against, 9 voted disapproving, 3 voted in favor, 3 voted in favor the teaching materials/teaching media used by students have varied and attracted students' interest in learning of the 32 respondents, 22 voted disapproving, 8 agreed, and two disagreed. learner abilities students are less able to solve problems in the form of problemsolving. of the 32 respondents, 25 voted in favor, and 5 voted overwhelmingly in favor. learning tools to be developed learners need student worksheets for engaging and easy-to-understand learning 32 respondents voted in favor learners use electronic media to search for material 32 respondents voted in favor students are interested in using learning videos and power point media to learn over and over again at home that can be easily understood 32 respondents voted in favor students are interested in the learning videos and ppt media provided are distributed through the website 32 respondents voted in favor students interested in using the learning tools used can improve problem-solving skills 32 respondents voted in favor based on the answers to the student questionnaire, it is known that students want more interesting learning resources at school; students want more learning time so that when they understand the material better; most students state that they have used and often use smartphones as learning resources, and most students stated that they are not able less able to solve problem-solving problems. based on the analysis of the characteristics during the researcher's time at school, a peer instruction type flipped classroom-based learning tool in the form of lesson study, student worksheets, learning videos, power point media, and the web can help students in learning at home needs to be developed so that the learning time is available more, and in face-to-face, it focuses more on group discussion activities and solving problems related to the materials that can be accessed on the web; also, learning videos and power point media as interesting learning media can motivate students to study harder. in addition, students volume 11, no 2, september 2022, pp. 325-348 333 with individual character prefer to learn individually in understanding the material and can use the learning tool anytime and anywhere. 3.5. literature review literature review; literature analysis was carried out by analyzing the learning resources used at one of the junior high schools in padang. the results of the analysis of learning resources are presented in table 3. table 3. literature analysis results no indicator valuation yes no 1 learning resources obtained from written sources. √ 2 learning resources are obtained from unwritten sources. √ 3 the learning resources used can be found in the library. √ 4 the learning resources used are in accordance with the indicators in peer instruction. √ 5 the learning resources used do not facilitate problem-solving skills. √ 6 the learning resources used are less attractive to students in learning. √ 7 the learning resources used are easy for learners to understand. √ based on the literature analysis carried out, it is known that the books used at one of the junior high schools in padang are mathematics books of the 2013 curriculum and student worksheets. the book and student worksheet have not facilitated students to solve problems and the question exercises presented on the worksheets have not facilitated problem-solving. educators are still looking for teaching materials that are suitable for use in learning, especially those that are in accordance with the independent curriculum but due to the constraints on the cost of procuring books, educators still use the 2013 curriculum books. in terms of appearance, the worksheets used have not been able to attract students' learning interest because the design is simple. in addition, the presentation of materials from the package books used is still difficult for students to understand, and the package books can only be borrowed for use in the school area. the boks cannot be brought home because they are limited. based on the literature analysis, student worksheets will be designed to attract the students and guide them in conducting an investigation and solving problems in learning mathematics, especially linear equation materials. the next is product development based on the analysis, curriculum needs, concepts, and students, a peer instruction-type flipped classroom-based learning tool was designed in the form of student worksheets, learning videos, power point media, and web on the topic of seven-grade linear equations. peer instruction type flipped classroom-based lesson plan is a learning tool specifically designed for junior high school students in seven grades. the lesson plan format is designed to contain components based on flipped classroom-type peer instruction. the flipped classroom-based lesson plan type of peer instruction is designed to consist of three components that have been arranged in the independent curriculum. (a) general identity. general information contains information about the identity of the author, initial competencies, facilities and infrastructure, target students, and the learning model used. (b) core competencies. core competencies contain learning objectives and learning activities. (c) appendix. the attachment contains the learner worksheet. the pictures of the lesson plan can be seen in figure 3. azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 334 figure 3. lesson plan characteristics of student worksheet based on flipped classroom type peer instruction the model used in the student worksheet developed is a peer instruction type flipped classroom learning model that has the following steps: pre-class activities (students watch learning videos at home), class activities where the teacher gives the first question test which should be done individually, discussions between students regarding the answers to the first question, then the second question test which is done in groups, measuring students’ understanding at the end of the lesson; all of which consist of questions from a problemsolving problem. the structure of the student worksheet that has been compiled can be seen in table 4. table 4. student worksheet structure no student worksheet section student worksheet structure 1 introduction 1. cover 2. title page 3. foreword 4. table of contents 5. learning objectives 6. instructions for using student worksheet 7. concept map 2 fill learning materials (meeting 1-5) 3 cover bibliography volume 11, no 2, september 2022, pp. 325-348 335 the writing of student worksheet based on flipped classroom peer instruction type is based on the arrangement of the student worksheet structure that has been made. the writing of student worksheet on prototype 1 that the researchers have designed can be seen in the figure 4. figure 4. student worksheet front cover the front cover page contains the student worksheet title, class, target user, and creator identity. the foreword page view can be seen in figure 5. figure 5. foreword in the foreword section, the introductory preface page contains a general review of the content of the student worksheet, forework acknowledgment from the author, and the azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 336 author's expectations. student worksheet based on flipped classroom type peer instruction compiled is expected to be a guide and help students in learning linear equation materials. the student worksheet usage instructions page can be seen in figure 6. figure 6. instructions for use of student worksheet instructions for use of the student worksheet contain things that need to be considered in studying the student worksheet. on the student worksheet, there is an explanation page for step-by-step peer instruction. this page contains an explanation of the steps of peer instruction which aims to make it easier for students to understand the steps of peer instruction. the step-by-step peer instruction can be seen in figure 7. figure 7. peer instruction steps volume 11, no 2, september 2022, pp. 325-348 337 the concept map contains a sequence of concepts or an overview of the material to be studied. the concept map can be seen in figure 8. figure 8. concepts maps in the mathematical figures of linear equations, to presents the history of mathematical figures of linear equations, related to biographies and experiences related to mathematics. the page view of the mathematical figures of linear equations can be seen in figure 9. figure 9. linear equation figure azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 338 learning materials. the learning materials in the student worksheet are designed in accordance with the peer instruction type flipped classroom component which is summarized as follows. test concept 1 this component is a problem in the form of the first test question regarding a basic concept of the material. learners are given this test to find out the extent to which they learners understand the material being studied. learners are given time to answer the questions individually. the test 1 concept view can be seen in figure 10. figure 10. concept test 1 discussion of test concepts 1 in this component, students are given the opportunity to discuss and argue with each other about the first test questions given. in this stage, the discussion is carried out in groups. after the students’ correct answer to the discussion is more than 80%, it is continued to the second question. the discussion page is shown in figure 11. figure 11. discussion of test concepts 1 volume 11, no 2, september 2022, pp. 325-348 339 test concept 2 in this component, the second test question is given to further strengthen the concepts that have been obtained by the students. the work on this question is carried out in groups. the test 2 concept page is shown in figure 12. figure 12. concept test 2 final assessment at the end of the discussion, students are given a final assessment test related to the evaluation of the linear equation material that has been studied. the final assessment can be seen in figure 13. figure 13. final assessment the design stage of learning videos, power point media, and wesite began by establishing the main concepts so that the preparation of the material can achieve learning objectives. the material is presented in the form of learning videos and also in powerpoint azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 340 media that is adapted to the independent curriculum and can help students develop problemsolving skills. the web was created by using google site which can be accessed at https://sites.google.com/view/tuti-azizahs-math-education/material site. the web that has been designed has several navigations, namely home navigation which contains the student's attendance list, menu navigation which consists of materials for each meeting (ppt slides contain learning objectives, materials, quizzes, and learning videos), and the student worksheet menu at each meeting which can be viewed directly or downloaded by students. here are some views of learning videos, power point media, and website that have been designed. the development of the homepage can be seen in table 5. table 5. learning video view, ppt, and web part footage homepage. the home page is a web start page that contains the identity of the author and there is a student attendance menu. learner attendance page. on this page, students can fill the attendance of the meeting according to the meeting carried out homepage. the home page is a web start page that contains the identity of the author and there is a student attendance menu. https://sites.google.com/view/tuti-azizahs-math-education/material volume 11, no 2, september 2022, pp. 325-348 341 part footage display materials on this page, students can select the material they want to learn at each meeting by clicking on the available meeting icon. meeting materials 1. on this page, students can learn about the material available in the ppt display by clicking on the ppt section. ppt displays. in the ppt, there is an identity menu, concept map, learning (containing materials), quizzes, and teacher profiles. ppt is equipped with audio that explains the functions of the available menus section. azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 342 part footage download ppt materials in addition, students can learn the material on ppt directly on the web. learners can also download ppt by clicking download learning videos. on this page, students can watch the learning video by clicking play. learners can also stop the video by clicking pause. students can also download videos by going to the video link uploaded on youtube student worksheet on this page, learners can view and download student worksheets at each meeting teacher profile on this page contains the identity and biography of the author volume 11, no 2, september 2022, pp. 325-348 343 based on the results of the evaluation with colleagues, there are revisions related to the preparation of menus; then, in the ppt, recorded audio is added by the researcher so that students are not confused in operating the menus on the ppt; then, improvements were made to the animations on the ppt slides, improvements on the explanations in the learning video that are a bit slow, and the web display. the results of the improvement can be seen in table 6. table 6. self-evaluation results repair suggestions before and after revision before the revision, there was only a linear equation menu that contained an introduction, ppt, learning videos after the revision, the menus provided are arranged according to the meeting, and are more complete before revision after revision before revision ppt cannot be accessed by other users after the ppt revision can be run on the web and the menu button is smooth before revision after revision azizah, fauzan, & harisman, “flipped classroom type peer instruction-based learning” based … 344 repair suggestions before and after revision before the revision, there was no student worksheet menu that students could download after the revision, the student worksheet menu is given for each meeting so that students can access it as needed based on the results of product development that have been described, it is hoped that this media can improve students' problem-solving abilities. student problem solving is influenced by how the teacher behaves (harisman et al., 2019b). if teachers can develop creative learning media such as videos, ppt, and the web, they will be able to change students' problem-solving to be more sophisticated (harisman et al., 2018, 2019a; harun et al., 2019). the background of teachers in teaching and the experience of the teachers in teaching also affect how the teachers develop media in the learning process (fauzan et al., 2019; harisman et al., 2020; kariman et al., 2019). the literature review was also carried out in previous research and found some information related to mathematics learning that can be used as a basis for designing peer instruction-type flipped classroom-based learning tools. flipped classroom-based mathematics learning model can maximize learning through one-on-one interactions through learning videos uploaded online and offline (abdelaziz, 2014; fraga & harmon, 2014; kim & jeong, 2016; rontogiannis, 2014). next, the effectiveness of the flipped classroom shows that learning at the university becomes more effective with flipped learning (ma et al., 2018; mccabe et al., 2017; wachira & absaloms, 2017; wong & chu, 2014). the theoretical study of flipped classrooms in mathematics learning shows that the flipped classroom learning model can help students learn both inside and outside the classroom, resulting in students being directly involved in the learning process and along with the development of information and technology that does require teachers to apply technology in learning activities and is based on the characteristics of mathematics learning (choi et al., 2015; iverson et al., 2017; lin & hwang, 2019; mccabe et al., 2017; wachira & absaloms, 2017). next, flipped learning model with peer instruction can improve problem-solving abilities (bokosmaty et al., 2019; gough et al., 2017; kim & jeong, 2016; matzumura-kasano et al., 2018; wang et al., 2019; zhang et al., 2018). 4. conclusion the development of this product is based on a preliminary analysis comprising needs analysis, analysis of student characteristics, curriculum analysis, concept analysis, and literature analysis. based on the preliminary analysis, the material arrangement of linear equations is arranged into 5 meetings, the first meeting is studying the relationship between two magnitudes (<, >, =, ≤, ≥) in a problem. understanding the correctness of mathematical sentences of equations when letters are substituted with numbers in a problem, the second meeting determines the solution of an equation without substituting numbers into letters in a problem; solving equations using the properties of equations in a problem, third meeting volume 11, no 2, september 2022, pp. 325-348 345 solves equations using the idea of moving tribes in a problem; fourth meeting solving equations in the form of decimals and fractions in a problem, solving problems by using linear equations; the fifth meeting is understanding ratio relationships by using linear equations in a problem and solving problems related to ratios by using linear equations. furthermore, based on the distribution of materials, a learning video with a duration of 1015 minutes was prepared based on the results of the preliminary analysis; the material is also provided in the form of ppt slides which have menus that are provided with audio features to make it easier for students to carry out ppt. in the end, there is a quiz that can be done by students to measure their ability of students after studying the materials, and the student worksheet contains practice problem-solving questions that are arranged based on a flipped classroom type of peer instruction which consists of the first test question, discussion of the first test, second test, and final assessment. the lesson plans are prepared following the curriculum used in schools and contain this activity with the steps of a flipped classroom type of peer instruction, namely pre-class students’ study first at home by accessing the web which includes learning videos, and ppt and providing the results of material resumes by uploading them to the menu. the main activities in the classroom are students completing student worksheets according to the steps of the flipped classroom type of peer instruction. acknowledgements we would like to thank the drpm directorate of research and community service of the republic of indonesia on a master's thesis scheme with a letter number 2076/un3513/lt/0022 and contract number 197/e5/pg0200pt/2022. references abdelaziz, h. a. 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(2018). spoc-based flipped classroom of college english: construction of an effective learning model. international journal of emerging technologies in learning (ijet), 13(1), 37-45. https://doi.org/10.15294/kreano.v7i2.7300 https://doi.org/10.1088/1742-6596/895/1/012079 https://doi.org/10.1109/icalt.2014.216 https://doi.org/10.15294/kreano.v6i2.5188 https://doi.org/10.33087/phi.v2i2.37 https://doi.org/10.1109/afrcon.2017.8095566 https://doi.org/10.1007/s10734-019-00366-8 https://doi.org/10.1007/978-3-319-08961-4_10 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p117-132 117 mathematical resilience, habits of mind, and sociomathematical norms by senior high school students in learning mathematics: a structured equation model samsul maarif1*, nelly fitriani2 1universitas muhammadiyah prof. dr. hamka, indonesia 2insititut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received dec 26, 2022 revised feb 18, 2023 accepted feb 27, 2023 the process of learning mathematics is determined by cognitive aspects and requires an affective domain. the affective domain is essential in developing mathematical abilities to solve mathematical problems. this study aims to analyze the effect of mathematical resilience (rm) and habits of mind (hom) on socio-mathematical norms (smn) in mathematics learning. the research method used is quantitative, with survey techniques and structured inquiry models. the sample in this study was 100 high school students in the dki jakarta area. data analysis was performed using the structured equation model (sem) using smartpls software. this research uses eight items of mathematics resilience instrument, ten items of habits of mind instrument, and 12 items of socio-mathematical norm instrument. each instrument has four alternative answers with a likert scale. the results of the study concluded: 1) there is a positive impact of mathematical resilience on socio-mathematical norms; 2) there is a positive impact of habits of mind on socio-mathematical norms; 3) there is a positive impact of mathematical resilience on habits of mind; 4) there is a positive impact of mathematical resilience on sociomathematical norms mediated by habits of mind. keywords: habits of mind, mathematical resilience, smartpls, sociomathematical norm, structure equation model this is an open access article under the cc by-sa license. corresponding author: samsul maarif, department of mathematics education, universitas muhammadiyah prof. dr. hamka jl. tanah merdeka, pasar rebo, east jakarta 13830, indonesia. email: samsul_maarif@uhamka.ac.id how to cite: maarif, s., & fitriani, n. (2023). mathematical resilience, habits of mind, and sociomathematical norms by senior high school students in learning mathematics: a structured equation model. infinity, 12(1), 117-132. 1. introduction the study of the affective domain in mathematics learning has been going on for a long time. this is intended to explore the positive contribution of the affective domain in the learning process. the scope of study of affective has begun to develop, not only examining attitudes but examining several aspects such as beliefs and emotional reactions (ignacio et al., 2006), mathematical resilience (hendriana et al., 2019; johnston-wilder et al., 2018; https://doi.org/10.22460/infinity.v12i1.p117-132 https://creativecommons.org/licenses/by-sa/4.0/ maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 118 johnston-wilder et al., 2015; kooken et al., 2013; thornton et al., 2012), habits of mind (costa & kallick, 2008; dwirahayu et al., 2017; matsuura et al., 2013; yellamraju et al., 2019), sociomathematical norm (güven & dede, 2017; maarif et al., 2022; sánchez & garcía, 2014; yackel & cobb, 1996; zembat & yasa, 2015), and so on. these affective aspects must be developed and optimized in teaching and learning activities, especially mathematics. it is undeniable that in the process of learning mathematics it is possible for students to experience failure and unpleasant experiences can occur (hutauruk & priatna, 2017). unpleasant experiences can be in the form of psychological pressure or high cognitive load or difficulties in understanding the mathematical concepts being studied (maarif et al., 2019; roth, 2019). the failure of the mathematics learning process experienced by students cannot be avoided, but the impact of this failure can be minimized or even eliminated. this is where the importance of mathematical resilience exists, to help students minimize the impact of student difficulties or failures in the learning process they experience (ishak et al., 2020; johnston-wilder et al., 2018).mathematical resilience is defined as a person's resilience to the difficulties encountered, being able to collaborate in collaboration, having language skills in communicating strengths and weaknesses, being resilient in dealing with difficulties related to learning problems (johnston-wilder et al., 2015), having a positive perspective on problems (gürefe & akçakin, 2018), easy to respond positively to the difficulties encountered (kooken et al., 2013), the ability to adapt to the challenges encountered for the continuity of work in the future (chirkina et al., 2020). someone who has good mathematical resilience will respond to problems in positive ways rather than prioritizing the anxiety they experience (gürefe & akçakin, 2018; kooken et al., 2013). developing mathematical resilience allows students to adapt the mathematical problems they face sustainably as a learning experience they have experienced (thornton et al., 2012). mathematical resilience allows a student to face difficult situations with the opposite situation which can have an impact on motivation gradually towards something better than the learning difficulties they face (johnston-wilder et al., 2018). therefore, it is important that in the process of learning mathematics, students develop mathematical resilience in order to foster productive thinking patterns in dealing with problems so that they are clear in their thinking processes to find solutions. the right action to build resilience is to cultivate a student's productive mindset which is called the habit of mind. habits of mind are organizing principles of how to think about mathematical concepts by resembling the way of thinking of previous mathematicians (cuoco et al., 1996; matsuura et al., 2013). it’s not about how theorems or algorithms are used, but rather about how mathematicians develop their thinking processes to find these mathematical theorems or algorithms (matsuura et al., 2013). for example: in learning mathematics a geometric theorem can be learned and used to solve a problem, but it will be more important for students to develop a thinking process on how mathematicians construct their thinking process to find the geometric theorems used. so, habits of mind are important to do so that students can fully understand mathematics with their thoughts by following the way mathematicians think, not directly applying a concept that can cause failure in comprehensive application (dwirahayu et al., 2017). levasseur and cuoco (2003) divides two groups of habits of mind, namely: 1) habits of mind which are common to all scientific disciplines which include determining patterns, experimenting, formulating, visualizing, creating and guessing; and 2) habits of mind that are specific to the field of mathematics include giving examples with examples, generalizing, abstracting, thinking in terms of functions, using several points of view and combining several experimental deductions. the two groups of habbit of mind are useful for someone to think, take action, behave in the learning environment and the surrounding environment volume 12, no 1, february 2023, pp. 117-132 119 (anggriani & septian, 2019). in addition, habits of mind will grow someone to be smart by knowing how to act to find solutions to the mathematical problems being faced (farida et al., 2019). costa and kallick (2008) explain that habits of mind are the foundation of students in ongoing learning. students are required to have good thinking habits in order to be able to respond to any problems that arise in learning and find solutions. habits of mind require a combination of attitudes, skills, previous experience and personality when deciding what to do in various situations. thus, in the process of forming habits of mind, toughness is needed in thinking and facing problems. mathematical resilience and habits of mind aim to develop thinking skills in dealing with mathematical problems (chusna et al., 2021; hodiyanto & firdaus, 2020). the process of solving mathematical problems in learning is inseparable from how the interaction process occurs in learning (wu et al., 2019). this is in line with the statement which revealed that güven and dede (2017) the interaction between students in the process of learning mathematics is very complex involving collective and interactive relationships. therefore, in developing resilience and habits of mind a process of social interaction norms of students is needed in the mathematical thingking process which is called the sociomathematical norm. yackel and cobb (1996) revealed that sociomathematical norms are an aspect of a person's normative understanding of the process of mathematical activity which is considered to be different mathematically, efficiently and elegantly. social interaction in learning mathematics is needed to develop their ideas in solving math problems (kang & kim, 2016). as long as students play an active role in building sociomathematical norms, they develop self-confidence and mathematical values that can be used as a basis for thinking actions in autonomous learning communities (dickes et al., 2020; zembat & yasa, 2015). talking about the relationship with socimathematical norms, resilience has a positive influence on students' self-confidence in the process of social interaction in the learning process (amelia et al., 2020). the results of nettles et al. (2000) revealed that students with good resilience skills will provide significant opportunities to interact with peers with a sense of optimism, participation and academic achievement in learning mathematics. laia's research also revealed that social interaction with peers in the learning process can be developed by mathematical resilience skills in the learning process (liew et al., 2018). furthermore, the results of research conducted by hodiyanto and firdaus (2020) reveal that habits of mind contribute to students' creative thinking abilities which are built by a process of social interaction in the mathematics learning process. research conducted by levasseur and cuoco (2003) found habits of mind determine interaction behavior that can reduce memory workload in the process of learning mathematics. agree with levasseur and cuoco, the results of research by anggriani and septian (2019) students' habits of mine will lead to an optimal pattern of social interaction, a spirit of togetherness and provide a pleasant new atmosphere in learning mathematics with a group discussion process where students give ideas. the results of research conducted by hutauruk and priatna (2017) revealed that mathematical resilience has a positive effect on positive responses in learning mathematics as indicated by students' thinking habits. other studies have also confirmed that mathematical resilience has a positive effect on thought processes for solving problems (fitriani et al., 2023). therefore, from some of these studies it is necessary to analyze in more depth the relationship between mathematical resilience, habits of mind and sociomathematical norms in the process of learning mathematics. from some of the previous explanations, it is necessary to analyze the relationship between mathematical resilience, habits of mind and sociomathematical norms from the process of learning mathematics. this study aims to determine the relationship between the three with research questions: (1) is there a positive impact of mathematical resilience on maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 120 sociomathematical norms? (2) is there a positive impact of habits of mind on sociomathematical norms? (3) is there a positive impact of mathematical resilience on habits of mind? (4) is there a positive impact of mathematical resilience on sociomathematical norms mediated by habits of mind? 1.1. conceptual framework the process of learning mathematics is inseparable from the process of social interaction to develop students' ideas and thought processes. kang and kim (2016) said that sociomathematical norms are an attitude of consideration from mathematical explanations to differences in mathematical understanding received by someone. the results of nettles et al. (2000) said that students with good resilience skills will provide significant opportunities to interact with peers with a sense of optimism, participation and academic achievement in learning mathematics. furthermore, levasseur and cuoco (2003) found habits of mind determine interaction behavior that can reduce memory workload in the process of learning mathematics. in line with levasseur and cuoco, the results of research by (anggriani & septian, 2019) found students' habits of mine will lead to an optimal pattern of social interaction, a spirit of togetherness and provide a pleasant new atmosphere in learning mathematics with a group discussion process in which students give ideas. the research results of hutauruk and priatna (2017) revealed that mathematical resilience has a positive effect on positive responses in learning mathematics as indicated by students' thinking habits. other studies have also confirmed that mathematical resilience has a positive effect on thought processes for solving problems (fitriani et al., 2023). therefore, from some of these studies it is necessary to analyze in more depth the relationship between mathematical resilience, habits of mind and sociomathematical norms in the process of learning mathematics. figure 1 shows the conceptual framework model and the hypotheses proposed in the study. figure 1. conceptual framework model note: rm : mathematical resilience; hom : habits of mind; volume 12, no 1, february 2023, pp. 117-132 121 smn : sociomathematical norm; h1 : there is a positive impact of mathematical resilience on sociomathematical norms; h2 : there is a positive impact of habits of mind on sociomathematical norms; h3 : there is a positive impact of mathematical resilience on habits of mind; h4 : there is a positive impact of mathematical resilience on sociomathematical norms mediated by habits of mind. 2. method this study uses a survey-based correlational research design with an inquiry model structural approach (karakus et al., 2021). this study aims to analyze the effect of mathematical resilience and habit of mind on the sociomathematical norms of senior high school students in dki jakarta. participants in this study were 100 senior high school students consisting of 54 (54%) male students and 46 (46%) female students in dki jakarta. samples were taken randomly which then responded by filling in the mathematical, habit of mind and sociomathematical resilience questionnaires which were distributed via the google form. the research instrument was compiled based on several article sources in determining the indicators. the instrument consists of a mathematical resilience questionnaire, habits of mind and sociomethanetical norms. this research uses 8 items of mathematics resilience instrument, 10 items of mind habits instrument, and 12 items of sociomathematical norms instrument. each instrument has 4 alternative answers with a likert scale. each instrument is composed of indicators adapted from several article sources. mathematical resilience instruments are structured based on indicators: persistent, work hard, have a willingness to discuss, look for various alternative solutions in solving problems, self-reflect, cooperate with peers, use failure experiences to build self-motivation, and have the ability to control oneself (hendriana et al., 2019; johnston-wilder et al., 2018; kooken et al., 2013). the habits of mind instrument refers to the following indicators: 1) persistent: being serious in solving a problem and not giving up easily in solving problems. someone who has; 2) flexible thinking: changing perspectives when receiving new information, knowing when to think big carefully and in detail, and using several alternative solutions in solving problems; 3) thinking about thinking: knowing what is known and what is not known, and being aware of the strategies used in solving problems; 4) apply existing knowledge to new situations: use experience in problem solving to apply to new problems; 5) critical response: detecting symptoms of doubtful solutions, statements, and arguments, as well as distinguishing several situations from the solutions that have been constructed (costa & kallick, 2008; dwirahayu et al., 2017; yellamraju et al., 2019). furthermore, for the sociomathematical norm instrument in this study, it refers to the following indicators: 1) experience of mathematics: contributing carefully and actively in discussion activities in the process of learning mathematics; 2) explanation of mathematics: understanding ideas and being able to explain their ideas from solutions systematically; 3) mathematical difference: identifying the similarities and differences in the ideas of several alternative solutions, as well as comparing the similarities and differences in the ideas of several alternative solutions that have been constructed; 4) mathematical communication: making sense as a basis for communication in the learning process and submitting statements to understand an idea in a language that is easy to understand; 5) mathematics effectiveness: finding the most effective alternative solutions and explaining the solutions to problems in a straightforward manner; 6) mathematical insight: interact in depth in discussion activities maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 122 and use various sources in the discussion process to solve problems (kang & kim, 2016; ningsih & maarif, 2021; widodo et al., 2020; yackel & cobb, 1996; zembat & yasa, 2015). analysis of research data to test hypotheses was carried out using partial least squares structural equation modeling (pls-sem) using smartpls 3 software. the use of the pls-sem method can be applied in various fields, including the field of mathematics education with reliable analysis results (xu & zhou, 2022). the use of smartpls software is because it is appropriate for analyzing a research model that integrates empirical theory and facts (wong, 2013). before testing the hypothesis, convergent validity and reliability tests were first carried out, as well as the discriminant external model (karakus et al., 2021). hypothesis testing was carried out to examine the relationship between latent variables, namely mathematical resilience, habits of mind and sociomathematical norms. 3. result and discussion 3.1. results a statistical description from research on mathematical resilience, habits of mind and sociomathematical norms is presented by showing the maximum score, minimum score, average, kurtosis and skewenes, as shown in table 1. table 1. statistical description variable item code min max mean stdev kuart. skew. mathematical relisience rm.1 1 4 2.990 0.877 -0.683 -0.341 rm.2 1 4 2.860 0.906 -0.670 -0.373 rm.3 1 4 2.890 0.958 -0.740 -0.468 rm.5 1 4 2.650 0.953 -0.850 -0.228 rm.6 1 4 2.940 1.037 -0.762 -0.642 rm.7 1 4 3.040 1.009 -0.609 -0.733 rm.8 1 4 2.820 0.953 -0.652 -0.474 rm.9 1 4 2.800 0.917 -0.424 -0.538 habits of mind hom.1 1 4 3.150 0.817 0.013 -0.733 hom.2 1 4 3.120 0.962 -0.084 -0.929 hom.3 1 4 3.110 0.904 0.303 -0.963 hom.4 1 4 3.130 0.945 0.050 -0.843 hom.5 1 4 3.090 0.850 0.387 -0.870 hom.6 1 4 3.070 0.941 0.131 -0.946 hom.7 1 4 3.170 0.906 0.537 -1.083 hom.8 1 4 3.190 0.857 0.711 -1.058 hom.9 1 4 3.110 0.937 0.341 -1.038 hom.10 1 4 3.040 0.916 0.300 -0.953 sociomathematical norm smn.1 1 4 3.170 0.849 -0.251 -0.737 smn.2 1 4 3.080 0.913 -0.126 -0.801 smn.3 1 4 3.080 0.902 -0.126 -0.801 smn.4 1 4 3.110 0.904 0.072 -0.880 smn.5 1 4 3.190 0.796 0.386 -0.841 smn.6 1 4 3.190 0.875 0.378 -0.961 smn.7 1 4 3.180 0.853 0.387 -1.045 smn.8 1 4 3.200 0.860 0.707 -1.072 smn.9 1 4 3.100 0.831 0.366 -0.829 volume 12, no 1, february 2023, pp. 117-132 123 variable item code min max mean stdev kuart. skew. smn.10 1 4 3.130 0.820 0.970 -1.020 smn.11 1 4 3.120 0.930 0.674 -1.151 smn.12 1 4 3.590 0.928 3.046 -2.133 table 1 shows that each item in the mathematical reliability, habits of mind and socimathematical instruments all have a kurtosis value between -7 to 7 and a skewness between -2 to 2 (levasseur & cuoco, 2003). that is, all items in each instrument are all distributed. the sem pls convergent validity test and discriminant validity were carried out on each item of the mathematical resilience instrument, habits of mind and sociomathematical norms to find out whether the instrument is valid or measures what it should measure. convergent validity is carried out by looking at the loading factor and average variance extracted (ave) values, while discriminant validity is done by looking at the fornell & larcker criterion values (hermanda et al., 2019).the results of the loading factor are as shown in figure 2. figure 2. pls algorithm results (modification) maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 124 the item criteria for each variable are said to be valid if the outer loading > 0.7 (wong, 2013). table 2 shows that each item the instrument of mathematical resilience, habits of mind and sociomathematical norm has a loading factor value of >0.7, which means that each item is valid as shown in table 2. table 2. the results of the modification on the validity of testing covergent variable indicator outer loading explanation mathematical relisience mr.1 0.762 valid mr.2 0.812 valid mr.3 0.764 valid mr.5 0.726 valid mr.6 0.802 valid mr.7 0.751 valid mr.8 0.820 valid mr.9 0.756 valid mathematical habits of mind hom.1 0.788 valid hom.2 0.841 valid hom.3 0.849 valid hom.4 0.830 valid hom.5 0.834 valid hom.6 0.887 valid hom.7 0.849 valid hom.8 0.849 valid hom.9 0.853 valid hom.10 0.836 valid socio-mathematical norm smn.1 0.838 valid smn.2 0.796 valid smn.3 0.810 valid smn.4 0.784 valid smn.5 0.841 valid smn.6 0.840 valid smn.7 0.855 valid smn.8 0.828 valid smn.9 0.785 valid smn.10 0.797 valid smn.11 0.757 valid smn.12 0.809 valid the validity of each instrument is determined by the average variance extracted (ave) value with the criterion of an ave value > 0.5 (wong, 2013). from the results of testing the instrument of mathematical resilience, habits of mind and sociomathematical norms have an ave > 0.05 as shown in table 3. this means that the indicators for each instrument are said to be valid. volume 12, no 1, february 2023, pp. 117-132 125 table 3. the result of average variance extracted (ave) variable ave rule of thumb explanation mathematical relisience 0.600 >0.500 valid mathematical habits of mind 0.705 >0.500 valid sociomathematical norm 0.659 >0.500 valid furthermore, discriminant validity testing was carried out with the fornell & larcker criterion on mathematical resilience instruments, habits of mind and sociomathematical norms. the results of discriminant validity testing are shown in table 4. table 4. the result of discriminant validity: fornell & larcker criterion mathematical relisience (rm) mathematical habits of mind (hom) sociomathematical norm (smn) mathematical relisience 0.775 mathematical habits of mind 0.763 0.880 sociomathematical norm 0.751 0.840 0.812 the criterion for discriminant validity with the fornell & larcker criterion is that the ave value on the diagonal (see table 4) is higher than the other values (karakus et al., 2021). so that the discriminant validity requirements are met. from testing convergent and discriman validity through the three criteria, all of them meet the requirements. thus, based on confirmatory factor analysis (cfa) concluded that the developed instrument of mathematical resilience, habits of mind and sociomathematical norms can be used to test the proposed model hypothesis. after the instrument of mathematical resilience, habits of mind and sociomathematical norms are declared valid. the next step is to test the pls sem reliability with cronbach's alpha. the results of the reliability test for the instrument are shown in table 5. table 5. the result of reliability test variable cronbach’s alpha composite reliability rule of thumb explanation mathematical relisience 0.905 0.907 >0.700 reliable mathematical habits of mind 0.953 0.954 >0.700 reliable sociomathematical norm 0.953 0.954 >0.700 reliable table 5 shows the reliability testing criteria are cronbach's alpha > 0.7 and composite reliability > 0.7 (wong, 2013). table 5 shows the results of reliability testing for each the instrument of mathematical resilience, habits of mind and sociomathematical norms have a cronbach's alpha value and composite reliability > 0.7. so it can be concluded that the instrument meets the reliability requirements. there are four hypotheses proposed in this study as previously mentioned in figure 1. to test the hypotheses of the structural model that has been proposed, the t-value can be used through a bootstrap procedure with 5000 repeated samples (hermanda et al., 2019). figure 3 shows the results of bootstrapping that has been done. maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 126 figure 3. bootstrapping test results. furthermore, the results of testing the structural model which shows the results of testing the hypothesis using the t-value as shown in table 6. table 6.the result of hypothesis testing hypotesis variable original sample standard deviation tvalue pvalue explanation h1 rm -> smn 0.181 0.085 2.112 0.035 accepted h2 hom -> smn 0.748 0.086 8.666 0.000 accepted h3 rm -> hom 0.763 0.047 16.193 0.000 accepted h4 rm ->hom -> smn 0.571 0.075 7.565 0.000 accepted criteria for the significance of the hypothesis by looking at the parameter coefficient values and the significance value of the t-statistic in the bootstrapping algorithm report. by looking at the t-table at alpha 0.05 (5%) = 1.96 and comparing it with the t-test we can conclude whether the hypothesis proposed is significant or not. if the t-test value is > ttable, then the proposed hypothesis is accepted (wong, 2013). table 6 shows h1, h2, h3 and h4 each having a t-value > 1.96, so it can be concluded that h1, h2, h3 and h4 are accepted. volume 12, no 1, february 2023, pp. 117-132 127 3.2. discussion the results of testing the hypothesis in table 6 conclude that there is a positive impact on socio-mathematical norms with a t-value = 2.222 > 1.96 with a p-value = 0.035. this shows that someone who has good mathematical resilience will influence sociomathematical norms in learning mathematics. this finding shows that each indicator on mathematical resilience supports the formation of indicators on socio-mathematical norms. mathematical resilience is supported by how a student has a willingness to discuss which can play a role in carrying out in-depth interactions in discussion activities or mathematical insights. interaction in depth in the consultation process will build social intelligence. this is in line with the wishes of sánchez and garcía (2014) that the experience of interacting with the surrounding environment or social interaction influences the development of mathematical thinking processes in solving problems. in addition, students with good resilience will try optimally to find various alternative solutions in solving problems. this is in line with the results of his research (kang & kim, 2016; maarif et al., 2022) which revealed that someone with good socimathematical norms would find the most effective alternative solutions and be able to explain solutions to problems straightforwardly. so that resilience can have an impact on the ability of sociomathematical norms. thus, it appears that mathematical resilience has an effect on the formation of students' societal norms. the results of nettles et al. (2000) revealed that students with good resilience skills will provide significant opportunities to interact with peers with a sense of optimism, participation and academic achievement in learning mathematics. the results of testing the hypothesis in table 6 conclude that there is an impact of positive thinking habits on socio-mathematical norms by showing a t-value = 8.666 > 1.96 with a p-value = 0.000. this shows that someone who has good thinking habits will influence socio-mathematical norms in learning mathematics. habits of mind allow students to think flexibly by using several alternative solutions to problem solving while using several alternative solutions is a characteristic of socio-mathematical norms. that is, habit of mind has a contribution to socio-mathematical norms. to find alternative solutions, interaction is needed to multiply ideas. this is what levasseur and cuoco (2003) found in their research that habit of mind determines interactive behavior that can reduce memory workload in the process of learning mathematics. in line with levasseur and cuoco, the research results of anggriani and septian (2019) students' habit of mine will lead to an optimal pattern of social interaction, a spirit of togetherness and provide a fun new atmosphere in learning mathematics with the process of student group discussions where give ideas. in addition, students with good sociomathematical norms will be able to explain solutions to problems in a straightforward manner. the ability to explain in a straightforward manner can be easily carried out by students when they are aware of the steps/strategies used in solving problems. students who are aware of problem solving strategies will master the solution and easily explain the solution. this is in line with the research of murtafiah et al. (2018) who revealed that explanations of problem solutions can help students understand concepts, procedures and be flexible in choosing information in solving mathematical problems. table 6 concludes that there is a positive impact of mathematical resilience on the habit of mind which is indicated by a t-value = 16.193 > 1.96 with a p-value = 0.000. this shows that someone who has good thinking habits will influence socio-mathematical norms in learning mathematics. this is in line with the research findings of hutauruk and priatna (2017) reveal that mathematical resilience has a positive effect on positive responses in learning mathematics as indicated by students' thinking habits. other studies have also maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 128 revealed that mathematical resilience has a positive effect on the thought process for solving problems (fitriani et al., 2023). the results of hypothesis testing in table 6 show a t-value = 7.565 > 1.96 with a pvalue = 0.000 which means that there is a positive impact of mathematical resilience on sociomathematical norms mediated by habits of mind. these findings have the meaning that apart from mathematic resilience directly influencing sociomathematical norms, it also has an influence based on mediation by aspects of habits of mind. this shows that in the process of learning mathematics, someone who has good resilience and a good habit of mind will support socio-mathematical norms either directly or indirectly. habits of mind can directly affect resilience or can also mediate students who have less math resilience. 4. conclusion the results of the study concluded: 1) there is a positive impact of mathematical resilience on sociomathematical norms; 2) there is a positive impact of habits of mind on sociomathematical norms; 3) there is a positive impact of mathematical resilience on habits of mind); 4) there is a positive impact of mathematical resilience on sociomathematical norms mediated by habits of mind. the results of this study indicate that the affective aspects of learning mathematics are related to one another. therefore, in the learning process it is important to develop affective aspects, especially mathematical resilience, habits of mind and sociomathematical norms. these three aspects can be taken into consideration in developing a learning strategy to improve mathematical competence. in addition, these three aspects can also be used as a reference in determining the success of learning in addition to cognitive aspects in order to create effective and efficient mathematics learning, especially at the senior high school level. acknowledgements the authors would like to thank muhammadiyah university prof. dr. hamka and institut keguruan dan ilmu pendidikan siliwangi for allowing the collaboration of research and publications between universities to be carried out. references amelia, r., kadarisma, g., fitriani, n., & ahmadi, y. 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(2015). using classroom scenarios to reveal mathematics teachers' understanding of sociomathematical norms. international journal of education in mathematics, science and technology, 3(3), 242-261. https://doi.org/10.54870/1551-3440.1285 https://doi.org/10.22342/jme.9.2.5388.259-270 https://doi.org/10.1080/10824669.2000.9671379 https://doi.org/10.30738/wa.v5i1.9966 https://doi.org/10.1002/sres.2590 https://doi.org/10.1007/s10649-013-9516-0 https://doi.org/10.11591/ijere.v9i2.20445 https://doi.org/10.1145/3329485 https://doi.org/10.33225/jbse/22.21.706 https://doi.org/10.5951/jresematheduc.27.4.0458 https://doi.org/10.1109/te.2019.2924610 maarif & fitriani, mathematical resilience, habits of mind, and sociomathematical norms … 132 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p85-100 85 how to develop an e-lkpd with a scientific approach to achieving students' mathematical communication abilities? wahyu hidayat*, usman aripin institut keguruan dan ilmu pendidikan siliwangi, indonesia article info abstract article history: received aug 24, 2022 revised jan 15, 2023 accepted feb 18, 2023 this research aims to look into the growth and effectiveness of e-lkpd on the mathematical communication skills of seventh-grade junior high school students. the design research development study was used. this study is divided into two parts: preliminary (preliminary study and design of e-lkpd) and formative evaluation (evaluation and revision). the research subjects included 49 smp negeri 3 ngamprah's class viii students. the research instrument had interview sheets, expert validation sheets, observation sheets, student response questionnaires, and five questions about mathematical communication skills. lines and angles were used in this study as the material. data processing techniques in the development process use the likert scale type of measurement scale to calculate the percentage of each indicator with the help of canva and test the effectiveness of the e-lkpd using inferential statistics with the wilcoxon test with the help of spss. according to the study's findings, the feasibility of lkpd is eligible. the study's results also show that an e-lkpd influences students' mathematical communication abilities, indicating that the developed e-lkpd has effectiveness. keywords: e-lkpd, mathematical communication, scientific approach this is an open access article under the cc by-sa license. corresponding author: wahyu hidayat, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi jl. terusan jenderal sudirman no. 3, cimahi, west java 40526, indonesia. email: wahyu@ikipsiliwangi.ac.id how to cite: hidayat, w., & aripin, u. (2023). how to develop an e-lkpd with a scientific approach to achieving students' mathematical communication abilities? infinity, 12(1), 85-100. 1. introduction mathematics is a crucial field in the advancement of science and technology. mathematics is studied at nearly every level of education. this occurs because many disciplines require practical mathematical concepts or calculations to advance. the international student assessment program is one way to show appreciation for mathematics (pisa). even in 2022, pisa's main focus will be on the math test, with the addition of a creative thinking ability test (almarashdi & jarrah, 2022; özaydin & arslan, 2022). this demonstrates that mathematics has grown in importance, even becoming a global concern, https://doi.org/10.22460/infinity.v12i1.p85-100 https://creativecommons.org/licenses/by-sa/4.0/ hidayat & aripin, how to develop an e-lkpd with a scientific approach … 86 to the point where studies on mathematical problems have become an urgent and top priority in the development of learning, particularly in indonesia. the current challenge is that students must be able to assess a problem, determine the appropriate representation based on existing data, and then represent it in a mathematical model (apsari et al., 2020; putri & zulkardi, 2018). mathematical communication skills are the ability to provide rational reasons, change the shape of a situation into a simpler model, and describe mathematical ideas or ideas in the form of relevant descriptions. mathematical communication is useful for completing, exploring, and investigating mathematics, as well as a social activity for exchanging ideas, and opinions, and sharpening ideas to persuade others. mathematical communication skills are critical for students to have when learning mathematics, both elementary and middle school students must have good communication skills (hendriana et al., 2022). mathematical communication is essential for students, according to several explanations. however, the facts show that students' communication skills remain relatively low. one of the causes of students' low mathematical communication skills is their inability to communicate mathematical ideas while learning mathematics. the problems that arise at this time present new challenges for teachers and researchers to find the best solutions to achieve learning objectives. the challenge for mathematics teachers in particular is to provide engaging, effective, and efficient learning under the current conditions of distance learning (barana et al., 2021; dhawan, 2020; haleem et al., 2022; russo et al., 2021). however, in reality, distance learning or online learning is still not optimally carried out, as research by suripah and susanti (2022), shows that many students still struggle to understand learning material, students lack the motivation to learn, and students do not master learning well. furthermore, problems with the internet network are common when learning online. this is a common stumbling block when not all students have adequate facilities to support online learning, and unstable internet networks affect the online learning environment (chu, 2010; eze et al., 2018; pham et al., 2019). zoom meetings are a platform that is quite attractive to teachers for the teaching and learning process during online learning (tsarapkina et al., 2020). this is because the zoom application drains a large quota and when the signal is less stable, learning becomes ineffective and information becomes incomplete, so the zoom application is preferred by the smallest percentage. this shows that online learning is not effective if it is not supported by adequate technological facilities and infrastructure (ardiansyah et al., 2021; chu, 2010; dharma et al., 2017; eze et al., 2018; pham et al., 2019). based on the problems described above, online learning requires special attention to minimize the obstacles that arise in order to achieve learning objectives and improve mathematical communication skills. conference applications such as zoom will be more effective once they are supported by adequate facilities with good devices and signals, but the availability of student facilities has not been achieved for the most part, so some of these conference applications are still in demand. the platform is an effective online learning tool that does not require a strong signal and is easy to implement. e-learning is a platform that does not require a stable network. it is also shown that learning through google classroom e-learning has a positive impact on students' reasoning abilities (ansong-gyimah, 2020). according to the preliminary studies conducted in this study, educators continue to use ready-to-use lkpd that only contain instructions and practice questions, making learning more passive. this demonstrates the importance of creating interesting, non-boring, and interactive worksheets. other field findings indicate that other supporting applications should be designed as creatively as possible in order for students to find learning more interesting, not boring, and meaningful. creating interactive student worksheets (e-lkpd) with the canva application is one way to support interactive e-learning that does not require volume 12, no 1, february 2023, pp. 85-100 87 internet signal stability. this application will convert the standard lkpd into audio-visual additions. the presence of audio tools in the e-lkpd will aid comprehension of each instruction in the module. video tools will be able to present contextual problems in a more interesting, real, and optimal manner. several studies show that using audio-visual learning media improves learning outcomes and results in higher achievement than those who do not use audio-visual (apriyanto et al., 2019; rachmavita, 2020). e-lkpd development must be accompanied by a learning approach that is relevant to the current curriculum, relevant to the material to be studied, and expected to improve mathematical communication skills. the scientific approach is one suitable approach for overcoming the various problems described earlier. the scientific approach makes learning more active by asking questions, making it less boring, and training students to communicate conclusions (hendriana et al., 2017; hendriana et al., 2018; muttaqin et al., 2017; nicol, 1998; shahrill, 2013). students can observe various problems in the e-lkpd, which can be packaged in a variety of formats, including not only text but contexts that appear more real, such as by inserting a video link. it is hoped that with these features, it will be able to attract students' attention and actively ask questions, as well as try to reason well so that they can construct and conclude mathematical concepts. several scientific studies on e-lkpd have been conducted. research by ulfah et al. (2020), researching on preliminary research of mathematics learning device development based on realistic mathematics education (rme). previous research has found no specific development of interactive electronic worksheets based on a scientific approach and testing their effectiveness in improving mathematical communication skills. as a result, researchers are interested in conducting research on developing e-lkpd using a scientific approach and the canva application and examining how effective it is in improving students' mathematical communication skills. 2. method this study is a research of development research, design research, or development studies. the subjects of this study were smp negeri 2 cimahi class ix students. the canvaassisted math student worksheets (lkpd) using a scientific approach to material outlines and angles are the subject of this development research. according to tessmer (hidayat et al., 2022; kurniawan et al., 2018), there are two stages: preliminary research and formative evaluation. the preliminary study stage includes an analysis stage (student analysis, curriculum, and teaching materials), a design stage (prototyping), and a formative evaluation stage that includes self-evaluation, prototyping (expert review, one-to-one or small group), and field testing. 2.1. preliminary design the researcher organized the location, subject, and other preparations, including the research schedule, at this point. the participants in this study were 49 students from smpn 3 ngamprah's classes viii and vii. with an adjusted time allocation for learning mathematics for the 2021/2022 school year. the research subjects included three students for the individual test in class viii, ten students for the group test in class viii, and 36 students for the pilot test in class vii at smp negeri 3 ngamprah. hidayat & aripin, how to develop an e-lkpd with a scientific approach … 88 2.2. formative evaluation design this stage is divided into three sub-stages: self-evaluation, prototype design, and field testing. 2.2.1. self-evaluation this stage is split into two categories: analysis and design. the analysis phase aims to examine the discrepancy between learning objectives and learning outcomes, evaluate the curriculum to determine core competencies and fundamental competencies as a basis for deciding learning objectives that will be accumulated in the form of teaching materials to be designed, and analyze the curriculum to assess core competencies and basic competencies. the researcher used a scientific approach to design a teaching material product in the form of a canva-assisted electronic worksheet (e-lkpd) to improve students' mathematical communication skills on lines and angles during the design stage. the e-lkpd design is based on five aspects of feasibility (see table 1): (1) material/content; (2) presentation; (3) language; (4) scientific approach suitability; (5) graphics table 1. indicators of development aspects of e-lkpd no aspects of development indicator 1 content / material a. compliance with the learning objectives b. compatibility with student characteristics c. the framework of the materials 2 presentation a. the topical relevance of the picture examples b. appropriateness of layout settings c. appropriateness of letter type and size 3 language a. communicative and interactive b. compliance with good and correct indonesian rules c. compatibility of sentences with students' thinking levels 4 in accordance with the scientific approach a. construction ideas b. according to the scientific approach's characteristics 5 technical compliance a. representative b. it can be used without any other media. c. convenient to be using 2.2.2. designing the prototype this stage has three stages: expert review, one-on-one, or small group. at this stage, the developed e-lkpd must be tested, reviewed, and evaluated. expert review validation is carried out at this stage by material and media experts who analyze the strengths and weaknesses and comprehensively assess and evaluate them. expert validation volume 12, no 1, february 2023, pp. 85-100 89 is performed to obtain an assessment of the teaching materials that are being developed, with the hope that the teaching materials will be suitable for testing. researchers have created an initial design for the developed lkpd. at the formative revision stage, the draft receives input from material and media experts, as well as practitioners, and this becomes material for consideration in the lkpd development process before the lkpd is implemented. one-on-one test at this stage, errors in planning and learning resources should be avoided, and an initial reaction from student stakeholders should be obtained. three students with good, medium, and poor criteria who had studied lines and angles material were selected. small-group experiment the goal at this stage is to determine the practicability of the e-lkpd that has been developed after being revised by the validator. students who have studied lines and angles will participate in this stage. at this stage, there were as many as ten respondents from class viii smp negeri 3 ngamprah. students in this stage are asked to observe the development of the e-lkpd, and then they are given a questionnaire to assess the e-practicability. lkpds the collected data is then summarized and used to revise the developed e-lkpd so that it can be processed and re-tested at a later stage to produce an effective e-lkpd. 2.2.3. field tryout formative evaluation concludes with field trials. the goal of field trials is to determine whether learning can take place in a given context. the results of the field trials will be used to determine whether learning is appropriate and effective to implement. the test was conducted on a larger number of students than the small group test, namely 36 class vii students from smpn 3 ngamprah. the validity and practicability of lkpd, as well as its effectiveness in use, were tested through data analysis. the first stage evaluated the product's validity. according to hidayat et al. (2023), the following formula is used to calculate the validation results from the validator. v= 𝑓 𝑛 ×100% the details: v : the final score f : score obtained n : maximum score the obtained results are interpreted by the product validity level criteria (see table 2). hidayat & aripin, how to develop an e-lkpd with a scientific approach … 90 table 2. product validity criteria grade (%) category 81 100 precise 61 80 valid 41 60 adequately valid 21 40 invalid < 21 null a likert scale is used in the second stage of the practicality test. the form of each statement is scored, and the total score of each student's answer is calculated. the value of the questionnaire data was assessed using the formula (mustami et al., 2019). p= (∑ 𝑓) 𝑁 ×100% the details: 𝑃 : the final score ∑ 𝑓 : sum score 𝑁 : maximum score after determining the final value of practicability, examine the criteria (see table 3). table 3. criteria for practicality grade (%) category 80% < 𝑃 ≤ 100% highly practical 60% < 𝑃 ≤ 80% practical 40% < 𝑃 ≤ 60% quite practical 20% < 𝑃 ≤ 40% less practical 𝑃 ≤ 20% impractical the third stage of testing the effectiveness of lkpd involves using spss software to examine the normality and difference between the two wilcoxon averages. because the data were not normally distributed, the wilcoxon test was used. the wilcoxon test compares the means of two paired samples to see if there is a difference. 3. result and discussion 3.1. preliminary the first step is to conduct a preliminary study to determine what is causing the disparity between learning objectives and learning outcomes. in this step, the researcher gathered data by conducting interviews with one of the mathematics teachers. the information gleaned from the interviews is then entered into the performance appraisal. performance evaluation reveals the reasons or causes of learning performance gaps. this performance evaluation includes the field learning process, the desired learning process, and the main causes of learning performance gaps. in addition, there is a performance evaluation as follows (see table 4). volume 12, no 1, february 2023, pp. 85-100 91 table 4. preliminary research the actual situation ideal objectives the primary cause the use of lkpd in schools when learning is still simple and relies relying on government learning resources the application of lkpd with modern technology. lack of teacher interest in using other media, limited time, and inadequate training the lecture and question and answer methods are still used in the learning process, which is centred on the teacher. student-centred education inadequate learning resources and facilities, overcrowded classes, and a concentration on pursuing material students are less able to construct mathematical ideas into images and graphics due to a lack of understanding of the material, one of which is lines and angles. students can construct mathematical ideas into pictures and graphs by understanding the material of lines and angles. limited student ability to translate mathematical concepts into images and graphics, and lack of qualified media digital learning media are still used infrequently. utilizing effective and efficient media in the learning process learning platforms with insufficient training table 4 shows that the use of lkpd in schools when learning was still simple and relied solely on government learning resources. the teacher still dominates the learning process through the lecture and question and answer method, resulting in the teacher having a greater influence than the students. due to students' lack of understanding of the material, one of which is lines and angles, students cannot construct mathematical ideas into images and graphics. 3.2. formative evaluation 3.2.1. self-evaluation following a preliminary study that identified gaps between learning objectives and reality on the ground, the researchers developed an e-lkpd design that is expected to support/become an alternative solution to the problems mentioned above. hidayat & aripin, how to develop an e-lkpd with a scientific approach … 92 translate: indicators of competence achievement 3.10.4 finding the concept of angle 3.10.5 find the measure of the angle formed by the clockwise 3.10.6 naming angle 4.10.3 solving mathematical problems related to angles activity 3.1 let's observe observe carefully figure 1.3 below! from the pictures above, explain the parts of each picture above that form an angle? ………………………………………… ………………………………………… figure 1. e-lkpd display using canva figure 1 depicts an e-lkpd created with the canva app on a smartphone. this canva application offers a variety of interesting tools for designing various types of designs; you can even create teaching materials that are effective and efficient in their use. the elkpd is distributed in the form of a link that can be opened on a smartphone or laptop, making the learning process easier, and more cost-effective because it does not need to be printed, and can be equipped with audio and visuals such as embedding a video link on google drive or a youtube link. 3.2.2. designing the prototype expert review the results of validation, individual trials, small group, field, and pilot tests were used to determine the feasibility of the e-lkpd on lines and angles using a scientific approach aided by canva. two material and two media expert validators validated phase i expert validation. table 5 shows the results of stage i expert validation. volume 12, no 1, february 2023, pp. 85-100 93 table 5. expert validation at the first stage no aspects observed validator percentage criteria ∑p �̅� 1 material expertise eligibility of content 1 67% 71% valid 2 75% 2 presentation eligibility 1 71% 73% valid 2 75% 3 language eligibility 1 73% 73% valid 2 73% 4 media professional graphic eligibility 1 68% 72% valid 2 75% average percentage of all aspects 72% valid according to table 5 of the results of the expert validation stage i that has been analyzed, the total average percentage value for all aspects with the "valid" criteria is 72%. as a result, it can be concluded that the developed lkpd is feasible and can be used in the learning activities of junior high school class vii students. the following are expert notes that have been revised and are ready for researchers to use (see figure 2). translate: it would be better if the problem is presented contextually. at the reasoning stage, empty instructions/steps should be made so that you can see patterns well and the reasoning process becomes more optimal figure 2. lkpd revision results 2 according to the results of the revised lkpd by experts and practitioners (see figure 2), the lkpd is good, but the problems presented contextually are better, the approach to the material taken is appropriate, particularly using a scientific approach where the stages help students in communicating the things or problems received so that participants can present their results in front of other groups, and there will be mathematical communication between groups. after completing stage 1, expert validation and making improvements to the product, stage 2 expert validation was completed. the results of the second phase of expert validation of student worksheet products on lines and angles using a scientific approach assisted by canva are presented in table 6. hidayat & aripin, how to develop an e-lkpd with a scientific approach … 94 table 6. validation stage 2 no aspects observed validator persentase criteria ∑p �̅� 1 material expertise eligibility of content 1 81% 82% very valid 2 83% 2 presentation eligibility 1 83% 85% very valid 2 88% 3 language eligibility 1 90% 93% very valid 2 95% 4 media professional graphic eligibility 1 92% 93% very valid 2 95% average percentage of all aspects 88% very valid according to table 6 of the results of the stage 2 expert validation that has been analyzed, the total average percentage value for all aspects is 88% with the criteria of "very valid." as a result, the developed product is extremely valid and ready for use. as a result, the developed lkpd is very feasible and can be used in class vii student learning activities. one-on-one evaluation to assess a product's viability, individual tests are conducted. at this point, the respondents included three smp negeri 3 ngamprah class viii students who were chosen randomly based on their aptitude. the following are the findings from student responses to the e-lkpd (see table 7). table 7. one-to-one test results no. indicator persentage 1. lkpd contents comprehension 76% 2. clarity of the study guidelines and data 76% 3. lkpd exhibit suitability 79% 4. motivation 79% 5. attractiveness 70% 6. curiosity 84% average for the category 77% practical based on the individual test results in table 7, the average percentage of students' responses to the developed lkpd is 77% in the "practical" category, indicating that the lkpd can be re-tested in small group tests. small group evaluation trials with a small group of people to evaluate the product's viability. respondents at this stage included ten class viii students from smp negeri 3 ngamprah who were chosen volume 12, no 1, february 2023, pp. 85-100 95 on a random basis based on ability. the following are the findings from student responses to the e-lkpd (see table 8). table 8. trial results in a small group no. indicator persentage 1. lkpd contents comprehension 76% 2. clarity of the study guidelines and data 73% 3. lkpd exhibit suitability 72% 4. motivation 74% 5. attractiveness 80% 6. curiosity 75% average for the category 75% practical according to the results of the small group trial for the e-lkpd using a scientific approach aided by canva (see table 8), the average percentage of scores obtained from 10 students is 75% in the "practical" category, indicating that the developed lkpd is testable. field try out the formative evaluation concludes with field trials. the effectiveness of the elkpd scientific approach on students' mathematical communication abilities on lines and angles was tested in trials. as a prerequisite for determining the types of parametric and nonparametric statistics, the normality test is used. the results of the normality test are shown in the table 9. table 9. test results for normality kolmogorov-smirnova shapiro-wilk statistic df sig. statistic df sig. pretest 0.211 36 <0.001 0.922 36 0.015 postest 0.149 36 0.043 0.933 36 0.031 a. lilliefors significance correction the results of the spss software-assisted normality test show that the sig. <0,05 then the decision is not normally distributed. furthermore, proceed with the non-parametric wilcoxon test assisted by spss software as an alternative to the paired sample t test. the wilcoxon test results, as aided by spss software, are presented in the table 10. table 10. wilcoxon test results postest pretest z -5.247b asymp. sig. (2-tailed) <0.001 a. wilcoxon signed ranks test b. based on negative ranks. hidayat & aripin, how to develop an e-lkpd with a scientific approach … 96 the wilcoxon test results (see table 10), as aided by the spss software, show that if the sig<0.05, then the decision is there is a significant influence on the mathematical communication skills of students who use e-lkpd with a scientific approach assisted by canva. if there is a significant "influence" on students' communication skills after the pretest and posttest and obtains "very effective" results. students are encouraged to actively study independently using this e-lkpd teaching material. e-lkpd makes it simple for students to gain access to it, allowing them to be more flexible in the learning process and efficient in their use. the canva application offers a variety of design features to assist researchers in creating visually appealing e-lkpds, as well as audio-visual features to entice students to study. according to another viewpoint, developing e-lkpd will enable students to participate actively, creatively, and independently in learning activities, allowing students to overcome their fear of learning mathematics (negari et al., 2021; susiana & renda, 2021). other studies' findings indicate that e-lkpd is useful and that students can be motivated and interested in learning activities because it makes it easier for students to understand the material and solve learning problems (wijaya & hidayat, 2022). students are also more enthusiastic about learning because they can work in groups, exchange opinions among friends, and are not afraid to express their opinions to the teacher when presenting the results of group work in front of the class (lydia & suparman, 2019). e-lkpd teaching materials take a scientific approach to guiding students in the construction of the line and angle concepts. the steps in learning are intended to guide and stimulate students to think about and discover the concepts of lines and angles. in accordance with current learning in the 2013 curriculum, which is oriented toward directing students to formulate problems and train analytical thinking so that students achieve higher learning outcomes than the traditional approach (indarti et al., 2018; nenotaek et al., 2019). after being given the e-lkpd, students' mathematical communication skills improved significantly. all aspects of the scientific approach, beginning with observing, questioning, reasoning, associating, and communicating, can be used to improve mathematical communication skills. when observing problems, students must comprehend the various contexts presented and interpret them based on their understanding. the ability to understand and interpret context is an indicator of mathematical communication ability. this is consistent with engel (2000), which states that the ability to understand, interpret, and evaluate mathematical ideas both orally and visually is an indicator of mathematical communication ability. furthermore, students can communicate something they know through dialogue events or reciprocal relationships that occur in the classroom environment (rahmi et al., 2017). 4. conclusion based on the results and discussion, it demonstrates the feasibility of student worksheets on lines and angles using a scientific approach aided by canva, according to experts and practitioners, paying attention to the results of expert and practitioner assessments can be categorized as "very valid" and the results of practical trials are in the "practical" category. the effectiveness of the e-lkpd on lines and angles using a scientific approach assisted by canva on the mathematical communication abilities of grade vii secondary school students is that there is a significant influence on the mathematical communication abilities students using the e-lkpd with a scientific approach assisted by canva and obtaining very effective results. as for additional research, the e-lkpd can be used to create a learning trajectory with a wider scope. volume 12, no 1, february 2023, pp. 85-100 97 acknowledgements the authors would like to thank institut keguruan dan ilmu pendidikan siliwangi for the permission for the research and publication. references almarashdi, h. s., & jarrah, a. m. 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(2022). development of e-lkpd based on real problems in theory statistics data class vi sdn 101868 sena village. widyagogik: jurnal pendidikan dan pembelajaran sekolah dasar, 10(1), 132-147. https://doi.org/10.21107/widyagogik.v10i1.16815 https://doi.org/10.21107/widyagogik.v10i1.16815 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p27-40 27 the sociograph: friendship-based group learning in the mathematics class sri adi widodo1*, turmudi2, jarnawi afgani dahlan2, somchai watcharapunyawong3, hasti robiasih1, mustadin4 1universitas sarjanawiyata tamansiswa yogyakarta, indonesia 2universitas pendidikan indonesia, indonesia 3 thepsatri rajabhat university, lopburi, thailand 4universitas islam negeri sunan kalijaga, indonesia article info abstract article history: received jan 10, 2023 revised feb 14, 2023 accepted feb 24, 2023 mathematics class can cause many problems if students do not organize diversity and habits correctly. using a sociograph to form a mathematics study group is one way to organize assortment in the mathematics class. sociograph is a friendship pathway that appears in a math class. in this sense, this study aims to determine the impact of forming study groups based on friendship in a mathematics class on problem-solving abilities. a quasi-experimental research design with 30 students was used. a friendship questionnaire and a problem-solving test were used as instruments. in addition, an independent ttest was used to analyze the data. the study results indicate that study groups formed through friendship pathways (sociograph) have a more significant effect than those formed through other means. as a result, the formation of heterogeneous groups based on friendship can be used as an alternative to the formation of study groups. keywords: friendship, group learning, mathematics class, sociograph this is an open access article under the cc by-sa license. corresponding author: sri adi widodo, department of mathematics education, universitas sarjanawiyata tamansiswa yogyakarta jl. batikan tuntungan uh-iii/1043, umbulharjo, daerah istimewa yogyakarta 55167, indonesia. email: sriadi@ustjogja.ac.id how to cite: widodo, s. a., turmudi, t., dahlan, j. a., watcharapunyawong, s., robiasih, h., & mustadin, m. (2023). the sociograph: friendship-based group learning in the mathematics class. infinity, 12(1), 27-40. 1. introduction character is defined as the psychological traits of morality or manner that set one individual apart from another (ganellen, 2007; kosinski et al., 2014). character is an identity that describes a person's qualifications (asch, 2005; fejes & köpsén, 2014). it does not necessitate quantitative evaluation tools, so its formation does not necessarily entail a separate subject (entwistle & ramsden, 2015; orne, 2006). various student characters from outside the classroom provide color-to-student interaction in the mathematics class. this character difference is one reason why learning in mathematics class will cause many problems (cheema & kitsantas, 2014; pekrun, 2014). students' personalities are very diverse or heterogeneous, and their habits differ. the impact of learning mathematics is that https://doi.org/10.22460/infinity.v12i1.p27-40 https://creativecommons.org/licenses/by-sa/4.0/ widodo et al., the sociograph: friendship-based group learning … 28 it can organize students' diversity and habits so that learning goals can be achieved while remaining unlikely to be affected (kereluik et al., 2013; killpack & melón, 2016; kövecsesgősi, 2018; widodo & purnami, 2018). social interaction is a pattern that teaches students how to analyze a phenomenon related to their life problems and experiences (russell & martin, 2014). social interaction is a relationship between two or more people or between one person and another (cacioppo & cacioppo, 2014). a reciprocal relationship develops between the two parties during that interaction (lewis et al., 2014; sprecher et al., 2013). students who engage in social contact are more likely to have an attitude of collaboration, to connect with others on an individual or group level, to communicate with one another, and to offer solutions to problems (cowie et al., 1994; cullen-lester & yammarino, 2016; ratts et al., 2016; tseng & kuo, 2014). by focusing on students' emotional and social needs, teachers can play a crucial part in fostering positive interactions in the classroom (bambaeeroo & shokrpour, 2017; glass et al., 2015). students can feel at ease and enjoy studying at school through these interactions, which will help them meet their learning goals (biesta, 2015). the students' situation and psychosocial state must be considered by the teacher before teaching, especially in math class (roeser et al., 2013). this seeks to help students effectively absorb the knowledge the teacher is trying to deliver (hattie, 2015; rasmitadila et al., 2020). this leads to patterns of social interaction between students and their environment, which can assist teachers and students in creating effective learning environments since teachers can recognize the diversity and habits of students. this notion is consistent with constructivism, which identifies the importance of social and interpersonal interactions, as well as an individual's ties with their social environment, as the starting point for knowledge (bozkurt, 2017; endres & weibler, 2017; galbin, 2014). instead of only remembering formulae or theorems, students are considered to grasp mathematical concepts if they can create cognitive links between new experiences and their prior comprehension of mathematics (bujak et al., 2013; esteban-guitart & moll, 2014; haylock & manning, 2018). students engage with other students and their groups, and as a result, the learning process involves social relationships (amineh & asl, 2015; argyle, 2017). one technique for organizing very diverse students is to divide them into study groups, as in cooperative learning in mathematics (capar & tarim, 2015; chan & idris, 2017; slavin, 1988; zakaria et al., 2013). some of the learning goals achieved through group work include allowing students to discover their unique talents and skills, making the material simpler to understand, giving students more roles and responsibilities for learning and understanding the materials, and raising students' awareness of cooperation, mutual tolerance, and respect (islamov et al., 2016; kauffman, 2015; nenotaek et al., 2019; zakaria et al., 2013). as of now, the process of creating study groups, such as in cooperative learning models for mathematics, is heterogeneous and uses several ways, such as counting techniques and random, peer-to-peer, or lottery processes (van ryzin et al., 2020). in reality, some earlier researchers who studied cooperative learning noted that the creation of groups in collaborative learning occurs randomly based on peers (ji et al., 2016; stigmar, 2016). the issue arises when some students encounter mathematical difficulties and are hesitant to approach the teacher or other group members who are not on the same "frequency" as them. this is true even when the research results indicated that students' cognitive abilities had improved. even though math class can be integrated into society, friendship groups are rarely formed in class (esmonde et al., 2013; esmonde & langer-osuna, 2013; fields & enyedy, 2013). in mathematics class, study groups built on friendships between students will reduce problems with students who are unwilling to ask the teacher questions or do not participate in group activities consistently (wang & tahir, 2020). volume 12, no 1, february 2023, pp. 27-40 29 in this regard, this study aimed to determine the impact of forming study groups based on friendship in mathematics class on students' cognitive abilities. this study defines the student's cognitive ability as solving mathematical problems. this ability allows students to use mathematical activities to solve problems in mathematics, other sciences, and everyday life (widodo, 2017; widodo et al., 2017; widodo et al., 2019b). in addition, a sociograph can be used to illustrate the dynamics of friendship in math class. the sociograph pathway can identify the math study group depending on how closely the students are interconnected. it is hoped that students will be able to ask their peers to understand complex content rather than coming to their supporting teacher. 2. method 2.1. research design because external factors that affect research results cannot be controlled entirely, this quasi-experimental study was chosen. a nonequivalent post-test control-group design was adopted for the investigation. in general, the experimental and control groups' student groups engage in the same type of learning, known as problem-based learning. the same teacher observed the learning process, the control and experimental groups attended the same class, and the subject matter was the same to prevent teaching-related bias from affecting the research findings. this questionnaire aims to discover if students will likely seek assistance when presented with mathematical problems. following that, four groups with four students each were chosen. the researcher verified that establishing two groups was based on friendships among students in the mathematics class using the four selected groups. the experimental group was later used to refer to these two groups. finally, students from the control and experimental groups took a post-test to gauge their cognitive capacities after learning the math material at the end of the lesson. 2.2. participant purposive sampling was used to select participants for this study. purposive sampling is a technique with defined goals and characteristics (etikan et al., 2016). the study aimed to see how group learning using a sociograph affected the ability to solve mathematical problems. as a result, the students used in this study were 16 students divided into four study groups. two of the four study groups were formed based on student friendship pathways, so they were assigned to the experimental group. the other two groups, in contrast, joined the control group because group formation was not based on student friendship pathways. a friendship questionnaire is required to determine who the closest friend is in order to determine the friendship network. this friendship questionnaire only has one question: "when faced with a mathematical problem, whom do you turn to for assistance? mention no more than two students! ". figure 1 depicts creating a friendship path or sociograph based on the questionnaire results. following all these, two study groups were formed based on the friendship pathway: group a, which consisted of 1, 2, 18, and 19, and group b, which consisted of 5, 6, 16, and 22. one group was compared to sociograph groups to determine whether study groups formed on sociographs have a positive effect; namely, study groups formed heterogeneously and not based on friendship pathways. group c consists of 18, 15, 23, and 24 students, while group d consists of 9, 15, 20, and 26 students. widodo et al., the sociograph: friendship-based group learning … 30 figure 1. sociograph in the mathematics class 2.3. research instruments a research instrument is a tool used to measure an object and gather data from a variable (taherdoost, 2016). a mathematical problem-solving test provided after each learning session served as the research instrument for this study. because of this, each learning session's test for research participants only consists of a one-word problem with a mathematical solution. the format of this test is based on the material quadrilateral aspects. this content contains squares, rectangles, parallelograms, rhombuses, kites, and trapeziums. the score criteria for the test of mathematical problem-solving are shown in table 1 (widodo, 2017; widodo et al., 2017; widodo et al., 2019b). table 1. indicators of the problem-solving from polya polya's step score indicators understanding the problems 3 students can explain in written form what they know and need from the clearly-stated problem. 2 students only write (express) what they know or what is asked of them. 1 students note down (disclose) data/concepts/knowledge that is unrelated to the nature of the problem, causing students to misunderstand the problem at hand. 0 students do not write anything, so they do not comprehend the nature of the problem. devise a plan (translate) 2 students write down/tell the sufficient and necessary conditions (formula) of the problem posed and use all the collected information. 1 students tell/write the steps to resolve the problem but must do it more coherently. 0 students do not tell/write the steps to solve the problem. volume 12, no 1, february 2023, pp. 27-40 31 polya's step score indicators carry out the plan (solve) 4 students carry out their plans and follow the steps to solve the problem correctly. there are no procedural errors, nor are there any algorithm/calculation errors. 3 students carry out their plans correctly, following the steps to solve the problem, and there are no procedural errors, but there are algorithm/calculation errors. 2 students carry out their plans, but errors in procedure occur. 1 students carry out their plans, but procedural, and algorithm/calculation errors exist. 0 students need help to carry out the plans that they have made. look back (check and interpretation) 1 students re-check their answers. 0 students do not re-check answers. 2.4. data collection and analysis examples of questions that are posed to subjects at the end of a learning session include: a square photo frame is rotated at 45°, with the axis of rotation at the point where the diagonals intersect. if the square's side length is 1 cm, determine the area of the slice between the photo frame before and after rotating it! the learning model used in both groups (control and experimental groups) in this study was problem-based learning. furthermore, the mathematics material given to both groups was the same, the teacher who provided the mathematics material to the two groups was also the same, and the treatment time (including learning time) assigned to the two groups was also the same. this was carried out to avoid biased research results. therefore, researchers make every effort to prevent non-observational variables that could tamper with the findings of this study. 2.5. data analysis the post-test provided to students at the end of the learning session is given 15 times since the math teacher conducts learning for around one month with 15 meetings. moreover, the post-test results on problem-solving were used to calculate the average level of problemsolving for each treatment group. in addition, the statistical package employed the average findings from each session of the social sciences (spss version 21) program to determine the data distribution, the overall average for each group, and the standard deviation in demographic data. finally, the average of each treatment group's data was examined in a paired t-test to determine the answer to this study question. using the independent t-test, it is feasible to conclude that study groups based on friendship paths (sociograph) influence problem-solving skills. t-value is obtained if the significance coefficient is less than 0.05. widodo et al., the sociograph: friendship-based group learning … 32 3. result and discussion 3.1. result geometry is the mathematical material used in this study to determine the area and perimeter of a rectangle. the teacher conducts learning 15 times to complete this material, and table 2 shows the average problem-solving ability for each session and group. table 2. the score of problem-solving skill session a score of problem-solving skill control group experiment group 1 7.250 8.000 2 7.250 8.375 3 8.375 7.125 4 7.250 7.875 5 7.625 7.750 6 7.375 8.000 7 7.625 7.625 8 7.625 7.875 9 7.500 7.125 10 7.000 8.625 11 7.500 7.250 12 7.500 8.125 13 7.000 8.375 14 7.750 8.125 15 7.250 8.625 average 7.458 7.925 the data in table 2 is used to calculate a t-test using ibm spss statistics version 25 software. an assumption test, namely the population normality test and variance homogeneity, is performed before calculating this paired t-test. the population normality test employs the chi-square test, assuming that if a significance coefficient (asymp. sig.) greater than 0.05 is obtained, the sample is drawn from a normally distributed population. the levene test is used in the population variance homogeneity test, which assumes that if a significant coefficient (sig.) of more than 0.05 is obtained, the two samples used have the same variance. table 3. the result of the normality test with chi-square exsperiment control chi-square 1.200 4.133 df 8 6 asymp. sig. 0.997 0.659 table 3 shows that the chi-square calculation results for the experimental and control groups were 1.200 with an asymp. sig of 0.997, respectively, and 4,133 with asymp.sig. volume 12, no 1, february 2023, pp. 27-40 33 of 0.659. these results indicate that the samples in the experimental and control groups come from normally distributed populations. the results of levene's test calculations using ibm spss statistics version 25 software showed that levene's test statistic was 2.042, df1 was 1, df2 was 28, and a significance coefficient was 0.164. these results indicate that the two sample groups used have the same variance. after performing the t-test assumption test, and then conducting the paired t-test using the ibm spss statistics version 25 software, it was discovered that the t-test is 3.039, the df is 38, and the sign coefficient is 0.005. if the t table for a df of 28 is 1.699, then the tob obtained exceeds the t table. furthermore, based on the obtained significance coefficient of less than 0.05, the formation of study groups based on the friendship path (sociograph) affects the ability to solve mathematical problems differently. table 4. the result of descriptive statistics group n mean standard deviation experiment 15 7.925 0.488 control 15 7.458 0.340 according to table 4, the average control group was 7.927, while the average experimental group was 7.458. based on these averages, study groups formed through friendship pathways (sociograph) have a more significant effect than those formed through other means. 3.2. discussion the findings revealed that study groups formed based on friendship paths (sociograph) had a more significant impact than those formed based on other criteria. this demonstrates that the process of social interaction during learning will produce a rule or agreement that must be followed. these agreements are called norms. there are two norms in learning mathematics: social and socio-mathematic. social norms are rules or patterns of social interaction that are not tied to topics or learning materials, such as tolerance of the surrounding environment in daily interactions, how to adequately express opinions, and respect for the views of others. socio-mathematical norms are explicitly linked to mathematical argumentation, namely how students engage in the process of interaction and negotiation with their surroundings in order to understand mathematical concepts so that the arguments expressed can be mathematically accepted by others (bonotto, 2010, 2011, 2013; cobb et al., 1989; lopez & allal, 2007; partanen & kaasila, 2015; yackel & cobb, 1996; yackel et al., 1991; yackel & rasmussen, 2002). when the explanations and justifications made in the mathematics learning class are acceptable to the environment, sociomathematical norms can be formed (mueller et al., 2014). unknowingly, teachers and students have used socio-mathematical norms during the learning process, such as encouraging students to ask questions and argue during the learning process, creating a creative and innovative learning environment, and employing learning methods that enable students to become more active. the problem is that learning mathematics places less emphasis on friendship in mathematics class. friendship is one factor contributing to the development of socio-mathematical norms and the acquisition of problem-solving abilities (lopez & allal, 2007; widodo et al., 2020; widodo et al., 2019a). along with the communication that develops between people, friendships play a role in how strong a student community is at school. if handled effectively, these communities can help teachers carry out the learning process in the classroom. widodo et al., the sociograph: friendship-based group learning … 34 a social network or networking is a collection of interactions influenced by friendship and communication. for example, the relationship that develops between companies in the industrial world is based on the existence of formalized social networks. this type of relationship is referred to as social network analysis (freeman, 2004; serrat, 2017). in social media, an image of a network or networking formed from interaction and communication between individuals is called a sociograph (barkley, 2012; liberatore et al., 2018). however, several studies state that the sociograph picture is called a sociogram (corbisiero, 2022; saqr et al., 2018). for example, a graph known as a sociograph represents the social network shown in figure 1 (al-fayoumi et al., 2009). the general goal of sociographs in social media is to discover networks in regular communication or interaction (campbell et al., 2013; de jaegher et al., 2010; liberatore et al., 2018). by adopting the sociograph's function on social media, the sociograph is portrayed in the social interactions in the mathematics classroom setting to describe or be familiar with social networks in mathematics learning. teachers can forecast the critical sources for solving mathematical issues by studying the sources of social networks in mathematics learning. the learning process in the mathematics class can aid the teacher's mathematics learning if the communities built based on social interaction networks can be handled by the teacher effectively. this explains why instructors must establish networks or friendship networks in math classes. as shown in figure 1, separating subjects 20, 22, 19, 23, and 27 allows for creating math study groups. this is because anytime a question relating to mathematics is posed, they serve as the go-to resource or command center for students to find solutions. the ability of students in subjects 11, 13, 21, and 24 to solve mathematical problems is precarious. this is evident because no students in one class attempted to approach him for assistance when he was given a mathematics problem requiring that subjects 11, 13, 21, and 24 be placed separately. as a result, subjects 20, 22, 19, 23, and 27 focus on mathematical learning activities, and weak subjects are separated. teachers can effectively control mathematics study by creating learning communities based on social interaction networks. by adopting the sociograph pathway, teachers can reduce social disputes during arithmetic lessons, resulting in efficient learning. because of this, study groups formed based on friendship, or sociographic lines benefit students' problem-solving abilities. 4. conclusion sosiograph is a friendship channel that can be used instead of forming study groups in math class. students in the mathematics class with the learning center category must be separated so that they can replace the teacher's role in conveying material to friends in the group. acknowledgements the authors would like to thank the universitas sarjanawiyata tamansiswa and universitas pendidikan indonesia, who have facilitated this research. volume 12, no 1, february 2023, pp. 27-40 35 references al-fayoumi, m., banerjee, s., & mahanti, p. 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(2013). effect of cooperative learning on secondary school students’ mathematics achievement. creative education, 4(2), 98100. https://doi.org/10.4236/ce.2013.42014 https://doi.org/10.5951/jresematheduc.27.4.0458 https://doi.org/10.5951/jresematheduc.22.5.0390 https://doi.org/10.1007/0-306-47958-3_18 https://doi.org/10.4236/ce.2013.42014 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p183-190 183 problem-based learning model in elpsa framework on mathematical learning process in junior high school sukasno 1 , drajat friansah 2 , lucy asri purwasi 3 1,2,3 stkip-pgri lubuklinggau, mayor toha st, air kuti, lubuklinggau, south sumatera, indonesia 1 sukasno@gmail.com, 2 drajatfriansah@stkippgri-lubuklinggau.ac.id, 3 asripurwasi@stkippgri-lubuklinggau.ac.id received: june 04, 2018 ; accepted: august 05, 2018 abstract the objective of this research was to find out the effectiveness of mathematical learning outcomes to the eighth-grade students after giving treatment by using the problem-based learning model. a preexperimental method was used in this research. the population of this research was all eighth-grade students of junior high school. the number of the population was 103 students, while the sample was chosen randomly. the sample of this research was 33 students. data collection was carried out using a test in the form of an essay. the results of this research showed that it was significantly effective to use problem-based learning model in elpsa framework to the eighth-grade students of junior high school. it was proved by the average score of the final test after giving treatment was 79.33 and the total number of students who completed the score was 78.79%. keywords: problem based learning, elpsa, mathematics. abstrak penelitian ini bertujuan untuk mengetahui ketuntasan hasil belajar matematika siswa kelas viii smp negeri muara kati kabupaten musi rawas setelah diterapkan model problem based learning menggunakan kerangka kerja elpsa. metode yang digunakan dalam penelitian ini preeksperimental. populasinya adalah seluruh siswa kelas viii smp negeri muara kati sebanyak 103 siswa, sedangkan sampelnya dipilih satu kelas secara acak sebanyak 33 siswa. pengumpulan data dilakukan dengan teknik tes berbentuk uraian. hasil penelitian menunjukkan bahwa hasil belajar matematika siswa kelas viii smp muara kati setelah mengikuti pembelajaran matematika dengan model problem based learning menggunakan kerangka kerja elpsa secara signifikan tuntas. ratarata nilai tes akhir setelah mengikuti pembelajaran sebesar 79,33 dan jumlah siswa yang tuntas mencapai 78,79%. kata kunci: problem based learning, elpsa, matematika. how to cite: sukasno, s., friansah, d., & purwasi, l. a. (2018). problem-based learning model in elpsa framework on mathematical learning process in junior high school. infinity, 7(2), 183-190. doi:10.22460/infinity.v7i2.p183-190. mailto:sukasno@gmail.com mailto:drajatfriansah@stkippgri-lubuklinggau.ac.id mailto:asripurwasi@stkippgri-lubuklinggau.ac.id sukasno, friansah, & purwasi, problem-based learning model in elpsa framework … 184 introduction basically, mathematics lessons are one of the required subjects which should be joined in the national examination (ne), so students are required to study mathematics at school. the fact shows that the cognitive abilities possessed by each student are different, teacher’s role is so crucial because it will help to get the ability to accept and absorb mathematics learning as expected by teachers. as it is stated yuhasriati (2012) mathematics is one of the basic sciences that plays an important role both in the development of science and technology and in shaping the human personality. however, mathematics is a complex subject and hard to be learned. based on prevoius study which was done to the eighth grade students and interview result to the teacher of mathematics, it was found out that there are several problems in learning process. they are students have not reached the minimum mastery criteria (mmc). the number of students who can achieve mastery learning in the daily math test only reached 46.67%. mathematical learning models was still conventional, and the lack of learning innovations used by the teacher in the learning process. in addition, faradhilla, sujadi, & kurwardi (2013) state that the selection of inappropriate learning approaches will lead to less effective learning development in the classroom so that it will become one of the causes of low learning outcomes. to achieve the success of the learning process with problem based learning model, it is necessary to use a systematic learning framework namely elpsa learning framework (experiences, language, pictures, symbols, application). lowrie & patahuddin (2015) suggested that the elpsa framework is a learning design which is cyclical. this design presents mathematical ideas through life experiences, mathematical conversations, visual stimuli, symbol notations, and knowledge applications. the argument is a benchmark for collaborating elpsa framework with problem-based learning learning model. both of problem based learning model and elpsa are two concepts that have similarities in several stages, especially in extracting student learning experiences, where problem based learning is a learning model that emphasizes two direction of learning and environment. according to maharani & laelasari (2017), one of characteristics of pbl is to give students full responsibility for experiencing their own learning process directly. as it is also stated by surya, putri, & mukhtar (2017) that the learning process which facilitates students to solve problems based on everyday context can improve problem solving abilities. according to arifin (2015), elpsa framework is a model of learning design as a reference for teachers in designing lesson plan which gives students the opportunity to express learning experiences (experience), use language to describe experience (language), visualize images to present experiences (pictures), written symbolization to express experience in general (symbols), and (application) as the application of knowledge gained in solving various situations. in this learning design, teachers are encouraged to introduce concepts from what students comprehend (lowrie & patahuddin, 2015). problem-based learning (pbl) model and elpsa framework have similarities in terms of experience content so that by combining both of them, it is expected to create innovative, meaningful and memorable learning. it can improve students’ learning outcomes in solving problems in mathematics learning. the adoption of the problem based learning model in elpsa framework consists of five main steps and each step in problem based learning volume 7, no. 2, september 2018 pp 183-190 185 learning is used by elpsa framework. the problem based learning model stages in elpsa framework can be seen in table 1. table 1. syntax of problem based learning in elpsa framework stage teacher’s behavior stage 1 students’ orientation on problem teacher explains the learning objectives, explains the logistics needed, raises problems, motivates students to be involved in solving selected problems. in this stage elpsa framework is used, namely the experiences that students have in relation to the goals and problems given by the teacher. stage 2 organizing students for learning the teacher helps students to define and organize learning tasks related to the problem stage 3 guiding individual or group investigations the teacher encourages students to gather appropriate information, carry out experiments, to get explanations and problem solving. in this stage the elpsa framework is symbol and application. • symbol aspects symbols involve students in presenting, constructing, and manipulating information in the form of symbols for problem solving in mathematics learning. • application aspects in aspects this application states how understanding of symbols can be applied to new situations to get explanation and problem solving in mathematics learning. stage 4 present and present the results of the discussion the teacher assists students in planning and preparing appropriate works such as reports, videos and models and helps them to share assignments with their friends. in this stage the elpsa framework is used, namely pictorial and languange. • this pictorial aspect relates to visual representation in presenting appropriate works such as reports and students presenting work in front of the class. • the languages aspect is used appropriately in reporting information, presenting and presenting the results of discussions for understanding. at this stage the teacher is only as a facilitator and provides input. stage 5 analyze and evaluate the problem solving process the teacher helps students to reflect or evaluate their investigations and the processes they use method the research method used in this study was a pre-experimental method. the experiment used was the pretest and postest group design (arikunto, 2013). sukasno, friansah, & purwasi, problem-based learning model in elpsa framework … 186 where : a: random sample o: pretest = posttest (test of learning outcomes) x: the treatment of problem based learning model learning using the eplsa framework the population of this study were all eighth grade students. data collection was taken by using test. there were 6 question items by using problem-based and the items has fulfilled validity, realibility, level of difficulty and differentiation criteria. results and discussion results recapitulation of data from the pre-test and post-test results is shown in table 2 table 2. result data of the pre-test and post-test no category pre-test post-test 1 highest score 44 94 2 lowest score 10 66 3 average score 22,79 79,33 4 classical completeness 0 % 78,79% the results of the pre-test showed that the highest score was 44, the lowest score was 10, and the average overall score was 22.79. descriptively, it can be concluded that the initial ability of students before the implementation of mathematics learning with problem based learning model in elpsa framework was incomplete. while the post-test results show that the highest score was 94, the lowest score was 66, and the average score after giving treatment was 79.33, and the classical completeness was 78.79%. so descriptively it can be concluded that the final ability of students after the implementation of mathematics learning by using problem based learning model in elpsa framework was included in the complete category. when it was compared with pre-test data, the average score obtained by students was an increase of 56.54. in the pre-test there were no or (0%) students who were completed and in the post-test there were 26 or 78.79% of students who completed after giving treatment by using problem based learning model, using the elpsa framework. the average score and completeness of the learning pre-test and post-test can be seen in the figure 1 a o x o volume 7, no. 2, september 2018 pp 183-190 187 figure 1. average score of pre-test dan post-test the results of the calculation of the normality test and t-test of the post-test data can be seen in table 3 and 4. table 3. result of normality testing data df conclusion post-test 6,72 5 11,07 normal table 4. result of t-test data df conclusion post-test 5,23 32 0,05 1,68 rejected table 3 shows that the experimental class final test data was declared to be normally distributed. because the data was normally distributed, then to test the hypothesis using the t test. based on the results of the t-test (table 4), t-obtained > t-table, it means that ho was rejected and ha was accepted. thus the tested hypothesis can be accepted to be true so it can be concluded that the mathematical learning outcomes of the eighth grade students of junior high school after the application of the problem-based learning model using the elpsa framework was significantly effective. discussion the research was conducted at junior high school. it was conducted in five meetings starting with the provision of a pre-test, followed by giving treatment by using problem based learning model in elpsa framework three times and giving of a post-test. in addition, in the initial learning activities, researchers first provided information about learning objectives and explain how to learn by applying problem based learning model in elpsa framework and explaining the learning steps to be implemented. researchers divided students into seven heterogeneous groups based on the results of the pre-test. students do activities by working on student worksheets (sw) that contain problems. then, students’ activities in groups with elpsa framework after the teacher explores the experiences of students as follows: 0 20 40 60 80 score average completenes 22.79 0% 79.33 78,79% pre-test post-test sukasno, friansah, & purwasi, problem-based learning model in elpsa framework … 188 a. students define and organize learning tasks related to the problem, in this case students begin to discuss the material contained in the worksheet. during group discussions students should actively seek information, construct new knowledge according to prior knowledge; b. students gather information that matches the form of symbols for problem solving, carry out experiments in expressing how the understanding of symbols can be applied to new situations such as everyday life, daily activities (application) to get explanation and problem solving; c. one of the students represent the group presented the work in front of the class using teaching aids or pictures with the right language (languange) in the presentation, namely the language of mathematics or in other words the students as well as expressing mathematical symbols in language and oral form. after working on student worksheets (sw) students are asked to present the results to the class in a random way to move forward so that all groups must be prepared. each advanced group is asked to hold discussions with other groups. to solve problems that arise during the discussion, the teacher appears as the main facilitator and resource person by explaining using existing media. after being treated with problem based learning model in elpsa framework, post-tests were given as a benchmark for learning success. based on the results of the post-test, the average value was 79.33. hypothesis testing shows that t-obtained (5.23) > t table (1.68). this proves the hypothesis in this study was accepted the truth. this result was in accordance with the results of research conducted by arifin (2015) entitled "elpsa framework-based lesson plan to build an understanding of the concept of addition and reduction of students' integers". the results of his research show significant results where students are able to make relationships between concepts, present concepts in various mathematical representations. it means that understanding the concept can increase significantly after giving treatment by using elpsa framework. elpsa framework starts from connecting students’ previous experience to the new learning, giving students the chance to express their own finding, and building visual thinking of students to the presentation of ideas using symbols (johar & hajar, 2016). that is, when teaching practice allows students to develop mathematical ideas that are associated with their experiences or knowledge, and engage in discussions of mathematical ideas with other people, then the possibility of introducing the concept is significantly greater. applicationbased learning is beneficial for students in solving new problems, it was also supported by yew & goh (2016) which its effectiveness on the quality of student learning and the extent to which its promise of developing self-directed learning habits, problem-solving and deep disciplinary knowledge. conclusion based on the results of the research and discussion, it can be concluded that mathematics learning outcomes to the eighth grade students of junior high school after giving treatment by using problem based learning (pbl) model in elpsa framework is significantly effective, with t-obtained (5.23) > t-table (1,68). the average score of the final test after giving treatment is 79.33 and the number of students who achieve learning completeness are 78.79%. on the other hand, the limitation was the students still have limitations in solving complex story problems. volume 7, no. 2, september 2018 pp 183-190 189 references arifin (2015). lesson plan berbasis kerangka kerja elpsa untuk membangun pemahaman konsep penjumlahan dan pengurangan bilangan bulat pada siswa. 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(2013). prosedur penelitian. jakarta: rineka cipta. faradhilla, n., sujadi, i., & kurwardi, y. (2013). eksperiment model pembelajaran missouri mathematics project (mmp) pada materi pokok luas permukaan serta volume prisma dan limas ditinjau dari kemampuan spasial siswa kelas viii semester genap smp negeri 2 kartasura tahun ajaran 2011/2012. jurnal pendidikan matematika solusi, 1(1), 67-74. johar, r., & hajar, s. (2016). implementation of elpsa framework in teaching integral using technology. international journal of science and applied technology, 1(1), 15-21. lowrie, t., & patahuddin, s. m. (2015). elpsa as a lesson design framework. journal on mathematics education, 6(2), 1-15. maharani, a., & laelasari (2017). experimentation of spices learning strategies with the method of problem based learning (pbl) to build motivation and the ability to think logically for vocational school students. infinity, 6(2), 149-156. surya, e., putri, f. a., & mukhtar (2017). improving mathematical problem-solving ability and self-confidence of high school students through contextual learning model. journal on mathematics education, 8(1), 85-94. yew, e. h., & goh, k. (2016). problem-based learning: an overview of its process and impact on learning. health profession education, 2(2), 75-79. yuhasriati (2012). pendekatan realistik dalam pembelajaran matematika. jurnal peluang, 1(1), 81-87. sukasno, friansah, & purwasi, problem-based learning model in elpsa framework … 190 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p1-10 1 a case study of students’ creativity in solving mathematical problems through problem based learning farrah maulidia* 1 , rahmah johar 2 , andariah 3 1,2 universitas syiah kuala 3 mtsn 1 banda aceh article info abstract article history: received may 30, 2018 revised oct 14, 2018 accepted nov 16, 2018 creativity could be interpreted as a person's cognitive abilities in solving problems by bringing up new ideas. the problems of students’ math achievement lows are math presented as a finished product, ready to use, abstract and taught mechanistically. this case can be lead to the creativity of the less developed students because students are not given the opportunity to think and use their ideas in solving mathematical problems. problem based learning model is a learning model that emphasizes the concept and information outlined from the academic discipline. the purpose of this study is to analyzed students’ creativity in solving mathematical problems through problem based learning model (pbl) in class viii-1 mtsn model banda aceh. data gained based on the students’ worksheet in groups. the data acquisition is categorized into 5 levels (highest level 4 and lowest level 0) which is analyzed descriptively. the results are three groups were at level 4 with very creative categories, one group is at level 3 with a creative category and another group is at level 2 with deeply creative enough category. to the conclusion is pbl model could cultivate the students’ creativity in solving mathematical problems. keywords: mathematical creativity problem based learning copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: farrah maulidia, departement of mathematics education, universitas syiah kuala, jl. tgk. hasan krueng kalee, kopelma darussalam, syiah kuala, banda aceh, aceh 23111, indonesia. email: farrahmaulidia@gmail.com how to cite: maulidia, f., johar, r., & andariah, a. (2019). a case study of students’ creativity in solving mathematical problems through problem based learning. infinity, 8(1), 1-10. 1. introduction mathematics is a science that was very closely related to the problems of everyday life. a student in solving a math problem must have a good understanding, skill, and have a variety of strategies that can be used to solve different problems (grootenboer & zevenbergen, 2008). problems that arise in everyday life often require students to think deeper to solve them. this can be done if students are able to understand the basic concepts of the content which is received. one of the most important factors in learning mathematics is the creativity. creativity is needed to assist students in solving various mathematical problems related to daily life and other science (brunkalla, 2009). maulidia, johar, & andariah, a case study of students’ creativity in solving … 2 creativity can be defined as a person's ability to create a new idea or incorporate an old idea to create something new that can be used in solving the problem. mathematical creativity is the ability to open avenues of new questions for other mathematicians (sharma, 2014). creativity in mathematics is defined as the ability to see and choose solutions in mathematics (sriraman & lee, 2011). creativity is a dynamic process within a person that can produce several strategies to solve problems. creativity is a very valuable thing that must be owned by students in him, so that in learning mathematics students are expected to grow up their creativity by solving various forms of mathematical problems (akgul & kahveci, 2016). the expected of mathematical learning is the learning that create the students more active and creative in solving the problems were given. learning like this can involve students directly in solving problems given. but nowadays reality students tend to memorize concepts and definitions which are given without understanding those are (hendriana, 2012). low math skills can lead to the appearance of students' disrespect for mathematics. the low math ability of students can be seen from the results of national examinations of mathematics subjects ranging from primary education to secondary education. based on results the evaluation of the thrend in international mathematics and science study (timss) and the program for international student assessment (pisa), it is known that indonesian students have low math skills, where indonesia is always got lower ranked (mullis, martin, foy, & arora, 2012). one of the biggest problems causing low mathematics achievement is that mathematics is presented as a ready-made, abstract, and mechanically-taught product (sembiring, hadi & dolk, 2008). this case can be lead to the creativity of the less developed students because students are not given the opportunity to think and use theirs ideas in solving mathematical problems. good creativity will facilitate students in solving any mathematical problems given, therefore creativity is a very important thing that must be owned by students. problem based learning (pbl) model is an instructional method that challenge a students to learn. problem based learning model (pbl) is a student-centered learning model, students learn about subjects in complex, diverse, and realistic contexts. among the several advantages of pbl models is that pbl will have meaningful learning, in pbl method students integrate knowledge and skills simultaneously and apply them in relevant contexts, and pbl can improve critical and creative thinking skills. this study does not see the work of each student individually but the views are the work of students in groups, the goal is that students can exchange creative ideas that have in solving any given mathematical problems. based on the forward case, there has been no previous research that discusses about student creativity in solving matematical problems with a groups. so, the formulation of the problem in this research is: "how is the students’ creativity in solving mathematical problems through problem based learning (pbl) model? 2. method this research employs a descriptive method to identify students answer of testing the implementation of problem based learning (pbl) learning model to know the level of creativity in solving math problems for 2x40 minutes. the participant of this research is the students of class viii mtsn model banda aceh. one of the authors as a teacher during the learning process. learning activities are carried out as follows. a. the teacher begins the learning by motivating students about the use fulness of learningto flat-3d shape. next, the teacher shows several pictures related to flat-3d shape such as book shelves, swimming pools, andothers. volume 8, no 1, february 2019, pp. 1-10 3 b. students are requested to ask questions about the surface area and volume of a flat-3d shape so that students obtain additional information from the material being studied. next, the students are divided into several small groups consisting of 4-5 people. c. the teacher distributes student worksheets forstudents. d. students solve problems with limited guidance from the teacher. e. each group presents answers alternately in front of the class. f. the teacher and students conclude the lesson about building a flat-3d shape g. students gather their answers, and the teacher ends the learning. assessment of student creativity can be seen from several aspects. according to siswono (2011) aspects that must be considered to assess the creativity of students are: a. fluency is an indicator of creativity in problem solving that refers to the diversity and correctness of answers that have been given by students. b. flexibility is an indicator of creativity in problem solving that refers to the ways used by students in solving problems and correctness in accordance with the problems given. c. originality is an indicator of creativity in solving problems that refers to the answers or ways given are not usually done students at the level of knowledge or it could be with the incorporation of ways done by students to produce a new data of students creativity in solving mathematical problems derived from the students worksheet in their groups during lesson. students' creativity in solving mathematical problems is assessed based on the rubric in table 1 which is analyzed descriptively. the examples of this research instrument are as follows. many things in daily life related to the matter of building a flat side room. look at the following picture !!! in the picture above there are several forms of bookshelves are very beautiful and interesting, as well as some books are arranged on the shelf. consider the following story! maulidia, johar, & andariah, a case study of students’ creativity in solving … 4 students are said to be creative if they meet the following three aspects of creativity bellows: first is fluency, student can solve the problem given correctly, precisely and clearly. second is flexibility, student can be provide interpretation and able to complete the calculation of the value of the problems provided in various ways (more than one) completion. the third is orisinality, student has a different settlement from other groups and can make the image clear, neat and unique (aizikovitsh-udi, 2014). creativity in solving math problems are divided into 5 levels they are, level 4 (very creative), level 3 (creative), level 2 (creative enough), level 1 (less creative), and level 0 (not creative) (siswono, 2011). the criteria for determining the level of students creativity as seen at tabel 1 below. table 1. characteristics of student creativity level characteristics level 4 (very creative) students are able to demonstrate fluency, flexibility, and orisinality or orisinality and flexibility in solving math problems level 3 (creative) students are able to show fluency and orisinality or fluency and flexibility in solving problems level 2 (creative enough) students are able to show orisinality or flexibility in solving problems level 1 (less creative) students are able to show fluency in solving problems level 0 (not creative). students are unable to show the three aspects of the indicators in solving the problem 3. results and discussion problem based learning (pbl) model requires the students to solve problems by using their own ideas, this requires students to explore the information and think how to solve problems given so that creative ideas that they have can be used. the idea used by students for problem solving were different from group to others, so the results are vary. this can be seen in the picture below. mr. yuda is a very famous interior designer. mr. yuda got a job to design a reading room at mr. bahrun's house. mr. yuda's main job is to make a bookshelves on one side of the room. bookshelves that must be made have an overall size of 2.2 m x 25 cm x 1.8 m. the tasks you should do are: a. help mr. yuda to design a bookshelves, you may choose one of the shelves of the drawings provided, or you can design your own shelves, so that the book on the shelf looks neat and the shelf design looks interesting! make a settlement of more than one way! b. determine the maximum number of books that can be loaded on the shelf you've designed! and specify the various sizes of books that you will fill in that shelves ! c. do you have to use the volume formula to determine how many books are arranged on the shelf? why? volume 8, no 1, february 2019, pp. 1-10 5 translation: note : √ : filled books with the same number of books without √ = unfilled books in one part of the shelf can be filled 11 books in 13 sections of bookshelves there are 143 books figure 1. student’s answer figure 1 shows that students illustrated the picture of the bookcase along with the contents of the shelf, the students also gave an explanation of the illustration of the image in accordance with the material of the flat side room that has been learned. in the picture above students explained about the number of books that can be placed on the shelves that have been designed with several different book sizes. based on the illustrations of drawings and book sizes that have been established then the bookshelf can be filled with 143 books, with some shelves not filled with books. tthe picture above shows that students have a good creativity in answering a given problem. maulidia, johar, & andariah, a case study of students’ creativity in solving … 6 translation: a. in the picture above there are 16 books of different sizes. every shelf is loaded with books and there is also an unfilled shelf full of books b. the average size of each book is 25 cm x 20 cm x 60 cm (less than 90 cm) c. no, to find the number of books placed on bookshelves do not have to look for rack volume because not all shelf parts are filled with books. if the shelf is full of books also do not meet the rack volume. to specify the size of a book only takes the length, width and height of the shelf. figure 2. student’s answer based on the figure 2 also shows that the students make illustrations of the problems provided and provide an explanation of the illustrations he did. at point a students explain the rack design illustrated, point b students determine the size of the book that can be arranged in the shelf, the size of the book specified adjusted to the size of the shelf given to lkpd. point c is the conclusion given to the problem being worked on. based on the work of the students together with each group then the conclusion obtained is to determine the number of books that can be arranged on a rack with a certain size does not have to use the formula of built up volume, in this case is to build a cube or block, but all it takes is the size of the book which corresponds to the size of each rack level in the design. from the students' worksheet in figures 1 and 2,there are three indicators of creativity are fulfilled is fluency, flexibility and novelty, so that based on the criteria in the assessment rubric, the two groups of students are at level 4 with very creative category. students' work in figures 1 and 2 shows that students taught by implementing problem based learning (pbl) model will have a good creativity in solving various problems given. the results obtained according to the theory forward by (padmavathy & mareesh, 2013) which states that one of the characteristics of learning with pbl model is volume 8, no 1, february 2019, pp. 1-10 7 to produce a new works that will be exhibited, this requires students to use creativity in solving problems given not all students have the ability and good creativity in solving a problem, there are any students who do not understand the material taught and have difficulty in doing the task given. some things that make the students difficult in doing tasks are because students are less accustomed in solving non routine problems, so that students experience some obstacles when doing the task. this is in line with the research conducted by yuliani, noer, & rosidin (2018) which is the less developed mathematical creativity in solving mathematical problems caused by the learning that has been going on in the classroom, the students are given definitions and examples of routine questions without being linked first with the problems in daily life. leikin (2013) suggest that students' creativity will develop when students can use a different thinking in solving a given problem. but the results obtained from this study, there are still some students who were not able to use a different thinking in solving the problems given. this can be seen based on the following picture. figure 3. student’s answer figure 3 shows that some students are only able to illustrate the problem, but are unable to give a proper explanation of the problem, so that the indicator of creativity that is maulidia, johar, & andariah, a case study of students’ creativity in solving … 8 fulfilled is only flexibility and novelty, then the students in this group are on level 2 with enough creativecategory. this is due to the fact that students in this group spend more time tidying up and perfecting the picture resulting the neatest and most beautiful picture in theclass. consequently, their mathematical explanation of the answer is incomplete . however, the teacher has reminded students as shown in the following passage. t: where is the answer? s: in a momentmiss, thepicture has notfinishedyet, afterthiswewillmaketheanswer t: two people should draw, two people should continue to answer, otherwise you will not be able to finish as the time goes on. s: well miss, a little more for the picture. the group’s mathematical answers were also not written on the answer sheets that were collected in stead they wrote numerical answers on the other sheets, and their solutions were not complete as shown in figure 4. translation: known: shelf leght: 2,2 m = 220 cm shelf width :25 cm shelf height : 1,8 m = 180 cm asked: the maximum number of books that can be filled on the bookshelf ? shelf volume : l x w x h shelf i : book leght :15 cm book width : 1,5 cm book height : 20 cm book that can be filled on bookshelf i are: ..... figure 4. student’s answer based on figure 3 and 4, it can be seen that the students’answers are not detailed and profound so that these students do not meet the criteria of fluency in their creativity as students in this group prefer to complete the picture entirely and neatly resulting in insufficient time to provide answer. this is in line with the research conducted by surya (2010) which states that visual type students prefer to solve a problem by using an image if the problem given is a new problem for them or the problem given is a matter of the story. in addition, other groups try to provide explanations and illustrations of the problem. however, the explanation given is not appropriate to answer the problems that exist in student worksheet, this is shown in figure 5, so it only meets the indicators of fluency and orisinality. based on criteria in the rubric of creativity assessment then this group is at level 3 with creative category. volume 8, no 1, february 2019, pp. 1-10 9 translation: volume bookshelf = 2,2 m x 25 cm x 1,8 m = 220 cm x 25 cm x 180 cm  shelf 1 = 10 book, with h = 17 cm and l = 3 cm  shelf 2 = 10 book, with h = 10 cm and l = 3 cm  shelf 3 = 10 book, with h = 8 cm and l = 3 cm  shelf 4 = 10 book, with h = 17 cm and l = 3 cm figure 5. student’s answer based on the results above, it can be generally be argued that there is 1 group achieved level 2, 3 groups achieved level 3, 3 groups achieved level 4, and no group in level 1 or 0. mathematics learning with problem based learning model (pbl) gives a positive impact in fostering student creativity. this is in line with finding by gunantara, suarjana & riastini (2014) state that the problem based learning (pbl) model can improve students' ability in solving mathematical problems. students who have good skills in solving mathematical problems will also have good creativity in solving various problems given, this is in line with the theory put forward by savery (2015) stating that the pbl model can increase students' ability in solving math problems. research conducted by pelczer & rodriguez (2011) also states that students' creativity will develop if students are given an open problem so that students can use their mind, imagination and instinct to solve the given problem. 4. conclusion based on the results of trials that have been done in mtsn 1 banda aceh with the subject of research is class viii-1 by applying the model of problem based learning (pbl) learning to cultivate students' mathematical creativity, then obtained the result that were three groups are in very creative category,one group was in creative categories and one group was in the creative enough category.these results reveal that pbl learning model can foster students' creativity in solving math problems. acknowledgements the authors would like to thank the lecturers for their support and guidance so far, as well as teachers and students of mtsn banda aceh model for their participation in realizing the implementation of this research. without them this research will not be done. maulidia, johar, & andariah, a case study of students’ creativity in solving … 10 references aizikovitsh-udi, e. 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(2018). guided discovery worksheet for increasing mathematical creative thinking and self-efficacy. international journal of trends in mathematics education research, 1(1), 30-34. infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 1, february 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i1.p61-68 61 design of learning materials on limit function based mathematical understanding muchamad subali noto 1 , surya amami pramuditya 2 , yudrick maulana fiqri 3 1, 2 ,3 universitas swadaya gunung djati, jl. perjuangan 1, cirebon, indonesia 1 balimath61@gmail.com, 2 amamisurya@gmail.com, 3 bimbingan16@gmail.com received: january 9, 2018 ; accepted: february 1, 2018 abstract this study aimed to analyze learning obstacles, designing learning materials based on the material mathematics understanding algebra limit function, determine teacher intervention during the implementation of learning materials and to analyze learning obstacle after the implementation of learning materials. this research is a design research using the form didactical design research. stages of research conducted: 1) adp before learning, 2) analysis of metapedadidatik and 3) the retrospective analysis. the instrument used was a matter cmat (comprehension mathematical ability test), interview, validation sheet, and documentation guidelines. research results obtained are: 1) students' difficulties in relating the material prerequisites to limit problems. 2) students can not write properly limit symbol, 3) students can not apply a limit theorem, 4) students are not able to determine the limit value at one point, and 5) students cannot determine the value of the limit at infinity. learning materials that have been made have validation level of (valid). the response was given when the student intervention, generally in accordance with response prediction so that interventions carried out in accordance with the design that has been made and learning obstacles implemented reduced. keywords: didactical design research, learning obstacle, limit functions algebra, mathematical understanding. abstrak penelitian ini bertujuan untuk menganalisis hambatan belajar siswa, mendesain bahan ajar berbasis pemahaman matematis pada materi limit fungsi aljabar yang valid, mengetahui intervensi guru saat implementasi bahan ajar serta menganalisis hambatan belajar siswa setelah implementasi bahan ajar. peneitian ini merupakan penelitian desain berupa didactical design research. tahapan penelitian yang dilakukan yaitu: 1) analisis situasi didaktis sebelum pebelajaran, 2) analisis metapedadidatik dan 3) analisis retrospektif. instrumen yang digunakan adalah soal tkpm (tes kemampuan pemahaman matematis), pedoman wawancara, lembar validasi bahan ajar, dan pedoman dokumentasi. hasil penelitian yang didapatkan adalah: 1) siswa kesulitan dalam mengaitkan materi prasyarat dengan permasalahan limit. 2) siswa tidak dapat menuliskan simbol limit dengan benar, 3) siswa tidak dapat menerapkan teorema limit, 4) siswa tidak dapat menentukan nilai limit di satu titik, dan 5) siswa tidak dapat menentukan nilai limit di tak hingga. bahan ajar yang telah dibuat memiliki tingkat validasi sebesar dengan kriteria sangat valid. respon yang diberikan siswa saat intervensi, secara umum sesuai dengan predisi respon sehingga intervensi yang dilakukan sesuai dengan rancangan yang telah dibuat. setelah bahan ajar diimplementasikan hambatan belajar siswa berkurang/terminimalisir. kata kunci: didactical design research, hambatan belajar, limit fungsi aljabar, pemahaman matematis. how to cite: noto, m. s., pramuditya, s. a., & fiqri, y. m. (2018). design of learning materials on limit function based mathematical understanding. infinity, 7 (1), 61-68. doi:10.22460/infinity.v7i1.p61-68. noto, pramuditya, & fiqri, design of learning materials on limit function … 62 introduction mathematics is one of the lessons that characteristics are abstract and difficult. most students at all levels of education believe that mathematics is a daunting and difficult lesson to learn because it is abstract. mindset or way of thinking like this that make the students feel pressure to learn mathematics. math is a subject that is not liked even hated subjects for children in general. according suhandri (2016), was difficult mathematics learning by students because most students have not been able to connect between the material being studied with the knowledge to use. therefore, the learning of mathematics is so important, because in the daily life of the students will encounter problems of mathematics where students are required to complete the math problems to master an understanding of mathematical concepts. wahyuni and kharimah (2017) state that the ability of mathematical understanding is an important factor in the study of mathematics and should be owned by the students to be able to solve the problems of mathematics. as said by ratnasari (2014) the ability of mathematical understanding is the ability to be owned by the students in achieving the goals of learning, where the material presented to students not just rote, but more than that the students should understand the concept of the subject matter itself and how the application of these concepts to solve problems. this is in harmony with bloom that an understanding not just of the fact, however, with regard to the ability to explain, explain, interpret or ability to grasp the meaning or the meaning of a concept. it can be concluded that the ability of mathematical understanding is one of the important goals in learning, because with control of students' mathematical understanding can develop learning ability in mathematics, students can apply the concepts they have learned to solve simple to complex. however, the importance of understanding mathematical ability is not fully developed in the process of learning mathematics, in particular, the limit with the subject matter limit algebra functions. based on the results of preliminary studies conducted by researchers obtained data from student learning in the classroom xi-grade ipa senior high school in cirebon based mathematical understanding of the subject of algebraic functions that limit students' average score was 47.53. the average value is still quite low if it is in the interval 1-100. in addition, if the result of mathematical understanding based learning is associated with the chief engineer at the school, it can be said to be the result of students' mathematical understanding based learning in the classroom xi-grade ipa senior high school in cirebon yet reached kkm, that are 75. these results indicate that mathematical understanding based learning outcomes of students at that school is still relatively low. the material limit function began to study at the senior high school level (sma) in xi-grade. it is an abstract matter, based on interviews with a math teacher at the school that some students in understanding the material limit function having trouble learning and solving problems limit function. this is in line with hidayat (2017), the limit function material students have difficulties in calculating the value of the limit function. in determining the limit of a function at a point and at the point of infinity students are still difficult to choose which way is appropriate to solve the problems because it can not examine the forms you see. in addition, students are still difficulties in factoring, multiplying by a factor of opponents, split with the highest rank, and apply the properties of the limit function to find the value of the limit of a function. based on the results of preliminary studies, interviews and relevant research it can be concluded that there are difficulties experienced by students in the study material algebra limit volume 7, no. 1, february 2018 pp 61-68. 63 function. difficulties and obstacles experienced by students in the study called the learning obstacle. according to brousseau (2006), learning obstacle is divided into three types of obstacle ontological (mental readiness to learn), didactical obstacle (teaching teachers) and epistemological obstacle (knowledge of students who have limited application context). if you paid attention, learning obstacles/ barriers to learning experienced by students in the study material that is limit function learning obstacle/ barriers to learning that are epistemological (knowledge of students who have limited application context). learning materials is very important for teachers and students, because the materials have a major contribution to the success of the learning process is implemented. the usefulness of the actual learning resources cannot be separated from the aim of learning resources that become meaningful. according to noto, hartono & sundawan (2016) the development of teaching materials must be based to the description of the abilities of the attention to the difficulty/confusion created by the students. function, purpose, and benefits of learning materials for teachers, among others, is to change the role of a teacher educator as a facilitator, helping students to learn something, and educators will have instructional materials that can assist in the implementation of learning activities. while the function, purpose, and benefits of learning materials for students, among others students, can learn the appropriate speed of each, helping students learn, and learning activities become more attractive. one type of resource is a module. module is a teaching material systematically arranged in an easily understood by students according to their level of knowledge and age so that they can learn on their own (independent) with minimal assistance or guidance from educators. by making the design of learning materials developed by learning obstacle is expected that students no longer have significant obstacles during the process of mathematical understanding. based on the identification of the above problems, the formulation of the problem of this study is as follows. method the research method is a qualitative research and design used didactical design research. according to cresswell (2012) qualitative research is an approach to building a knowledgebased statement-constructive perspective, perspective-participatory or both. according to suryadi (2013), study design didactic basically consists of three stages: (1) analysis of the situation didactic before learning that his form is in the form of design didactic hypothetically including adp; (2) analysis metapedadidaktik; and (3) analysis of retrosfektif the analysis linking the results of analysis of didactic hypothetical situation with metapedadidaktik analysis results. subjects in this study are divided into two, namely the subject of early identification of learning obstacle and the subject of learning materials design implementation. early identification subject of learning obstacle is the student who has received the material in xigrade limit algebraic function in ktsp 2006 that xi-grade ipa 1 at one of senior high school in cirebon and subject-based teaching material design implementation capabilities mathematical understanding on the material limit algebra functions are students who do not get the material limit function, which is in xi-grade ipa 2 in the same school. data collection techniques used were tests, interviews, questionnaires (questionnaire), and documentation. the instrument used was a matter cmat (comprehension mathematical noto, pramuditya, & fiqri, design of learning materials on limit function … 64 ability test), interview, validation sheet materials, and documentation guidelines. instruments of test trials used in the form of the ability of understanding mathematical description consist of 8 questions later identified his learning obstacle.to find out the merits of a matter that has been tested then have to do the analysis item, which is to see the validity, reliability, difficulty index, and the distinguishing features of the matter. instruments such interview guides for teachers and students. guidance teacher interview, containing questions about the learning materials used by teachers as well as the difficulties experienced by students when studying algebra limit function. while the student interview guidelines contain students about the difficulties experienced when working on the problems limit function-based algebra and mathematical understanding of the students' instructional materials used in the learning process. instruments of expert validation sheet are given to the experts after the design is completed instructional materials. validation sheet materials used to determine the level of validity of a decent teaching material and whether or not a resource is used. validator experts chosen are two lecturer of mathematics education unswagati cirebon and a senior teacher of mathematics at senior high school. instrument documentation in the form of documentation guidelines. guidance documentation did when researchers conducted a variety of activities associated with the research. documentation is also used as physical evidence of the activities of researchers. forms of documentation such as photos and videos when researchers make the learning process in the classroom. through video, documentation can be analyzed by descriptive prediction of a student's response, by linking the response and anticipation that has been made previously by the student responses that occur when implemented. results and discussion the results will be presented of which include learning obstacle before the implementation of the learning materials, learning materials validation results from three validators, intervention teachers in the implementation of learning materials and analytics learning obstacle after the implementation of learning materials. learning obstacle pra implementation of instructional materials based on the stages of the research, the authors compile learning materials appropriate learning obstacles found while testing the matter. based on the previous explanation of the difficulties experienced by students, it appears that the difficulties that occur are of learning obstacle the type epistemology where students have limited knowledge in a particular context. according to brousseau (2006), to find resistance epistemology one of them by finding common mistakes made repeatedly and are grouped under the concept. from the analysis of learning obstacle that has been done there is an error, namely 1) the student is not able to associate the material prerequisites to problems limit algebraic function, 2) students can not write symbols limit correctly, 3) students can not apply a limit theorem, 4) students are not able to determine the limit value at one point, and 5) students cannot determine the value of the limit at infinity. anticipations of the teacher when learning and presented in modules that have been made, is expected to overcome learning obstacles volume 7, no. 1, february 2018 pp 61-68. 65 experienced by students related to the ability to understand mathematical concepts on material addition and subtraction of integers. students are not able to associate the material prerequisites to limit the problems of algebra functions. the material limit function began material closely related functions, one example is the rational functions, irrational. in addition, the material algebra operations, rationalizing the root is also closely related to the limit algebra functions. in this case, the student is not able to link the concepts of equations or these functions on the material limit. this is evidenced by the student's answers to question number 1, 2, 3, 5, and 6. based on the students' answers on the number, students can not associate the material prerequisites to limit problems with either. this is due to students' understanding of the material is poor. students are not able to associate the material prerequisites to limit the problems occur because students do not understand the material previously associated with the material limit. the material is a very important prerequisite for understanding the material terms of the next well. topic or concept as a basic prerequisite for understanding the topics or concepts hereinafter. if students want to understand the material well, then he should better understand the material prerequisites again. students cannot write properly limit symbol occurs because students do not understand the concept of mathematical symbols. symbol or symbol used to represent a number. according to warsitasari (2015), student proficiency in using algebraic notation is highly dependent on their understanding of symbols. so it can be said that the students' understanding of a very important symbol in mathematics. if students are able to understand the symbols properly, then the ability of student understanding in solving the problems associated with the notation/symbols can be increased. results validation of learning materials on limit function material based on mathematical understanding designability-based learning materials are made in the form of mathematical understanding of mathematics modules. this module is based on learning difficulties experienced by students. modules are made according to the learning objectives and stages on the theory of jerome s. bruner through three stages of cognitive developments that enactive stage, the stage of the iconic and symbolic stage. the module also contains a didactic situation and anticipation of didactic to anticipate the various learning obstacles that arise. three stages of cognitive development according to bruner is applied to the module that stage enactive poured on sampling related to everyday life in the material limit algebraic function, phase iconic set forth in modules in the use of illustrations to instill the concept of limits, and the stage of the symbolic on the module with presented the concept of illustrations to form symbols or words. modules that have been prepared and then validated by four experts. the results obtained validation of 89.50%, this indicates that learning materials have been prepared otherwise very valid or can be used without revision. this can be evidenced by the results of the validation every aspect, even one of the 9 aspect indicator obtain a high value, namely the aspect of completeness dish. aspects of completeness dish obtain validation value of 93.75% with a very valid criteria. components completeness dish consists of standards of competence, basic competence, learning indicators, table of contents, bibliography and summary. syllabus ktsp 2006 used as a reference in selecting a standard and basic competencies. then from the basic competence is developed further in making learning indicators in accordance with the kd. noto, pramuditya, & fiqri, design of learning materials on limit function … 66 table of contents page contains related information contained in the module. while bibliography contains reference source of learning materials created. akbar (2013) suggested that good learning materials mentioned competencies that must be mastered readers, providing benefits to the reader the importance of mastering competencies, presents a table of contents, and presents a bibliography. presentation of the material should be complete, systematic, grain conformity with the demands of a student-centered learning and the manner of presentation that makes easy to read and learn to cultivate students' interest and desire to learn. however, the inputs of each validator into consideration for the improvement of learning materials module. the validation results of four expert learning materials as follows. table 1. results of validation expert no. validator percentage validation criteria validation 1 validator1 93.52% very valid 2 validator2 82.41% valid enough 3 validator 3 92.50% very valid intervention teacher in implementation subjects student responses are given when intervening, generally, correspond to the predictions response to interventions made in accordance with the design that has been made. intervention teachers conducted during the implementation depicted in anticipation pedagogical. the previous anticipation has been made by the teacher to solve problems beyond prediction response that does not cause learning obstacle. a new based on the analysis found that the situation 1 discussed on how to determine the limit of the left and right limits are assisted using a number line. efforts that teachers do is give guidance to students to better understand the concept of the limit of the left and right limits to then conclude the answer in the form of a mathematical model related limit. based on the results of this intervention turned out to be able to improve students' mathematical understanding that barriers to learning can be minimized. situation 2 discusses how to apply the limit theorem. efforts teachers provide guidance related to the properties and remind limit theorem regarding the material prerequisites are a form of intervention teacher while learning materials are implemented. based on the data obtained, there is a response beyond prediction that the student cannot perform arithmetic calculations division operation. the author did not expect that students will have difficulty in calculating the arithmetic division operation for long-sharing concepts students are learning arithmetic, namely when class vii even while still in elementary school. according to sarah, suryadi and fatimah (2017) if the response is beyond prediction, the anticipation of pedagogical given to this response was spontaneous anticipation. one of the efforts the teachers that students are guided in the calculation of the distribution operation, the teacher guides the students to remind students about the calculation of the arithmetic properties of numbers, especially in arithmetic division operation. situation 3 discusses the limit symbol write correctly. efforts teacher to give a stimulus to the students related writing the correct symbol and remind limit regarding the material volume 7, no. 1, february 2018 pp 61-68. 67 prerequisites are a form of intervention teacher while learning materials are implemented. based on the data obtained by the students tend according to predictions. while the situation 4 shows the results of a student's response that there is a response beyond prediction, the student is still wrong in the calculation of the concept of algebra operations. efforts that teachers are the students reminded again of the calculation of algebraic operations. according to fauzi (2012), students error for understanding the concept can be minimized with the intervention of teachers based on the situation and the right time, to be reminded of such material in the form of questions that lead and dig for students to understand the concept without teacher give the final shape for granted. the teacher's role in facilitating the students when students are faced with learning difficulties in a relationship pedagogical module in accordance with one component of the didactic triangle proposed by suryadi (2013). to encourage optimum student learning process is influenced by several things including the relationship between teachers and students (hp). the pedagogical relationship is the teacher's role in implementing the learning materials and to anticipate the response of students to the learning materials that are not in accordance with the prediction of the previous teacher. analysis of learning obstacle after implementation of subjects having implemented instructional materials, learning obstacle identified at the outset can be reduced. learning obstacle that most greatly reduced obstacle learning students cannot write properly limit symbol. anticipation given are anticipating anticipation didactic and pedagogical. anticipation didactic learning materials presented in the form of problems related to limit the right of writing symbols. on these issues the student is given three options to determine the limit of writing correct, these problems are discussed with group members. the anticipation of pedagogical given when there are students / groups who are confused in determining the correct limit writing. anticipation pedagogical provided in the form of stimulus, the stimulus-related writing teacher gives the correct symbol limit by providing questions that lead students to be able to write symbols limit properly. the question in the form of the material prerequisites (in this case is to determine the value of the function) and the concept of writing procedural limit. to cope with students who are still not able to write the limit symbols correctly, more students are expected to be able to understand more regarding the material prerequisites function, precisely in determining the value of the function (hidayat, 2017). rizki & syutaridho (2014) both concluded that the design of learning materials ddr implemented to minimize the barriers to student learning. conclusion based on the discussions that have been outlined, the research concluded the learning obstacle are: (a) the student can not understand the concept of limit and left limit right, (b) the student can not write symbols limit correctly, (c) students cannot apply a limit theorem, (d) the student cannot perform the procedure calculate the limit at infinity, (e) the student cannot perform factoring and determine the form of the adjacency to solve the problems limit function. based on the results validation obtained from three validator, overall percentage of with a very valid qualification. intervention provided during the learning intervention limit algebraic function is didactic and pedagogical intervention. didactic interventions that do focus on how the teaching material is presented to students by taking into account prerequisite material and the stages of learning the learning theory of bruner, while a given pedagogical intervention focuses on how researchers deliver teaching the material to students. during the implementation of these materials more researchers using the method of group discussion. learning obstacle experienced by students in the study of algebraic functions limit noto, pramuditya, & fiqri, design of learning materials on limit function … 68 the material obtained from the test results about the instrument at the time of implementation. learning obstacle after the implementation of teaching material on the material-based limit function mathematical understanding of algebra can be reduced/minimized. references akbar, s. d. (2013). instrumen perangkat pembelajaran. bandung: pt remaja rosdakarya. brousseau, g. (2006). theory of didactical situations in mathematics: didactique des mathématiques, 1970–1990 (vol. 19). springer science & business media. creswell, j. w., & creswell, j. d. (2017). research design: qualitative, quantitative, and mixed methods approaches. sage publications. fauzi, k. m. a. (2012). kemampuan koneksi matematis siswa dengan pendekatan pembelajaran metakognitif di sekolah menengah pertama. jurnal paradikma, 6(1), 49-74. hidayat, w. (2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. noto, m. s., hartono, w., & sundawan, d. (2016). analysis of students mathematical representation and connection on analytical geometry subject. infinity journal, 5(2), 99-108. ratnasari, i. (2014). perbandingan kemampuan pemahaman matematis antara siswa yang menggunakan model pembelajaran problem based learning (pbl) dan model pembelajaran group investigation (gi) pada siswa kelas viii smp negeri 2 jalaksana. euclid, 1(1). rizki, s., & syutaridho, s. (2014). efektivitas bahan ajar bangun ruang sisi datar menggunakan 5e instructional model terhadap aktivitas dan hasil belajar. aksioma: jurnal program studi pendidikan matematika, 3(2), 1-9. sarah, s., suryadi, d., & fatimah, s. (2017). desain didaktis konsep volume limas pada pembelajaran matematika smp berdasarkan learning trajectory. journal of mathematics education research, 1(1), 31-42. suhandri, s. (2016). meningkatkan kemampuan pemahaman matematis siswa smp/mts dengan menggunakan strategi konflik kognitif. jurnal penelitian dan pembelajaran matematika, 9(2), 240-249. suryadi, d. (2013). didactical research (ddr) dalam mengembangkan pembelajaran matematika. prosiding seminar nasional pendidikan matematika. fmipa unnes. wahyuni, i., & kharimah, n. i. (2017). analisis kemampuan pemahaman dan penalaran matematis mahasiswa tingkat iv materi sistem bilangan kompleks pada mata kuliah analisis kompleks. jnpm (jurnal nasional pendidikan matematika), 1(2), 228-240. warsitasari, w. d. (2015). berpikir aljabar dalam pemecahan masalah matematika. apotema: jurnal program studi pendidikan matematika, 1(1), 1-17. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p307-322 307 infinity teaching measurement: the role of mathematics teachers’ enacted pck on learner outcomes hanlie botha*, corene coetzee, liezell zweers university of pretoria, south africa article info abstract article history: received apr 6, 2023 revised jun 9, 2023 accepted jul 2, 2023 published online aug 21, 2023 teaching is a challenging profession where teachers must create valuable learning opportunities to enhance learners’ conceptual understanding. apart from mathematical content knowledge, teachers use their pedagogical content knowledge (pck) that develops as they reflect on previously taught lessons and learner responses in assessments. international and national assessment studies showed that south african learners perform poorly in, among other topics, measurement. thus, we determined the gain in learner outcomes as revealed in a pre-and post-test on measurement and studied one of the pck domains, namely teachers’ enacted pck as informed by the baseline assessment learner outcomes. the aim was to determine how teachers’ enacted pck relate to learner outcomes. underpinned by a social constructivist paradigm, the study used a mixed-method research approach. data were gathered from a pre-and post-test written by 124 grade 9 learners taught by two experienced mathematics teachers in a city school in south africa. findings revealed that although some improvements are evident after the topic has been taught, the test was still experienced as difficult by almost all the learners. however, from the observations, there is little evidence that the experienced teachers extensively used the baseline assessment outcomes to inform their teaching. keywords: baseline assessment, enacted pedagogical content knowledge, learner outcomes, teaching strategies this is an open access article under the cc by-sa license. corresponding author: hanlie botha, department of science, mathematics and technology education, university of pretoria groenkloof campus, corner of george storrar drive and leyds street, groenkloof, pretoria, south africa. email: hanlie.botha@up.ac.za how to cite: botha, h., coetzee, c., & zweers, l. (2023). teaching measurement: the role of mathematics teachers’ enacted pck on learner outcomes. infinity, 12(2), 307-322. 1. introduction one of the core aspects of maximising learners’ learning opportunities, is quality instruction using tuition time effectively. the driving forces behind teachers’ quality instruction are appropriate goals, developed knowledge, and positive beliefs regarding mathematics and the teaching thereof (artzt et al., 2015). regarding a teacher’s knowledge, hill et al. (2008), distinguish between mathematical content knowledge and pedagogical content knowledge (pck). other components that enhance teachers’ instruction are their https://doi.org/10.22460/infinity.v12i2.p307-322 https://creativecommons.org/licenses/by-sa/4.0/ botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 308 ability to meaningfully use assessment, but also to create a positive classroom environment, create opportunities for meaningful discourse in class, and use of purposive activities (artzt et al., 2015). good instruction starts with teacher clarity on teaching intentions, and continue by using appropriate teaching and learning strategies, well-designed tasks, meaningful classroom discourse, multiple representations, various resources and manipulatives (fisher et al., 2017). these aspects are encompassed in hill et al.’s (2008) components of pck namely a teacher’s knowledge of the learners, knowledge of teaching mathematics, and knowledge of the curriculum. apart from mathematical content knowledge as a knowledge base, it is sound teacher’s pck, and in particular, topic-specific pck that makes for good instruction. the importance of exploring topic-specific pck is underlined by kirschner et al. (2015, p. 234) when saying that “… one major, if not the main theoretical premise behind studying pck, is that teachers with higher levels of pck are better able to help students learn”. geddis (1993) purported that pck is embedded in a teacher’s knowledge of learner understanding and misconceptions and strategies towards sound conceptual understanding by using different representations. these are accepted as knowledge components of pck at topic level (mavhunga, 2019). a valuable source to inform a teacher’s knowledge about learner understanding is the outcome of baseline assessment. sadler and sonnert (2016) found in a study conducted to 620 senior phase science teachers in 589 schools, that teachers having sound topic-specific pck regarding their learners’ misconceptions, are more likely to increase their learners’ knowledge, than those teachers lacking that knowledge. pck is a knowledge base that distinguishes the teacher from the mere subject specialist (shulman, 1986). shulman (1986, p. 9) described it as knowledge about “the ways of representing and formulating the subject that make it comprehensible to others”. in the years following shulman’s conceptualization of pck, researchers have embarked on studies using pck as a lens to investigate teaching and learning. veal and makinster (1999) developed a hierarchical taxonomy of pck in which they suggested that pck can be investigated at different levels, namely general, domain specific, and topic-specific pck. for investigating pck at topic level, mavhunga and rollnick (2013) proposed five components from which the transformation of teaching content develops. the components that are most relevant to the current study are: teachers’ knowledge of (i) learners’ understanding of the topic at hand, and (ii) representations, such as diagrams and models that can be used in a conceptual teaching strategy. in 2019 a revised consensus model of pck was developed that places what happens in the classroom at the centre of the model (carlson et al., 2019) where the enacted pck (epck) of the teacher is situated. the epck of a teacher is a subset of the personal pck (ppck) of the teacher which is informed by the collective pck (cpck); a pck belonging to the profession. pedagogical reasoning, that “takes place during all aspects of the teaching are unique to each teacher and every teaching moment” (carlson et al., 2019, p. 83), shapes a teacher’s enacted pck and resonates with dewey’s (1933) idea of reflective thought. dewey (1933) introduced the concept of reflective thought, describing reflection as “an active and deliberative cognitive process which involves the sequence of interconnected ideas that take into account the underlying beliefs and knowledge” (pedro, 2006, p. 130). reflection before teaching (reflection for action), when a teacher employs pck for the personal and the enacted realms of pck, can be done when a teacher is planning a lesson, to anticipate the possible outcomes of the lesson or how learners will act in the lesson. the learner outcomes of a baseline test could inform the teacher’s instruction and epck in terms of the learners’ pre-concepts and prior knowledge, appropriate modes of representation, teaching strategies, and integration with other mathematical topics and disciplines. findings from the trends in international mathematics and science study (timss) (reddy et al., 2020), and the southern and eastern africa consortium for monitoring volume 12, no 2, september 2023, pp. 307-322 309 infinity educational quality iv (sacmeq, 2017), two international assessment studies conducted in 2019 and 2013 respectively, revealed general poor performance from grade 9 mathematics learners. this was in line with the outcomes of the grade 9 annual national assessment (ana) (department of basic education, 2014a, 2014b), a terminated national assessment study. according to ijeh and nkopodi (2013, p. 473), the findings from the timss and sacmeq assessments concluded that factors influencing the quality of mathematics and science education “are likely to be deeply rooted in the learner, national curriculum, subject matter and pedagogical flexibility of teachers”. it has been posited that the problem lies in teachers having “limited content knowledge and ineffective teaching approaches and unprofessional attitudes” (kriek & grayson, 2009, p. 185) and this is still echoed in the timss report (reddy et al., 2020). 1.1. purpose of study to shed light on the role of teachers’ enacted pck on learner understanding, the curriculum topic of measurement has been chosen for this study, being identified in the timss item diagnostic report: south africa grade 9 mathematics (mosimege et al., 2016), as well as the annual national assessment of 2014 diagnostic report intermediate and senior phases mathematics (department of basic education, 2014a), as one of the topics south african learners find most challenging. this poor performance in measurement is not restricted to south africa, but is a global tendency. sisman and aksu (2016, p. 1294) referred to several international studies indicating that learners “have poor and superficial understanding of length, area, and volume measurement”. in their study conducted on grade 6 learners, they found learners specifically lack comprehension of the concepts; they confuse the perimeter with the area formula, and the volume with the surface area formula. even first-year students in tertiary institutions revealed the same kind of errors and misconceptions. a study conducted in the united kingdom, investigating 326 first-year bioscience students’ mathematical errors and misconceptions, revealed that “a high proportion (52–95%) of students encountered difficulties with individual questions involving the calculation of volume or surface area, the conversion of units of measurement, working with ratios, proportions and powers of 10, and determining magnification or magnitude” (tariq, 2008, p. 889). tariq (2008, p. 889) further mentioned that “many of the errors and misconceptions students exhibited were similar to those reported previously as made commonly by 13–14-year-old children”. in the south african school curriculum, two-dimensional (2d) shapes and threedimensional (3d) objects form a prominent part from the beginning of the intermediate phase (grades 4-6) to the end of the senior phase (grades 7-9). the department of basic education’s (2011a) reason for its prominence is that “it relates directly to the learner’s scientific, technological and economic worlds, enabling the learner to make sensible estimates and be alert to the reasonableness of measurements and results” (p. 6). from the report about the findings of grade 9 mathematics learners’ performances in international assessments, mosimege et al. (2016) pointed out that due to an overload of content in the intermediate phase curriculum, there is not enough teaching and consolidation time. due to limited tuition time, teachers do not involve learners in activities building 3d objects from their nets and having the opportunity to “touch and see the object to remember” (mosimege et al., 2016, p. 72). consequently, learners’ poor prior knowledge and conceptual understanding are carried over to the senior phase, and seemingly not addressed in the first two grades of the senior phase. the annual national assessment 2014 diagnostic report intermediate and senior phases mathematics (department of basic education, 2014a) indicated two areas of weaknesses. the first area was knowledge of properties of 2d shapes botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 310 and 3d objects, where learners “had difficulty in identifying regular and irregular 2d shapes” (department of basic education, 2014a, p. 21), “could not name a common 3d object and displayed limited knowledge in finding the number of faces and naming the shapes” (department of basic education, 2014a, p. 34). the second area was conversion of units in measurement where learners generally failed in “converting meters and centimeters to [only] centimeters” (department of basic education, 2014a, p. 36), and to “convert the given mass from kilograms to grams” (department of basic education, 2014a, p. 49). finding the perimeter and area of shapes, as well as using terminology and definitions in geometry, are also areas of weakness in grade 9. learners were, for example, required to “demonstrate knowledge of and skills in finding the perimeter and area of 2-d shapes” (department of basic education, 2014a, p. 53). this article is part of a larger study, of which one of the aims was to shed some light on the role of teachers’ enacted pck on learner outcomes. in this paper, the first objective was to determine the gain in learner outcomes as revealed in a baseline (pre) and a post assessment. the second objective associated with this aim was to study teachers’ enacted pck, as informed by baseline assessment learner outcomes, when teaching measurement. the components of teachers’ enacted pck under investigation were teachers’ knowledge and skills related to learner understanding of the topic, and teachers’ use of conceptual teaching strategies. the research questions we posed were: 1) how did learners’ performance change after instruction? and 2) how can the teachers’ enacted pck in relation to learner outcomes be described? 1.2. conceptual frameworks the refined consensus model (carlson et al., 2019) identifies three realms of pck: collective pck (cpck), personal pck (ppck) and enacted pck (epck). collective pck constitutes the canonical pck and is informed by research and discussion amongst peers. at the centre of the model is the enacted pck of the teachers which a teacher reveals in planning and executing lessons. both the ppck and epck of a teacher develop as a teacher reflects and reasons about lessons previously taught, learners’ responses in assessments and evidence of learning (carlson et al., 2019). in this study the focus was on the enacted pck of the teachers as revealed through classroom observations after they had access to learners’ responses in a baseline assessment on measurement. the components of pck that were particularly under investigation were teachers’ knowledge and skills related to student understanding of the topic and knowledge and skills related to conceptual teaching strategies (chan et al., 2019). knowledge and skills related to student understanding involves a teacher’s understanding of the pre-conceptions, naïve ideas and possible misconceptions learners come to class with and the teacher’s competence in addressing those. knowledge and skills related to conceptual teaching strategies refers to any appropriate and valuable strategy a teacher uses to create meaningful learning opportunities and a teacher’s ability to select and use appropriate representations (examples, models, diagrams etc.). 2. method the study was conducted in a social constructivism research paradigm where we constructed our knowledge of the phenomenon by being actively involved in two grade 9 mathematics teachers’ practices from the same school. the purpose was to explore teachers’ enacted pck, as informed by baseline assessment learner outcomes when teaching measurement. the assumption was that, knowledge of the outcomes of the baseline test, will inform teachers’ knowledge of learner understanding of the topic which will, in turn, inform volume 12, no 2, september 2023, pp. 307-322 311 infinity their instructional strategies. a mixed method was used where the quantitative part of the study used a rasch analysis to determine the change in learner performances after instruction. the other part consisted of a qualitative exploratory case study where an indepth investigation was done on the teachers’ enacted pck with focus on teachers’ knowledge of learner understanding and their use of conceptual teaching strategies. we used purposive and convenient sampling in selecting the school. through purposive sampling schools were selected that conform to the following inclusion criteria: (1) diverse in terms of social, economic and racial backgrounds and consequently also divers in terms of first language, (2) had an average performance in the 2014 ana results, and (3) follow the dbe’s curriculum. from this list, the school nearest to our working place was conveniently chosen. the two teachers were purposively chosen based on their experience, alice having four years’ and mary having nine years’ grade 9 mathematics teaching experience. two classes per teacher with a total of 124 participating learners were purposefully chosen based on their performances – one poor performing and one average/high performing class. this serves as a good representation of the group of grade 9 mathematics learners. the data were collected using a baseline test, classroom observations, and a formative test. 2.1. tests the two tests were similar, except for given measurements in three of the questions. the tests were marked by us and handed back to the teachers prior to their instruction. the test items that were set by us and approved by the teachers, were taken from standardized tests for this grade based on the official department of basic education’s curriculum and assessment policy statement (caps). the purpose of the tests was to test learners’ knowledge and understanding, prior and after instruction, of: a) certain pre-concepts, by explaining in their own words the meaning of surface area, volume and capacity; b) nets of a rectangular container and being able to say in words how the net is used to calculate the surface area, to use that explanation to write down a formula for surface area, and to finally calculate the surface area of the container with given measurements. c) conversions between si units: 1 m3 to cm3; 1 m3 to liter; and a certain number of ml to cm3, as well as finding the volume of the container, volume not occupied after a certain amount of water is poured in, and lastly the height of the water in the container. 2.2. observations there were five classroom observations for alice and three classroom observations for mary enabling them to complete the teaching of surface area, volume and capacity of prisms. unlike mary, alice did not teach in her mother tongue. these lessons were video recorded, and afterwards transcribed verbatim. a deductive data analysis approach was used, where the observation data were analysed according to the components of enacted pck as indicated in the conceptual framework. 2.3. rasch analysis learner performance in the preand post-test were analysed using rasch analysis employing the rasch unidimensional measurement models (rumm2030) software. in this study the attribute being measured is learners’ performance (person ability) before and after instruction of the topic. the person ability of a participant refers to his/her competence as measured by the instrument. during rasch analysis person ability and item difficulty are analysed simultaneously and placed on the same numerical scale (wright & mok, 2004). botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 312 the ability measure of a person is defined by the position on the numerical scale. this information is displayed on a person-item map (see figure 1) generated by the rumm software, with person locations on the left-hand side and item locations on the right-hand side. a person with average ability has a 50% chance of getting the item at the 0.00 item location right. such a person will have a higher chance of getting the items below 0.00 correct. in general, it shows that the items at the bottom of the diagram (low item location) are experienced as easy by the average student and the ones at the top of the diagram as difficult. in a research field such as mathematics and sciences, it is important that the instrument used for measurement does not change. however, in a study such as this, one expects that participants’ perception of the instrument changes because of the intervention, in the sense that, although the test stays the same, the students may find the items easier. a technique in rasch analysis called racking the data enables the researcher to determine the change in item-difficulty from the pre-test to the post-test as perceived by the participants (wright, 2003). with this analysis, inferences can be made about what knowledge was acquired and what was not acquired. data is racked when the preand post-test are analysed simultaneously as one test with labels to distinguish between preand post-test items (for example 1a and 1b). the preand post-test were designed such that items with the same number assess knowledge about the same concept. consequently, with racked data the item location (and thus the perceived difficulty) of an item before and after the intervention can be tracked and compared. 3. result and discussion in this study data were collected by a baseline test (pre-test) and a post-test consisting of 13 items. after the teachers had insight into the learners’ responses to the baseline test, they taught the topic (considered as an intervention) after which the test was repeated (posttest). the responses were analysed using rasch analysis and discussed qualitatively. based on the assumption that learner outcomes should improve after instruction, we were interested in analysing teachers’ enacted pck when addressing content related to items that showed little or no improvement. items were categorised in four categories, namely: (a) conversions of units (items 9, 10, 11); (b) understanding concepts (items 1, 2, 3, 4, 5, 6); (c) calculations (items 7, 8); and (d) higher order application (items 12, 13). before inferences can be made from the rasch analysis we needed to establish that the research instrument and the sample fit the rasch model to ensure the validity of the inferences. with the first run of the rasch analysis, items 5 and 13 were flagged as extreme items as almost no participants obtained any marks for these items in both the preand posttest. item 5 expected learners to explain in their own words how to use a net to calculate the surface area of a container, while item 13 expected learners to determine the height of a given amount of water in a container of which the total capacity (in ml) was known. consequently, these items were removed from the test post hoc. although these items were not included in further rasch analysis, the fact that they were labelled as extreme, indicated that learners experienced serious difficulties with these items and they will be discussed qualitatively. the rumm software also provides individual item and person fit residuals. a residual is related to the difference between the expected value and the observed value for a particular person or item. in the rumm software, these values are set to be highlighted when they fall outside the -2.5 to 2.5 interval. a value outside this interval indicates substantial deviation from the model. the individual item fit residuals for all items in this volume 12, no 2, september 2023, pp. 307-322 313 infinity analysis (except items 5 and 13 that were labelled as extreme) lie between -2.17 and 1.50 and the individual person fit residual between -1.31 and 1.41 with two extreme persons removed; indicating that items and persons behaved as expected by the model. response dependence between items can be a threat to validity of the analysis because these items may measure the same concept and one of them can be considered as redundant. the response correlation matrix provided by the rumm software indicated correlation between 0.5 and 0.6 for items 6 and 7. this is to be expected since item 6 required learners to give a formula for calculating surface area and item 7 required learners to use the same formula by substituting appropriate values. it was decided not to delete these items for the analysis because of the differences detected in the responses between the pre and post-test for these items. the person-item-map obtained with racked data after the deletion of the extreme items is shown in figure 1. this map enabled us to compare the item difficulties as experienced by students before and after the intervention. it can be seen that all but five participants’ person location is below 0.00, which means that the learners in the sample performed poorly in both the preand the post-test. the obvious conclusion is that the test was far too difficult for this particular group of learners and that the expectations were unrealistic. however, keeping in mind that test items were taken from standardized tests for this grade based on caps, the investigation of the problem of poor performance leads us to investigate what happened in the classroom and with teachers’ enacted pck. figure 1. person-item map generated by rumm2030 in the rumm software we were able to distinguish between learners from the two teachers by using person factors. we investigated the difference in performance of the two teachers’ groups per item by comparing the item characteristic curves (icc) (see figure 3) of the pre -and post-test items. the rasch program group the participants (persons) in three class intervals according to the person locations (person ability). the class interval structure for the current sample (n=124) is shown in figure 2. for example, cint1 is the group of participants with lowest person location and can therefore be considered as the participants with lowest performance in the test. botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 314 figure 2. class interval structure of the sample (n= 124) 3.1. conversions (item 9, 10, and 11) in these items, learners were required to convert from 1 m3 to cm3 (item 9); 1 m3 to litre (item 10); and 500 ml to cm3 (item 11). for item 11 learners could obtain either 1 or 0 marks. in the icc curve (see figure 3) the person location is given on the horizontal axis while the average performance in the test for the item is on the vertical axis. the light grey curve represents the performance as expected by the model. mary’s learners are shown with circles and alice’s learners with crosses. the position of three markers (circles and crosses) on the lines show the performance of the three class intervals of each teacher. it can be seen that the learners in the firstand second-class interval in alice’s group received zero out of one for this item in the pre-test (item 11a). the average for the learners in the highest-class interval was 0.2 out of 1. mary’s group performed slightly better but an anova (p = 0.842) shows no significant difference between the learners of the two teachers. the icc curve for item 11 in the post-test (item 11b) shows that the learners in the highest-class interval of alice’s group performed better (with an average just above 0.6 out of 1) than in the pre-test and also significantly better than mary’s highest group (p = 0.000). this was however, only for the highest group; the learners in the firstand second-class intervals still did not obtain marks for the item. figure 3. icc curves for item 11 in the preand post-assessments volume 12, no 2, september 2023, pp. 307-322 315 infinity this tendency that only the highest-class interval of alice’s group shows improvement, is also visible in items 9 and 10. a qualitative discussion of what transpired in the classes about conversions, follow. alice addressed conversions in lessons 1 and 5. she started using a rhyme: “kids hate doing maths during cloudy mondays” and wrote the symbols used for the units of measurement of length on the board: k h d m d c m, indicating kilometres, hectometres, decametres, meters, decimetres and metres. she continued: “if you go from millimetres to metres, you go left and each jump counts one zero. if you work with square metres (and she added the squares above), each jump counts two zeros (indicated it on board). now this was also in the test, if you have a cm by a cm by a cm, it is 1 ml. (and she wrote c3 d3 m3 below c2 d2 m2 and wrote ml below c3). now 10 cm by 10 cm by 10 cm, that is a litre (wrote l on the board). and then a metre by a metre by a metre is a kl. so, if you have to go from cubed cm to ml, it is exactly the same. so, if you have to go from millilitres to cubed meters, in other words, from small to large (demonstrate with arms crossed to arms open), you have to divide with three, six zeros (wrote ÷1 000 000 on the board).” in lesson 5 she revised the conversions between capacity and volume and told them to memorise that and to remember that each jump counts three zeros. she did three more examples in the same manner as in lesson 1: convert from cubed cm to kl; from l to cubed cm; and 3 l to cubed centimetre. mary attended to this question only in lesson 3. a similar homework problem that she discussed, required the volume of a rectangular prism and afterwards the capacity of the same container. she demonstrated the solution on the board, ending with an answer of 832000 cm3. to find the capacity, she said: “we need to do a conversion now and a 1000 cm3 is 1 litre (wrote 1000 cm3=1l on the board). this you need to know please. how will i convert from cm3 to l?” learners: “divide with a 1000.” teacher: “so, the answer is 832000 divided by a 1000, and it is 832 l.” although conversions is prescribed in the curriculum for grades 7 to 9, the focus in the classrooms was on procedural knowledge and memorisation of rhymes and rules, resulting in learners lacking conceptual understanding thereof. although alice used a table, the information on the table was confusing as the symbol m was used for both metres and millimetres. analysis shows that only the learners in alice’s most abled group showed improved outcomes and that learners in the least and average abled groups of both teachers showed no improvement at all. if we compare the teaching of the two teachers, the difference may be due to the fact that alice spent more time on the concept and have more elaborate explanations and examples of conversions. although alice spent more time on explaining conversions, we do not have enough evidence to conclude that this is the reason why her highest class interval performed so much better, because the lower class intervals still performed very poorly. the teachers had access to the responses of learners in the baseline test and, as such, to their prior and alternative conceptions. according to duit and treagust (2003), the learners should become dissatisfied with their prior conceptions during instruction and be willing to replace it with an intelligible and plausible idea. only then accommodation of the new concept will follow. the explanations of both teachers did not support conceptual development for most of the learners. apart from our finding that learners find conversions challenging, our experience at a tertiary institution also reveal that undergraduate students find conversions difficult. this concurs with the finding of tariq (2008) who found that a high percentage of first year bioscience students had difficulties with the conversion of units of measurement. botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 316 3.2. understanding the concepts (item 1, 2, 3, 4, 5, and 6) these items assess learners’ understanding of the concepts surface area, volume and capacity where they had to explain the concepts in their own words, draw the net of an open rectangular container and write down the formulae. all items were experienced difficult by the majority of learners. there was a slight improvement after the intervention in item 4 where learners had to draw a net of a rectangular container with an open top. when looking at the item map (see figure 1), it seems that item 6 has the biggest improvement, but it should be mentioned the improvement was from zero to six learners answering it correctly in the preand post-test respectively. even though responses to the items seem to have improved, all these items were still experienced as very difficult by most learners. the two questions with the lowest improvement were items 3 and 5, with item 5 being flagged as an extreme item. in an attempt to understand this poor improvement, we qualitatively analysed the two teachers’ enacted pck on these two items. in item 3, learners had to explain capacity in their own words. alice spent about half a minute in both lessons 4 and 5 to revise the definition of capacity. in lesson 4 she said: “volume is the space it takes up. capacity talks about fluids. it’s the same thing, it is also what is inside, we calculate it the same way, it’s just the unit that is going to change. so, if i fill it up with chocolates, its volume, and if i fill it up with melted chocolate, its capacity.” only a few learners actually listened as there were a number of poor behaving learners disrupting all her lessons. the result was that the lack of discipline caused the teacher to be very irritated. later in lesson 5 she said: “if we look at capacity, as we said in the beginning, volume is the space it wells up. capacity is the amount that wells inside. capacity is fluids. its measured in millilitres, litres, kilolitres.” one may consider the fact that alice is teaching in her second language. in this explanation she attempted to use the word ‘well’ as a verb and it could be understood as follows: volume is the space [in which] it [the liquid] wells up. capacity is the amount [of liquid] that wells up inside. it was in lesson 3 that mary marked a homework problem based on volume and capacity that she mentioned the difference between capacity and volume by saying: volume is the amount of space the box occupies, then, how much fluids or whatever fits in the box, is the capacity. even though alice gave a more extensive explanation than mary of the difference between capacity and volume, there is a slight, but not significant difference between the two teachers’ groups, with mary’s class slightly better. while alice is teaching in her second language, mary is teaching in her mother tongue and alice’s explanation may have been confusing to the learners. in item 5, learners were required to explain how the net of an open rectangular container, asked in item 4, will be used to calculate the surface area. alice covered the concepts related to these items in the first three lessons. regarding the surface area, she said in lesson 1: “it is the area of all the faces added together. so, it is all the sides (showing the different faces), we add it together. in lesson 2 she repeated how to find the surface area of a cube with length l, saying the area of one face is l x l, and there are 6, therefore 6×l×l.” alice then had the formula of a rectangular prism on the board: l×w×2+l×h×2+w×h×2, and by using a box, she explained the formula. a learner then asked: “if there are different rectangular prisms, will the formula change?” alice answered: “when it does not have a top, then it is going to change.” in lesson 3 she had the formula written on the board: surface area = 2(lw+lh+wh), saying: “you must know, if this one isn’t there (showing a side of the box), you have to take one of them out”, and she drew a cross over w×h inside the brackets and added w×h after the bracket. this explanation was however built on using the formula and not a net to calculate the surface area. mary covered concepts related to this item only in the first lesson. the quotes are direct translations from afrikaans to english. learners had volume 12, no 2, september 2023, pp. 307-322 317 infinity to write rectangular prism in their books. teacher: “it consists of different faces (and she indicated to the steel cabinet in the corner of the class).” teacher: “how many?” learners: “six.” teacher: “are all the same size?” learners: “no.” the teacher then used a net and showed that the opposite faces are the same and said: “there are three sets of rectangles that look alike. and remember, the rectangle is 2d. how do i calculate the area of a rectangle? it is l×b. but i cannot say times six because they are not all alike.” she then wrote three separate formulae on the board under the headings: top and bottom; front and back; and two sides, and then combined it in a final formula for the surface area. responses to the test item that required learners to draw the net improved significantly from the preto post-test, but despite of this the learners did not improve on using that net to explain in words how the net will be used to calculate the surface area. a possible reason can be that learners lack appropriate language to express their reasoning. there are two concepts involved in this question: how to use the net to calculate the surface area; and to do this for an open container. to conclude, conceptual understanding of a concept, as required in the last two items, is the foundation for learners to develop further knowledge and skills. the document mathematics teaching and learning framework for south africa: teaching mathematics for understanding (department of basic education, 2018), emphasizes the necessity that learners should be involved through activities, communication and the use of concrete material to develop conceptual understanding. furthermore, gooding and metz (2011, p. 36) claimed that teachers should “facilitate their [learners’] learning and encourage student discourse, individual reflection, metacognition, and acceptance of alternatives”. however, there was no such evidence of the teachers attempting to establish learners’ understanding of the topic. there was no meaningful oral discourse or any activity where learners were actively involved in doing and discussing in order to enhance their understanding, instead, only the teachers were talking and demonstrating. regarding representations, the teachers used either just words, or words and models respectively when addressing these two questions. 3.3. calculations (item 7 and 8) the iccs for items 7a and 7b (see figure 4) show a similar improvement from pre to post-test for both teachers. item 8 followed the same tendency, but was perceived as one of the easiest items. it is evident that the greatest improvement was shown by participants in the highest class-interval but that the average achievement was still far from full marks. both items involved the substitution of given values into a formula. item 7 expected learners to determine the surface area of a rectangular container with an open top. the learners used the formula of the surface area of a rectangular container but did not know how to deal with the open top as can be seen by the fact that the most able learners obtained only one out of two marks (see figure 4). figure 4. iccs for items 7a and 7b botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 318 item 8 required learners to calculate the volume of the same container where the open top was not an issue to be considered when applying the formula. in the post-test learners performed the best in this item with only 25 learners out of 214 still not able to achieve any marks for the item. these items resort under the first (item 8) and second (item 7) cognitive levels described by caps (department of basic education, 2011b) where only knowledge recall and routine procedures are expected. 3.4. applications (item 12 and 13) in item 12, learners had to determine the volume of a container not fully occupied with water. neither mary nor alice addressed this issue in their teaching. both teachers only discussed and marked homework where calculation of the entire volume was required and the three dimensions given. evidence shows that learners in the least and average abled groups of both teachers showed no improvement at all, obtaining no marks in the question in both the pre-and post-tests. although the group of most abled learners in alice’s class showed a slight improvement, it was actually only six learners who improved in this question. this was one of the most difficult items for the learners as it requires learners to analyse the question and consider the relevance of previous answers. they should have realised they had to subtract the answer from item 11 (volume of water converted from capacity) from the answer to item 8 (volume of container), thus resulting in a higher order question. to find the volume of the container not occupied by water, learners needed to realise the question is about volume and not capacity and that the unit of measurement is cm3. not one of the teachers referred to item 13 in the baseline test. this problem integrates with algebra where the subject of the equation needs to be changed. learners further need to know that height is one-dimensional, and the unit is cm. this item was flagged as an extreme item, because there was almost no correct answer in the preor posttest. it is evident that learner performance in higher order application problems did not improve during the teaching of the topic. learners were not challenged with such level of application in the classroom. these items are considered to be at the highest cognitive level namely problem solving according to caps (department of basic education, 2011b). mosimege et al. (2016) report that south african learners find questions difficult that require detailed reading, interpretation and problem-solving. they conclude that problem-solving is not sufficiently addressed in many mathematics classrooms in the country. 4. conclusion the mean person location (an indication of learner ability) improved from -2.37 (sd =1.27) for the pre-test to -1.42 (sd = 1.30) for the post-test. the baseline performance was extremely low despite the fact that the content for the grade 9 learners had been taught in increasingly difficulty levels since grade 4. although some improvement is evident after the topic has been taught, the test was still experienced as difficult by almost all the learners. we were interested in exploring the teachers’ enacted pck and learn from the creative and conceptual teaching approaches teachers would use to address these errors and misconceptions. as such, the study explored how the baseline assessment informed teachers’ instruction after they have had access to the outcomes of the test. our working assumptions were that experienced teachers possess rich pck regarding their learners’ common errors and misconceptions, but also learners’ understanding of difficult concepts. we assumed that they will reflect on the responses received in the baseline assessment and that they will draw volume 12, no 2, september 2023, pp. 307-322 319 infinity from their ppck to address the difficulties. the baseline assessment outcomes were intended to provide the teachers with an opportunity to identify specific aspects or concepts where their learners lacked prior knowledge and demonstrated poor understanding. this practice of interpreting and using baseline assessment is expected from south african mathematics teachers as underlined in the caps document (department of basic education, 2011b, p. 223), “knowing learners’ level of proficiency in a particular mathematics topic enables the teacher to plan her/his mathematics lesson appropriately and to pitch it at the appropriate level. baseline assessment, as the name suggests, should therefore be administered prior to teaching a particular mathematics topic”. however, from the observations there is little evidence that the experienced teachers extensively used the baseline assessment outcomes to inform their teaching. for example, the two most difficult items, items 12 and 13 were never addressed in any of their lessons. although mary’s learners showed a greater overall gain in learner outcomes after instruction, it is minimal. evidence demonstrated during their instruction, reveal that their ppck does not inform their epck or that there are filters present (carlson et al., 2019) that prevented them from drawing from their ppck when teaching this topic; an aspect we did not explore in this study. it should be mentioned that enacted pck is subject to various complexities and challenges embedded in the context of mathematics teaching in south africa. if it is true that the overload in the intermediate phase mathematics curriculum causes insufficient time for teachers to develop learners’ conceptual understanding (mosimege et al., 2016), it suggests that different tuition approaches are required where learners can be creative, and be involved in discovery in order to learn with impact and understanding. similar to the general findings of the timss 2019 results (reddy et al., 2020), the emphasis in these two classes was also on developing procedural instead of conceptual knowledge. teachers should realise that a learner-centred approach where learners are actively involved in activities, allowing them the opportunity to discuss and discover, results in more sustainable knowledge development. although the teachers used physical objects and nets of 3d objects to demonstrate, the learners should be engaged in meaningful tasks where they had to physically work with the nets of prisms. this kind of learner engagement will contribute to their curiosity and interest, allowing for meaningful discourse as they can ask questions, listen to others’ thinking, justify their reasoning, answering questions, all contributing to developing conceptual understanding (botha, 2012). according to the caps for intermediate phase grades 4-6 (department of basic education, 2011a), developing an understanding of the concepts of surface area and volume of 3-d objects begins in grade 4. the relationship between surface area and volume of rectangular prisms (asked in the baseline test), is addressed from grade 6 (department of basic education, 2011a) and continues to be described in the caps senior phase grades 7 to 9 (department of basic education, 2011b). it should be borne in mind that grade 9 is the last year this topic is explicitly taught for learners who continue with mathematics to grade 12. with this in mind, it is evident that teachers’ epck, the use of appropriate and effective teaching approaches and their use of the opportunities afforded by baseline assessment outcomes need to receive serious attention. in the current study, certain aspects emerged that have a negative impact on the instruction and consequently learner understanding, namely, the language proficiency of a teacher and the lack of discipline in class due to bad learner behaviour. these factors may act as filters for transfer of ppck into epck (carlson et al., 2019). in alice’s class some of the explanations were unclear since she was explaining in her second language. language of instruction, especially when teachers are not teaching in their mother tongue, plays an influencing role on learner performances (blömeke et al., 2011). the role of language is botha, coetzee, & zweers, teaching measurement: the role of mathematics teachers’ … 320 therefore pivotal and it is the teachers’ responsibility to “motivate learners in the use of comprehension skills in class when complex language and terminology are used” (mosimege et al., 2016, p. 98). language proficiency does not only refer to teachers, but to learners too. learners need to develop their mathematical language skills and neither of the participating teachers gave learners the opportunity to express their understanding of a concept in their own words and it can therefore not be expected of learners to perform well in such items. the important aspects that need to receive attention in pre-service and in-service teacher training are: (a) the use of the correct mathematical language in the class and the awareness teachers should have about the challenges learners have in learning in their second language; (b) the practical interpretation and use of baseline and formative assessments and how this should inform their instructional strategies; and (c) learner centered strategies, discourse in the mathematics classroom, contextualized learning and problem-solving skills. in alice’s classroom, the difficult and challenging group of learners continually disrupting all her lessons, caused her to be irritated and angry, and she struggled to keep her calm. according to carlson et al. (2019), learner attributes and the classroom environment influence the teaching and learning taking place, and timss results (reddy et al., 2020) also reported on the negative role of poor discipline in classrooms. according to hodgen (2011), the resulting affective emotions experienced by a teacher, also have an influence on the teacher’s enacted pck. the outcomes of the current study and emergent findings therefore suggest further study in the amplifiers and filters that impact the transfer of a teachers’ personal pck to the pck enacted in the classroom. the study points to aspects such as language proficiency of the teacher, discipline in the classroom and overload of the curriculum. to conclude, although depaepe et al. (2013) found that several studies reported on the positive relation between teachers’ pck and learners’ learning outcomes, we realise that the gain in learner outcomes depends on more than only the teaching intervention and that this leaves scope for further research. references artzt, a. f., armour-thomas, e., curcio, f. r., & gurl, t. j. (2015). becoming a reflective mathematics teacher: a guide for observations and self-assessment. routledge. blömeke, s., suhl, u., & kaiser, g. (2011). teacher education effectiveness: quality and equity of future primary teachers’ mathematics and mathematics pedagogical content knowledge. journal of teacher education, 62(2), 154-171. https://doi.org/10.1177/0022487110386798 botha, j. j. 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(1986). those who understand: knowledge growth in teaching. educational researcher, 15(2), 4-14. https://doi.org/10.3102/0013189x015002004 sisman, g. t., & aksu, m. (2016). a study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. international journal of science and mathematics education, 14(7), 1293-1319. https://doi.org/10.1007/s10763-015-9642-5 tariq, v. n. (2008). defining the problem: mathematical errors and misconceptions exhibited by first-year bioscience undergraduates. international journal of mathematical education in science and technology, 39(7), 889-904. https://doi.org/10.1080/00207390802136511 veal, w. r., & makinster, j. g. (1999). pedagogical content knowledge taxonomies. the electronic journal for research in science & mathematics education, 3(4). wright, b. d. (2003). rack and stack: time 1 vs. time 2. rasch measurement transactions, 17(1), 905-906. wright, b. d., & mok, m. m. c. (2004). an overview of the family of rasch measurement models. introduction to rasch measurement, 1(1), 1-24. https://journals.co.za/doi/abs/10.10520/ejc32199 https://doi.org/10.1007/978-981-13-5898-2_5 https://doi.org/10.1080/10288457.2013.828406 https://doi.org/10.1080/00228958.2006.10516449 https://doi.org/10.3102/0013189x015002004 https://doi.org/10.1007/s10763-015-9642-5 https://doi.org/10.1080/00207390802136511 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p99-108 99 undergraduate students self-concept and their mathematics procedural knowledge: the relationship muhammad win afgani* 1 , didi suryadi 2 , jarnawi afgani dahlan 3 1 universitas islam negeri raden fatah 2,3 universitas pendidikan indonesia article info abstract article history: received sept 11, 2018 revised feb 10, 2019 accepted feb 27, 2019 this study aimed to explain the relationship between self-concept and mathematics procedural knowledge among undergraduate students of mathematics education. this study investigated the affective and cognitive aspect of students in the learning of mathematics. the method in this study surveyed with non-probability sampling technique. the subjects were 133 undergraduate students of mathematics education from public and private university in palembang, indonesia. 66 of them were undergraduate students in public university. the rest of them were undergraduate students in a private university. the instruments that were used are questionnaire of selfconcept and essay test of mathematics procedural knowledge. the result from the spearman rank correlation showed sig. = 0.006 < 0.05. from that result, we conclude that there is a significant relationship between undergraduate students’ self-concept and their mathematics procedural knowledge. keywords: self-concept, mathematics procedural knowledge, relationship. copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: muhammad win afgani, department of mathematics education, universitas islam negeri raden fatah, jl. prof. k. h. zainal abidin fikri km. 3.5 palembang 30126, indonesia. email: muhammadwinafgani_uin@radenfatah.ac.id how to cite: afgani, m. w., suryadi, d., & dahlan, j. a. (2019). undergraduate students self-concept and their mathematics procedural knowledge: the relationship. infinity, 8(1), 99-108. 1. introduction the relation of affective and cognitive aspect to students in learning mathematics is an interesting issue to discuss. that issue involve not only about how students learn mathematics, but also how his/her psychology when learn it. assessment that only observes cognitive dimension is not enough yet to describe students’ mathematics achievement (suryanto, 2008). affective dimension also needs to be observed to students, because the characteristic can give an image to a teacher about students’ psychology, such as attitude when he/she faces mathematics subject. positive attitude to mathematics is an important part of mathematics education (reyes, 1984). one of the affective aspects that describes an individual and relates to his/her cognitive dimension is self-concept (ghazvini, 2011). afgani, suryadi, & dahlan, undergraduate students self-concept and their mathematics procedural … 100 self-concept is the way of individual views about he/she abilities in various aspects, such as academics, athletics, and social interactions. in social life, individual behavior corresponds to his/her self-concept. it determines individual reaction to people surrounding him/her (mishra, 2016). according to shavelson & bolus (1982), self-concept can be categorized as two, that is, academic self-concept and non-academic self-concept. self-concept that views from academic can be defined as academic self-concept. an academic self-concept is an individual psychological relate to belief about his/her ability in academic (e.g. mathematics) (flowers, raynor jr, & white, 2013). individual experiences in academic achievement related to the ways that he/she has learned to view his/herself and his/her relationships with others people. that individual view refers to his/her academic self-concept (oluwatosin & bamidele, 2014). according to fin & ishak (2014), an individual non-academic self-concept also influences his/her academic achievement, but indirectly. they explained that academic self-concept as a mediator between non-academic self-concept and academic achievement. there are many aspects relate to individual self concept, that are, physical characteristic, individual concern, social identity, individual personality (reyes, 1984), view of the future (barongo & nyamwange, 2013; deshmukh, 2015; lone & lone, 2016; rajeswari & kalaivani, 2016), the influence of others who become role models (andinny, 2015), and the influence of surrounding environment (singh, 2015). self-concept in academic, especially mathematics, according to reyes (1984), students’ view of the mathematics is an important aspect in learning mathematics. the studies that concern to the relationship between self-concept and mathematics achievement have been investigated by ayodele (2011), seaton et al. (2014), andinny (2015), jamaldini et al. (2015), singh (2015) and timmerman, van luit, & toll (2017). they argued that there is a relationship between both variables. on the other previous studies, there is no correlation between self-concept and academic achievement (yahaya & ramli, 2009; afuwape, 2011; berg & coetzee, 2014; nalah, 2014; yengimolki, kalantarkousheh, & malekitabar, 2015). that result also occurred to adebule (2014), awai & ogori (2016) studies’ in mathematics. we argued that inconsistence phenomenon because of mathematics achievement in their studies was unclear about what mathematical ability that they observed. students’ academic achievement, according to zulnaidi & zamri (2017), can be interpreted as students learning ability. there are many aspects of learning ability in mathematics, that are conceptual, procedural, problem-solving, communication, reasoning, and representation ability. to make that problem clear, we investigated self-concept and mathematical procedural ability. we defined mathematical procedural ability as procedural knowledge in mathematics. we focused on that ability because it is fundamental that students should have in learning mathematics. zulnaidi & zamri (2017) stated that students need to acquire procedural knowledge at least. mathematics procedural knowledge as a knowledge that knows about the symbols in a system and its structure and knows about the rules or procedures to solve mathematics problem (star & stylianides, 2013; joersz, 2017; zulnaidi & zamri, 2017). mathematics procedural knowledge is a part of mathematics understanding. mathematical understanding, according to skemp (1976), consists of instrumental and relational understanding. a student that has instrumental understanding, if he/she can apply mathematics formula to a problem, whereas a student that has relational understanding, if he/she knows a mathematics rule, apply it, and the reason. it shows that instrumental understanding corresponds to procedural knowledge or action conception in apos theory perspective (afgani, suryadi, & dahlan, 2017). mathematics procedural knowledge that was measured in this study is about calculus preliminary materials that are, simplification volume 8, no 1, february 2019, pp. 99-108 101 and factorization of algebra form in r system, quadratic equation, inequality, absolute value, linear equation, and function. one of the example of the items is "determine the linear equation that through (2, -5) and perpendicular with 2x – 4y = 3". a solution of that mathematics problem requires mathematics procedural knowledge of students. firstly, they must transform the equation to become y = ax + b. the new equation shows that "a" is a gradient of the first line. the gradient that perpendicular to the first line is negative reciprocal of "a". finally, they must substitute it and (2, -5) to a formula of the linear equation through a point with gradient is known and it produces only one answer. the procedure to solve that mathematics problem connects with some mathematics concept. this means that students also need to know mathematics concepts. in this situation, students must know the concepts that are needed to solve it so that they can operate the procedure completely. according to kusuma & masduki (2016), a student cannot choose a right procedure to solve mathematics problem due to he/she does not know the concepts that appropriate to solve the problem. in mathematical procedural ability, students only need to know or remember a mathematics concept and apply it and he/she does not have to understand the concept very well. as the previous explanation, there is an inconsistence result about students’ selfconcept and their mathematics achievement. in this study, we still investigated both variables, but the scope of mathematics achievement is focused on mathematics procedural knowledge. the research question is tried to answer; is there a relationship between selfconcept and mathematics procedural knowledge among undergraduate students of mathematics education. 2. method the method was used in this study is survey. the subjects were 133 undergraduate students of mathematics education from public and private university in palembang, indonesia. they were chosen through non-probability sampling. 66 of them were undergraduate students in public university which consists of 12 males and 54 females. the rest of them were undergraduate students in private university which consist of 9 males and 58 females. the instruments that were used are questionnaire of self-concept and essay test of mathematics procedural knowledge. self-concept instrument was developed from instruments that were used by ferla, valcke, & cai (2009), matovu (2012), flowers et al. (2013) in form likert scale. we used that form, because it easy to understand respondents’ perception (subedi, 2016). we developed the instrument to be appropriated with the condition to undergraduate students of mathematics education that take calculus of differential class. for test instrument, we used problems referenced from calculus book. before we applied both instruments to field test, we did validity and reliability test to the instruments. the respondents are non-experiment subject. according to subedi (2016), likert scale can be assumed as a set of data arranged in sequential category and continuous. the data can be treated as interval data, so according to joshi et al. (2015) implied that the validity test from interval data can use pearson product moment and the reliability test use cronbach's alpha. hence, the validity and reliability test in this study used both formulas and the result presented in table 1. afgani, suryadi, & dahlan, undergraduate students self-concept and their mathematics procedural … 102 table 1. the results of validity and reliability test instrument type of instrument number of items number of respondents critical value of pearson ( = 0.05) cronbach's alpha value number of valid items self concept non-test 43 65 0.244 0.92 42 mathematics procedural knowledge test 16 40 0.312 0.88 14 the result from table 1 showed that one item of the non-test instrument and two items of test instrument were invalid because the correlation values were less than each of its critical value of pearson. it because there was several respondents obtain same total score, but score of the items was different. that means a validity of the items was less convince (arikunto, 2012). hence, only 42 items of self-concept questionnaire and 14 items of mathematics procedural knowledge test were valid and reliable. the procedures to investigate self-concept and mathematics procedural knowledge of undergraduate students of mathematics education are, firstly, we collected the data of both variables. we examined mathematics procedural knowledge about calculus preliminary to the subjects during 2 hours. after they finished it, they were requested to fill self-concept questionnaire. the data of the questionnaire in this study were treated as interval according to joshi et al. (2015), because the result of construct validity test from four experts in mathematics education implied each item is arranged logically, interconnected, coherent, and measure different elements from aspects of self-concept. secondly, we processed the data. because type of both data vairiables was interval, we converted the score from both variables become value ranging 0 – 100. the formula is . ideal score for the self-concept questionnaire was 168 and mathematics procedural knowledge test was 54. after that, we seek the average and standard deviation value of self-concept score in terms of overall. we categorized the average value of students’ self-concept and their mathematics procedural knowledge become 5 categories, that are, very good (81 – 100), good (61 – 80), good enough (41 – 60), adverse (21 – 40), and very adverse (0 – 20). to test the hypothesis, there is a relationship between selfconcept and mathematics procedural knowledge among undergraduate students of mathematics education, we do correlation test. the data analysis that was used in this study is descriptive statistic to categorize the data and pearson product-moment to test the correlation between self concept and mathematics procedural knowledge among undergraduate students of mathematics education. to assist all that analysis, we used spss 16.0. 3. results and discussion descriptively, the results of data analysis showed that mean value of students’ selfconcept is 82.63 with standard deviation is 6.39. from 0 – 100 scale, it means that students’ self-concept was categorized good. this result is also supported by a study of suryanto (2008). mean value of students’ mathematics procedural knowledge is 21.31 with standard deviation is 14.94. it means that students’ mathematics procedural knowledge was categorized adverse. according to barongo & nyamwange (2013), the result reveals volume 8, no 1, february 2019, pp. 99-108 103 incongruence between students self-concept with their mathematics procedural knowledge. it will congruence if both aspects in the same category. to answer the research question, is there a relationship between students’ selfconcept and their mathematics procedural knowledge, firstly, we did preliminary analysis test to both variables. from table 2, the normality test of self-concept (sc) value has sig. = 0.200 > 0.05 and mathematics procedural knowledge (mpk) value has sig. = 0.002 < 0.05. it means distributions of the data are normal, except mpk value. table 2. summary of normality test of self concept and mathematics procedural knowledge value variable skewness kurtosis statistic sig. sc 0.123 0.661 0.068 0.200 mpk 1.266 2.558 0.101 0.002 for the homogeneity test between public and private university, the sig. value of sc was 0.746 and the sig. value of mpk value was 0.448. it shows that both sig. value > 0.05. it means that the data is homogenous variance. the result is summarized in table 3. table 3. summary of homogeneity test of self concept and mathematics procedural knowledge value variable levene statistic sig. self concept 0.105 0.746 mathematics procedural knowledge 0.580 0.448 from that result, to test the hypothesis there is a significant relationship between selfconcept and mathematics procedural knowledge of undergraduate students of mathematics education used spearman rank correlation. table 4. summary of relationship between self-concept and mathematics procedural knowledge value variable n mean standard deviation coefficient correlation sig. sc 133 82.63 6.39 0.238 0.006 mpk 133 21.31 14.94 the result from table 4 showed that sig. = 0.006 < 0.05. it means there is a significant relationship between students’ self-concept and their mathematics procedural knowledge. furthermore, the correlation coefficient showed 0.238 and its determination coefficient was 5.66%. it means that the correlation between both variables is positive but low, moreover student self-concept only influenced his/her mathematics procedural knowledge equal to 5.66% and the rest by other variables. this result also equivalents with timmerman et al. (2017) study. according to andaya (2014), self-directed learning, attitude towards mathematics, motivation, and time spend in learning mathematics were the other factors that can influence students’ success in mathematics. the results of the study had shown that there is a significant relationship between students’ self-concept and their mathematics procedural knowledge. the result could be convinced, because type of both data variables is not different. this result was relevant to the study of leonard & supardi (2010). they reported that students’ self-concept influence his/her mathematics learning outcome, because positive view, perception, and his/her confidence will improve afgani, suryadi, & dahlan, undergraduate students self-concept and their mathematics procedural … 104 mathematics learning outcome. othman & leng (2011) reported little different from the previous study. they claimed that students’ self-concept relate to his/her academic achievement, but weak. from our analysis, that because an academic achievement involve many school subjects area. in line with leonard & supardi, ayodele (2011) implied that students’ self-concept relate to his/her mathematics performance moderately, that because students thought or felt about, attitude, perception to mathematics was moderately positive. in seaton et al. (2014) perspective, academic self-concept has reciprocal relation with achievement. according to andinny (2015), self-concept influence mathematics performance in form extends an opportunity to students to solve a mathematics problem. jamaldini et al. (2015), and singh (2015) also reported that students’ self-concept relate to his/her academic achievement in mathematics. singh (2015) added an argumentation that self-concept was a consequence from high achievement. it means students’ academic ability influence his/her self-concept. more detail, timmerman et al. (2017) reported that the correlation between math self-concept with math achievement was weak in measurement and relations subject and moderate in numbers and scale subject. he argued that students’ attitude and perception about his/her mathematical ability is a main factor of self-concept that influence students’ math achievement. that means the result of this study supports the result of previous study, because students’ mathematics achievement can be observed through their mathematical abilities. one of that ability is mathematics procedural knowledge. in this study, it covered simplification and factorization of algebra form in r system, quadratic equation, inequality, absolute value, linear equation, and function. nevertheless, the result also showed the correlation between both variables was low and incongruence. it means that student with high self-concept does not certainly have high mathematics achievement. this is because undergraduate students’ self-concept was categorized good, whereas their mathematics procedural knowledge was categorized adverse. in this study, most of the undergraduate students difficult to solve mathematics problems related to linear equation and function. 4. conclusion based on the result of this study, we conclude that there is a significant relationship between self concept and mathematics procedural knowledge. besides that, this study has a few limitations. first, the sample was used non-probability sampling and a number of samples were too limited, so the conclusion cannot be generalized to all undergraduate students majoring mathematics education in palembang, indonesia. second, both instruments only valid and reliable to sample has characteristic that resembles the sample involved in this study. for next studies, we recommended to investigating self-concept and others mathematical ability or relation between mathematics procedural knowledge and others affective aspect to an amount of representative samples. so that, the relationship between affective and cognitive dimension in mathematics education can explain clearly. acknowledgements author extends gratitude to prof. dr. kasinyo harto, m.ag. and prof. dr. aflatun muchtar, ma. which have given opportunity and scholarship, so that author could participate as student in universitas pendidikan indonesia. last but not least, author render thanks and gives respect very grateful to all other individuals who have supported author to writing this paper. volume 8, no 1, february 2019, pp. 99-108 105 references adebule, s. o. 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(2017). the effectiveness of the geogebra software: the intermediary role of procedural knowledge on students’ conceptual knowledge and their achievement in mathematics. eurasia journal of mathematics, science and technology education, 13(6), 2155-2180. afgani, suryadi, & dahlan, undergraduate students self-concept and their mathematics procedural … 108 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 1, february 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i1.p179-192 179 doing mathematics intelligently and having good character through simpatik learning euis eti rohaeti1*, heris hendriana1, muhammad irfan2 1institut keguruan dan ilmu pendidikan siliwangi, indonesia 2universitas sarjanawiyata tamansiswa yogyakarta, indonesia article info abstract article history: received dec 10, 2022 revised feb 14, 2023 accepted feb 27, 2023 society’s openness to science, technology, and information development cannot be stopped. to face the development of technology and information, we must be intelligent in assessing, accommodating, and filtering the development of technology and information so that we can survive in constantly changing, uncertain, and competitive conditions. besides intelligence, the character is also important to face the development of science, technology, and information. individuals who have good character will be wiser in utilizing the development of science, technology, and information. therefore, in this analytical research paper, we introduce simpatik learning to increase the effectiveness of learning mathematics at school. through simpatik learning, students can do mathematics intelligently and have good character. simpatik learning can be defined as learning that can develop sociocultural, innovative, independent, caring, creative, and active abilities in mathematics learning based on information, communication, and technology. we believe that simpatik learning can increase the quality of mathematics learning and can be used by teachers to develop the intelligence and character of students. simpatik learning is also expected to contribute science to knowledge in mathematics education worldwide. keywords: character education, ict, intelligence, mathematics, simpatik learning this is an open access article under the cc by-sa license. corresponding author: euis eti rohaeti, department of mathematics education, institut keguruan dan ilmu pendidikan siliwangi jl. terusan jenderal sudirman no. 3, cimahi, west java 40526, indonesia email: e2rht@ikipsiliwangi.ac.id how to cite: rohaeti, e. e., hendriana, h., & irfan, m. (2023). doing mathematics intelligently and having good character through simpatik learning. infinity, 12(1), 179-192. 1. introduction society’s openness to the development of science, technology, and information cannot be stopped, which is in line with the swift currents of globalization sweeping human life today (carnoy, 2016). for this reason, the academic world must take a role to anticipate the impact it causes. it means that education must prepare students for the constellation of a global society. to face the development of technology and information, we are required to be intelligent in assessing, accommodating, and filtering the development of technology and https://doi.org/10.22460/infinity.v12i1.p179-192 https://creativecommons.org/licenses/by-sa/4.0/ rohaeti, hendriana, & irfan, doing mathematics intelligently and having good character … 180 information so that we can survive in conditions that are always changing, uncertain, and competitive. the knowledge obtained from the results of human thinking, which is processed in the world of rations, is processed through analysis and synthesis with the reasoning in the cognitive structure. it is hoped that mathematics taught in schools can be tools for students to practice their thinking skills. as we know, “mathematica” comes from latin. mathematics was originally taken from the yunani word “mathematike” which means science and knowledge. the word “mathematike” is also related to other words that are almost the same, such as “mathein” or “mathenein” which means learning or thinking. so, based on the word's origin, mathematics means knowledge obtained by thinking or reasoning. mathematics emphasizes activities in the world rations or reasoning, not emphasizing the results of experiments or observations of mathematics formed because human thoughts are related to ideas, processes, and reasoning. because the mathematics can be a tool to train thinking skills, children who are good at mathematics are often branded as smart children in all knowledge fields, as if there is a correlation between mathematics and overall brain intelligence (skemp, 2002). while children are good at other knowledge fields, for example, children who are good at drawing, some people say that children are just good at drawing but are not smart in all fields. this assumption is certainly wrong because many variables determine whether someone is smart or not. human intelligence comes from the term “intelligent,” which means intelligent and clever, responsive in facing the problem, and quick to understand when hearing information (sternberg, 2018). intelligence is the human ability to solve problems that are faced which requires the ability to think. in the latin language, the term “intelligent” means to connect or unite one another (to organize, to relate, and to bind together). intelligence is a concept that can be observed, but it is the most difficult to define. this happens because intelligence depends on the context or environment. stankov and cregan (1993) explains that intelligence can be defined quantitatively and qualitatively. as quantitatively, intelligence defines a learning process to solve problems that can be measured by an intelligence test. while qualitatively, intelligence defines thinking skills to construct how to connect and manage the information from outside that is adapted to oneself. so, it can be concluded that intelligence is a human ability to solve problems that require thinking skills and can be measured quantitatively and qualitatively. psychologists map human intelligence into four parts: intellectual intelligence, emotional intelligence, and spiritual intelligence, and one type that has recently emerged is the adversity quotient which defines as human intelligence to overcome difficulties and to be able to survive, not easily give up facing every difficulty in life. these four kinds of intelligence are inherent in a person’s personality. many people think that smartness and intelligence are the same despite being different because smart people are clever and intelligent people are brainy and accurate in analyzing something. furthermore, the difference between clever people and intelligent people is; first, smart people rely on knowledge, while intelligent people not only rely on knowledge but also rely on logic; second, smart people are more disciplined and organized, while intelligent people are more creative; third, smart people can be hated if they cannot share their knowledge with their surroundings, while intelligent people can be loved because they can adapt and be more friendly with their surroundings; fourth, smart people always think long because they always think of the significant risks, so they do not take those risks, while intelligent people think short because they always think of the small risk and they try to make the risk can be small; fifth, smart people always think, while intelligent people always try. when failing, smart people think about how to rise from failure, while intelligent people always try to rise from failure. therefore, intelligent people can rise faster from failure than smart people, because volume 12, no 1, february 2023, pp. 179-192 181 trying is better than just thinking continuously, which will take much time, so, it is better not just to be smart or intelligent people, but to be smart and intelligent people. for example, the result of schoenfeld’s experiment on elementary school students in the united states is that many students are less able to do math intelligently. then the students are given the task as follows: if there are 26 sheep and 10 goats on a ship, how old is the captain? the results were surprising because 76 out of 97 students solved this problem by adding, subtracting, multiplying, and dividing the numbers. they felt demanded to solve the problem as soon as possible and did not try to understand first the substance of the problem. another case occurred with one of the ikip siliwangi lectures when his child, who was still in elementary school, was given the task by the teacher, as follows: ..... + ..... = 10 the child thinks that the problem in the task does not exist, and the child’s father thinks that the questions are too difficult to give to the child who is still in elementary school. the problem is not too difficult to give to children who are still in elementary school, the problem can trigger students’ intelligence. intelligent students will have creative answers and are not afraid to try to answer the question. in contrast, smart students usually will be stuck and afraid to answer the question, afraid if the answer is wrong, so they will choose not to answer the question or blame if it is too difficult. on the other hand, in the education world, which still adheres to an orthodox method that requires students only to accept what the teacher or parents say to them. in the education world today, which still adheres to an orthodox method is very difficult to expect individuals can be able to give their thoughts, especially if their thinking is unique and different from other people’s. besides intelligence, the character is also important to face the development of science, technology, and information. individuals who have good character will be wiser in utilizing the development of science, technology, and information. etymologically, the character comes from latin, which means behavior, psychological traits, personality, and morals. ryan and bohlin (1999) explain that the character is a pattern of behavior that is individual or represents the individual’s moral condition. the individual character is related to the behavior around them. lastly, lickona (2009), professor of education at cortland university, stated that the quality of the nation’s character is closely related to the nation’s development. he explains that ten signs must be attention related to the nation’s character because if the ten signs are already happening in a country, the nation is heading for destruction. the ten signs are; first, increasing violent and destructive behavior among teenagers or students; second, the use of words or languages tends to get worse; third, stronger peer group influence from parents or teachers; fourth, increasing self-destructive behavior such as drug, alcohol, and free sex; fifth, declining moral behavior and increasing personal egoism or selfishness; sixth, decreased sense of pride, love of the nation, and love of the motherland or patriotism; seventh, lower respect for others, especially parents and teachers; eighth, increased behavior undermining the public interest; ninth, dishonesty occurs everywhere; and tenth there is mutual suspicion and hatred among each other. by analyzing these ten signs, it’s appropriate that the indonesian government hastened to adopt a policy that integrates character into education learning at schools. the indonesia ministry of education and culture has proclaimed four main character values that spearhead the implementation of character among students in schools. the four main characters are honest (from the heart), intelligent (from thought), tough (from sports), and rohaeti, hendriana, & irfan, doing mathematics intelligently and having good character … 182 caring (from feeling and intention). for example, the character, which is integrated into the mathematics question, is as follows. mr. syahid needs money rp. 450.000 to pay off his debt. he has strong-willed to pay off his debts by selling clean water in a residential area where clean water is difficult to get. the profit from selling clean water is used to pay off the debt. if mr. syahid makes a profit of rp. 2500 per drum, how many drums of water must he sell? from the questions above, the integrated character values are the values “though” in facing difficulties, “honest” paying off the debts by working hard using his effort, “intelligent” seeing opportunities for clean water business in an environment that needs it, and “caring” for the giver dept that has been kind to lend money. several studies have reviewed and analyzed intelligence in mathematics education and the integration of character into mathematics education in indonesia (mahfudy et al., 2019; nur et al., 2018; sukestiyarno et al., 2019). however, the results show that intelligence has not been optimally developed in mathematics education in indonesia. these studies also have shown that the integration of character into mathematics education in indonesia is still unsatisfactory. this is evidenced by the fact that many students depend on teachers and parents, which impacts their independence, discipline, sense of responsibility, curiosity, and critical and creative thinking skills, which are difficult to increase. based on the description above, we introduce simpatik learning to increase the effectiveness of learning mathematics at school. through simpatik learning, students can do mathematics intelligently and have good character. simpatik learning is a term from the initial of some indonesian words, such as “s” from the word “sosiokultural” (sociocultural), “i” from the term “inovatif” (innovative), “m” from the word “mandiri” (independent), “p” from the word “peduli” (caring), “a” from the word “aktif” (active), “t’ from the word “teknologi” (technology), “i” from the word “informasi” (information), and “k” from the word “komunikasi” (communication). 2. method this qualitative research uses descriptive analysis techniques with a literature review in which this research attempts to describe existing phenomena that are taking place now or in the past. this research highlights students' intelligence and character and the effectiveness of mathematics in schools in terms of socio-cultural, innovative, independent, caring, active, and creative elements in technology-based learning, information, and communication. 3. result and discussion simpatik learning can be defined as learning that can develop sociocultural, innovative, independent, caring, creative, and active abilities in mathematics learning based on information, communication, and technology. 3.1. sociocultural the sociocultural or social cognitive theory emphasizes how a child or learner includes culture in their reasoning, social interaction, and self-understanding. the teachers can teach mathematics while introducing indonesian culture. for example, as follows: indonesia has about 300 types of traditional dance. but among all the traditional dances, only 10 are the most popular in society. if a = the set of 10 most popular traditional dances in indonesia = {kecak, jaipong, pendet, zapin, gambyong, yapong, leleng, volume 12, no 1, february 2023, pp. 179-192 183 piring, tor-tor, ratoh-jaroe} and b = the set of regional names = {jakarta, jawa barat, jawa tengah, jawa timur, aceh, sumatera utara, sumatera barat, kalimantan timur, riau, bali}. please, make an arrow diagram that pairs member a with member b. is the relation a function? please, explain! two experts who created an idea about sociocultural theory are piaget and vygotsky. piaget reveals that learning is determined by individual initiative, meaning that knowledge comes from individuals (inagaki, 1992; piaget, 1976). students interact more with their peers than with people who are more adults in their social environment (tudge & rogoff, 1989). the primary determinant of learning is the individual concerned, while the social environment is a secondary factor. according to piaget, students’ activity is the primary determinant and guarantee of learning success, while the arrangement of conditions only facilitates learning. cognitive development is a genetic process followed by biological adaptation to the environment so that equilibrium occurs (piaget, 2013). to achieve equilibrium, adaptation processes (assimilation and accommodation) are needed (bormanaki & khoshhal, 2017). vygotsky created the learning theory known as the constructivism approach. he stated that the cognitive development of individuals is determined by themselves actively. besides that, also determined by an active social environment as well. the development of child cognition can occur through collaboration between members of one family generation with another. child development occurs in culture and continues throughout life in collaboration with others (hausfather, 1996). from this perspective, adherents of the sociocultural school argue that judging someone without considering the important people in their environment is impossible (matusov & hayes, 2000). there are three essential concepts in vygotsky’s theory of sociogenesis regarding cognitive development by the sociocultural revolution in learning and learning theory, as follows (wertsch & tulviste, 1992): a. genetic law development the law of developmental genetics states that each person’s abilities will grow and develop through two levels, which are inter-psychological or internment and intrapsychological. b. zone of proximal development vygotsky divided proximal development into two levels; first, the actual development level can be seen from a person’s ability to complete tasks or solve various problems independently; second, the level of potential development can be seen from a person’s ability to complete tasks and solve problems when under adult guidance or when collaborating with more competent peers (interment). c. psychological tools vygotsky explained that all psychological actions or processes that are uniquely human are mediated by psychological tools or psychological tools in the form of language, signs, symbols, or semiotics. there are two types of mediation; first, metacognitive mediation uses semiotic tools that aim to do self-regulation, which includes self-planning, self-monitoring, self-checking, and self-evaluating. this metacognitive mediation thrives in interpersonal communication; second, cognitive mediation uses cognitive tools to solve problems related to specific knowledge or subject-domain problems. cognitive mediation rohaeti, hendriana, & irfan, doing mathematics intelligently and having good character … 184 can be related to spontaneous concepts (which are fallible) and scientific concepts (which are more guaranteed to be confirmed). 3.2. innovation according to experts, innovation is an example where creativity, inventiveness, and strong initiative can produce something materially better than previous inventions. to develop students' spirit of innovation, brown (2008) presented ten methods to increase innovation, which are likely to be adopted and adapted in the context of developing innovation in schools. the ten methods to improve students' innovation abilities are first, encourage students to know the goals to be achieved in the future, so they must have the vision to change; second, motivate students against the fear of change; third, teach students to think like an investor who dares to take risks; fourth, educate students to focus on plans to solve a problem or task, and believe that each plan is easy to implement, resources are available properly, responsive and open to all; fifth, to achieve radical innovation, students are taught to have the courage to challenge various assumptions that exist around the environment because innovation is like art, in which there are many opportunities to think laterally, to be able to create new methods. sixth, allow each student to work on at least two tasks or methods, which is solving and completing tasks or questions in the usual method, and at the same time, they are also asked to find new methods to solve and complete the tasks or questions; seventh, conditioning students to collaborate because collaboration is seen as the key to success in innovation; eighth, for the formation of innovative students, teachers must encourage the formation of an experimentation culture. every student must be taught that every failure is the first step on a long journey to success; ninth, students must be taught to dare and try a new idea that costs and risks are relatively low in the real world, then see what the reaction is from people; and tenth, students must be educated to focus on everything they want to change. ready and always passionate and enthusiastic in facing and overcoming various challenges. 3.3. independent self-study does not mean self-study, often people misinterpret self-study as selfstudy. independent learning means learning with initiative, with or without the other’s help (brookfield, 1981). one of the principles of independent learning is that students can know when they need help or support from other parties (meyer et al., 2008). this understanding includes knowing when students need to meet with other students, study groups, teachers, or others. help or support can be in the form of mutually motivating activities to learn, for example, chatting with neighbors who study at other universities, can often motivate students to study hard. help or support can also mean dictionaries, books of supporting literature, cases from newspapers, news from radio or television, libraries, information about tutorial schedules, and other things that are not related to people. hiemstra (1994) stated that student is considered independent learning if each individual tries to increase responsibility for making various decisions, independent learning is seen as a trait that already exists in everyone and learning situations, independent learning does not mean separating oneself from others, with independent learning, students can transfer their learning outcomes in the form of knowledge and skills to other situations, students who carry out independent learning can involve various resources and activities, such as reading alone, group study, exercises, electronic dialogue, and correspondence activities, the influential role of teachers in independent learning is still possible, such as volume 12, no 1, february 2023, pp. 179-192 185 dialogue with students, finding resources, evaluating results, and providing creative ideas, several educational institutions are developing independent learning into more open programs (such as the open university) as an alternative learning individual and other innovative programs. 3.4. caring students can have the ability to empathize with others and live a life based on compassion, love, and compassion for those around them by building care (cooper, 2004). caring is not only emotional or motivation and a form of thinking but also the behavior or orientation of the entire internal life, such as emotional, rational, or intentional actions (van hooft, 1995). building care can students do with taught to be willing to listen, understand that someone needs help, and provide support for others without expecting appreciation. to build greater student care, these are three methods that teachers can do in teaching as follow: a. method i 1) develop a more empathetic perspective of students 2) develop students' sensitivity to other people's feelings 3) students to consider the impact of their actions on others 4) students dare to determine attitudes 5) students to respect others 6) students learn to reduce selfishness 7) students learn to pay attention to others b. method ii 1) students to be polite 2) students to share affection 3) students to learn to listen to others 4) students to be more generous 5) treat others as he would like to be treated 6) students should always try to be kind to others c. method iii 1) students to help people who need help 2) ask others about the difficulties they are experiencing 3) apologize when you must 4) students do good for others 5) students to share 6) students contact other people 7) students remember in detail what other people tell 8) students become volunteers 3.5. active learning active learning is usually fun, motivating, and effective in completing life tasks (lombardi et al., 2021). active learning tends to increase students’ egos, whereas passive learning does little to help students active learning usually stimulates pride, boosts selfconfidence, and gives credibility in front of teachers, friends, and parents. besides that, active learning can stimulate students’ curiosity about a broader and deeper understanding of future academics and makes learning enjoyable and personally satisfying. active learning rohaeti, hendriana, & irfan, doing mathematics intelligently and having good character … 186 increases student activity in accessing various information sources to be discussed in the learning process to gain various experiences that increase knowledge and analytical and synthesis abilities. there are seven dimensions of the learning process that result in active learning; first, student participation in setting the goals of learning activities; second, pressure on the affective aspect of learning; third, student participation in learning activities, especially in the form of interaction between students; fourth, teacher's acceptance of students' actions and contributions that are less relevant or even completely wrong; fifth, class cohesiveness as a group; sixth, freedom is given to students to make essential decisions in school life; seventh, the amount of time spent tackling student problems both related and unrelated to learning. schools that do active learning must have characteristics, such as student-centered learning, teachers guide in the occurrence of learning experiences, the purpose of activities is not just pursuing academic standards, management of learning activities and assessment. 3.6. information, communication, and technology (ict) information, communication, and technology (ict) combine tools and technology resources to manipulate and communicate information (thomas & knezek, 2008). globalization makes the role of ict important and growing rapidly in education. nowadays, ict has an essential role in integrating technology into learning activities. ict learning integrates ict and learning, an ethical study and practice to facilitate learning and improve performance by creating, using, and managing appropriate technological processes and resources (kaware & sain, 2015). the importance of ict is to develop student-centered learning, support the construction of knowledge, increase motivation to learn, develop the problem-solving ability, and create interest in learning. ict acts as a tool, not the main subject in learning. ict connects the medium to transfer knowledge from educators to students (suryani, 2010). two essential elements in the knowledge transfer process are media elements and messages conveyed through the media. the media element describes ict as an infrastructure network that connects educators with students, while the message element describes digital learning content. ictbased learning does not eliminate the initial context of face-to-face learning in the classroom but instead goes through several stages of evolution according to school conditions. ictbased learning can be described at the following levels: a. level 1 in schools that have just pioneered ict-based learning, learning is described as a face-to-face process in the classroom with digital content as a supplement. at this stage, the teacher is the deliverer of the material. the digital content submitted is only added, so it is not required to be submitted. the learning process is limited by space and time. b. level 2 this level is higher than level 1, where ict-based learning is described as a face-toface learning process in the classroom with digital content as a complement. in this condition, the teacher is still the conveyer of the material. some digital content must be delivered because it is included in the curriculum structure, while the learning process is still limited by space and time. volume 12, no 1, february 2023, pp. 179-192 187 c. level 3 at this level, ict-based learning is described as a learning process that has integrated ict advances into the learning process. all learning content is digital and must be delivered because it is included in the curriculum structure. students can access learning content without being limited by space and time, and the teacher acts as a tutor. management of learning does not use ict, so there is still interference in managing learning manually. d. level 4 this level is the highest level, where ict-based learning is described as a learning process that has merged with ict progress (unified like an infusion that cannot be distinguished between infusion fluids and blood). in this condition, students carry out independent and online learning that is not limited by space and time. teachers at this level act as tutors. there are several kinds of learning models based on information and communication technology as follows. a. blended learning singh and reed (2001) define blended learning as learning with more than one model to optimize learning. in line with graham (2013), blended learning combines a traditional learning system and a dissemination system, which emphasizes the role of computer-based technology centers. blended learning combines classroom learning and online learning (mosa, 2006). blended learning effectively combines several learning techniques, technology, and methods of delivering material to student needs. blended learning can also be interpreted as an educational approach that combines various face-to-face models with distance education and uses various types of educational technology (graham, 2006). combination models and methods of learning in ict learning have several advantages such as, first, can improve academic ability; second, can be applied to students with diverse and independent learning styles and enable cost savings and lower educational costs; third, using the variety of learning techniques also can attract the attention of students; and fourth, using a combination face-to-face education and other processes makes student can access knowledge every time and everywhere (hoic-bozic et al., 2008). blended learning can facilitate optimal learning by providing various learning media that attract student attention to learn and develop their knowledge. in blended learning, teachers are facilitators and media in the learning process (graham, 2006). teachers give instruction or learning materials and offer guidance to students in learning activities and utilizing technology used in learning. blended learning has three characteristics; first, traditional learning can support a virtual learning environment; second, the learning process is supported by an in-depth explanation of the material; third, using all kinds of technologies to support its learning process. b. computer-based learning computer based learning is fully computerized learning for students face-to-face and interacting directly with the computer. this interaction between computers and students occurs individually and learns independently without the teacher’s help. computer-based learning is a learning program using computer software in the form of computer programs that contain learning content, including titles, objectives, learning materials, and learning evaluations. ellington et al. (1993) refer to the triangle of technologies used in education: rohaeti, hendriana, & irfan, doing mathematics intelligently and having good character … 188 hardware, software, and under ware which are pedagogy materials. it means that computer systems can convey learning individually and directly to students by interacting with subjects programmed into computer systems, this is what is called computer-based learning. computer-based learning has the following principles first, oriented towards learning objective; second, oriented to individual learning; third, oriented towards independent learning; fourth, oriented towards complete learning. c. multimedia-based learning multimedia-based learning is considered necessary and will continue to be an essential learning platform soon, especially in skills-based learning programs (nazir et al., 2012). one cognitive principle in multimedia-based learning is to support the human brain in making reasonable mental representations of learning materials (mayer & mayer, 2005). the task of the human brain is to understand new material as an active participant and finally build new knowledge. multimedia-based learning materials impact students’ cognitive and active learning. clark et al. (2003) reveal that multimedia can encourage learners to engage in active learning by mentally representing the material in words and images and then mentally making connections between images and verbal representations. multimedia used in multimedia-based learning can be in the form of television, radio, social media, computers, and others that can be medium of learning. multimedia has many benefits in learning using multimedia-based learning; first, multimedia can be used as a learning presentation. multimedia presentations are used to describe material that is theoretical and has the purpose of visualizing the material with a projector. presentations with teaching materials presented with multimedia can be more complex because the content can be audio, visual, and text; second, multimedia is a simulation tool. some learning requires real experience so that students can understand the material provided better. some learning materials cannot be presented in real terms but can be simulated through multimedia. multimedia can simulate some learning materials that are difficult or even impossible in conventional learning; third, multimedia can combine video and learning materials. video content and learning materials can be made as interesting as possible and able to present new experiences that are relevant to the learning material in accordance with essential competencies; fourth, multimedia can present some information or cases that are up-to-date and relevant to the situation and environmental conditions of students, this can make students feel that they can connect the material learned at school with the situation around them. simpatik learning is mathematics learning that can develop intelligent mathematics and students’ good character, which emphasizes sociocultural, innovative, independent, caring, creative, and active abilities in information, communication, and technology-based learning. the term simpatik is formed by some indonesian words such as “s” from the word “sosiokultural” (sociocultural), “i” from the term “inovatif” (innovative), “m” from the word “mandiri” (independent), “p” from the word “peduli” (caring), “a” from the word “aktif” (active), “t’ from the word “teknologi” (technology), “i” from the word “informasi” (information) and “k” from the word “komunikasi” (communication). simpatik learning aims to invite students to be intelligent in mathematics and have good character. students are not only smart in mathematics but intelligent. students can learn math critically, creatively, and innovatively and have good character. to realize these aims, simpatik learning is implemented by emphasizing how students include their culture in their reasoning, social interaction, and self-understanding to develop their sociocultural ability. teachers can apply simpatik learning with an indirect learning volume 12, no 1, february 2023, pp. 179-192 189 model where students can build their knowledge based on their experiences, cultures, or ethnomathematics. besides that, teachers can also organize learning into groups and use peer tutors in their groups, and this can encourage students’ innovative, independent, caring, and active abilities. teachers also can trigger students to increase student’s innovation abilities by adopting the ten methods of brown (2008) to increase students’ innovation abilities. 4. conclusion teachers can implement learning not only in the classroom but also outside the classroom, this is to trigger the independence, activeness, and caring of students not only to their peers or group friends but also to the surrounding environment in simpatik learning. teachers can also set up learning that makes students more active, for example, by centering learning activities on students, so learning becomes more fun, motivating, and effective. learning that is not only in the classroom and uses the context of things around students can foster student curiosity and develop students' critical, creative, and innovative thinking skills, so that students can be mathematics intelligent, not just smart. with the rapid development of information, communication, and technology, the integration of learning with information, communication, and technology becomes necessary. therefore, simpatik learning must be integrated with ict by applying it to learning using ict-based methods. the learning models used can be various such as blended learning, computer-based learning, or media-based learning, adapted to the needs and conditions of students and schools. the rapid development of globalization is also rapidly developing advances in science, information, communication, and technology. therefore, there is a need for integration between education and ict and for good character development in students to use science, information, communication, and technology wisely and intelligently. through this paper, we propose simpatik learning that can trigger students to do mathematics intelligently and have a good character to use mathematics well and wisely. simpatik learning emphasizes sociocultural, innovative, independent, caring, creative, and active abilities in information, communication, and technology-based learning. simpatik learning in this paper is still an idea and needs further research regarding its implementation in school learning. it is hoped that simpatik learning can increase the quality of mathematics learning in indonesia and can be used by teachers in indonesia to develop the intelligence and character of students. this idea is also expected to contribute science to knowledge in mathematics education in indonesia and the world. acknowledgements the authors would like to thank institut keguruan dan ilmu pendidikan (ikip) siliwangi for incentive funds in the professorship program at ikip siliwangi, cimahi, indonesia. references bormanaki, h. b., & khoshhal, y. 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(1992). ls vygotsky and contemporary developmental psychology. developmental psychology, 28(4), 548-557. https://doi.org/10.1037/0012-1649.28.4.548 https://doi.org/10.1037/0012-1649.28.4.548 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p165-172 165 the students’ mathematical understanding ability through scientific-assisted approach of geogebra software euis eti rohaeti 1 , martin bernard 2 1,2 institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, indonesia 1 e2rht@ikipsiliwangi.ac.id, 2 pamartin23rnard@gmail.com received: may 08, 2018 ; accepted: july 05, 2018 abstract this research is motivated by the low ability of mathematics understanding of junior high school students, with the aim of the study is to examine the achievement, improvement, implementation, and difficulty of mathematics understanding ability of junior high school students through learning using geogebra-assisted scientific approach. the method of this research is an experimental method with two class groups namely experimental class and control class. the population of this research is all junior high school in west bandung regency, randomly selected and selected smp krida utama padalarang. samples were taken randomly class viii chosen d as an experimental class, and class viii c as control class. the experimental level received learning with the geogebra-assisted scientific approach, and the control class gained regular learning. the results showed that the achievement and improvement of the ability of mathematical understanding between students who received learning with scientific approach assisted geogebra software better with students who get regular learning. learning with geogebra-assisted scientific approach can be implemented in the experimental class by the steps of the scientific method of observing, asking, exploring, associating, and communicating. keywords: geogebra, mathematical understanding, scientific approach. abstrak penelitian ini dilatarbelakangi oleh rendahnya kemampuan pemahaman matematik siswa smp. tujuan penelitian ini adalah menelaah pencapaian, peningkatan, implementasi, serta kesulitan kemampuan pemahaman matematik siswa smp melalui pembelajaran menggunakan pendekatan saintifik berbantuan software geogebra. metode penelitian ini adalah eksperimen dengan dua kelompok yaitu kelas eksperimen dan kelas kontrol. populasi penelitian ini adalah seluruh smp di kabupaten bandung barat, dipilih secara acak dan terpilih smp krida utama padalarang. sampel diambil secara acak kelas terpilih kelas viii d sebagai kelas eksperimen, dan kelas viii c sebagai kelas kontrol. kelas eksperimen memperoleh pembelajaran dengan pendekatan saintifik berbantuan software geogebra, dan kelas kontrol memperoleh pembelajaran biasa. hasil penelitian menunjukkan bahwa pencapaian dan peningkatan kemampuan pemahaman matematik antara siswa yang memperoleh pembelajaran dengan pendekatan saintifik berbantuan software geogebra lebih baik dengan siswa yang memperoleh pembelajaran biasa. pembelajaran dengan pendekatan saintifik berbantuan software geogebra dapat diimplementasikan di kelas eksperimen sesuai dengan langkah-langkah pendekatan saintifik yaitu mengamati, menanya, mengeksplorasi, mengasosiasi, dan mengkomunikasi. kata kunci: geogebra, pemahaman matematik, pendekatan saintifik. how to cite: rohaeti, e. e., & bernard, m. (2018). the students’ mathematical understanding ability through scientific-assisted approach of geogebra software. infinity, 7(2), 165-172. doi:10.22460/infinity.v7i2.p165-172. mailto:e2rht@ikipsiliwangi.ac.id mailto:pamartin23rnard@gmail.com rohaeti & bernard, the students’ mathematical understanding ability through … 166 introduction mathematics is a compulsory subject in every formal education from the lowest level to the highest level of education, be it simple math or complex mathematics. mathematics is a subject universal (comprehensive) because according to chotimah, bernard, & wulandari, (2018) mathematics needs to be given to students from elementary school to a high level of cultivation. in the 2006 ktsp enhanced in the curriculum 2013 that the purpose of learning mathematics is understanding the concept of mathematics; using reasoning on patterns and attitudes; students can manipulate mathematics and generalize, present evidence, or explain mathematical ideas and statements; students can solve mathematics problems; 4) communicating mathematical symbols; 5) have an appreciation of the usefulness of mathematics in the life of curiosity, attention, and interest in learning mathematics, such as resilience and confidence in problem-solving (hidayat & sariningsih, 2018). referring to the objectives of mathematics learning above shows why the ability of mathematical understanding is important to be possessed by students because the knowledge of understanding is the underlying ability of a student in education. mathematics is not about how students memorize formulas that can have a negative impact on students, but mathematics is about how students can understand basic concepts that support their knowledge in learning new, more complex materials (bernard & rohaeti, 2016). in curriculum 2006 (sugiman, 2008), that indicator ability of understanding that is: 1) reiterating a concept; 2) classify objects according to certain traits (in accordance with the idea); 3) provide examples and non examples of ideas; 4) presents concepts in various forms of mathematical representation; 5) develop sufficient requirements or sufficient terms of a concept; 6) using, utilizing, and selecting certain procedures or operations; 7) apply the idea or problem-solving algorithm. referring to an indicator of the ability to comprehend, it indicates that the comprehension ability needs to be possessed by each student, be it elementary school students as well as intermediate level. based on the four stages of the theory of cognitive development according to piaget (ruseffendi, 2010) that the formal operation stage is the age range of children from 11 years to adulthood, showing junior high school students is an early stage students must be able to understand something abstract, and process information it becomes a concrete matter that can be recognized. difficulties students to mathematics in general due to the number of formulas that need to be known by students to be able to solve a problem. this happens on the basis of the students' habits, either from themselves or from the teacher's encouragement to memorize the formula instead of understanding the concepts of the material being studied, so that students find it difficult when given the exercise due to forgetting the formula that has been memorized, then the statement that mathematics is difficult to be asked by students . according to sumarmo (fuadi, johar, & munzir, 2016), that students' scores of comprehension and mathematical reasoning are still low. students are still experiencing difficulties in relational understanding and second-degree thinking. this indicates that students' understanding ability is still quite low. to overcome the reality in the field and to realize the objective of mathematics learning in ktsp 2006 which is enhanced in curriculum 2013, there is a need to improve the learning activities which are considered not reasonable and promote the good through innovation learning can be given a learning approach that can hone all the abilities that exist in the student self (bernard, 2014). volume 7, no. 2, september 2018 pp 165-172 167 based on one of the indicators of the ability of understanding that students are expected to be able to present the concept in various forms of mathematical representation by involving potential cognitive processes in stimulating intellect development, from simple ideas to more complex, a structured learning approach is needed discussing simple concepts to more complex concepts, ranging from students to new learning materials until they can communicate their learning outcomes in presentation activities in front of the classroom in order to facilitate the students to understand the learning materials presented by the teacher. the scientific approach is a scientific framework promoted by the 2013 curriculum. according to hendriana, rohaeti, & hidayat (2017), the steps of the scientific approach in the learning process involve digging information through observation, inquiring, experimenting, then processing data or information, presenting data or information, followed by analyzing, reason, they conclude, and create. through the instructional steps that the expert conveys in a scientific approach, students are directed to observe directly what is being learned, ask questions based on observations, try/gather information, process information, deliver, even at the end of the student's learning to create/produce a product. encouraged by the rapid development of technology and information at this time, learning by using scientific approach can be collaborated with computer media application geogebra, because through this application students can learn various mathematics materials at once, even students can try to operate itself because this application is free application not paid, and can use bahasa indonesia and comfortable in operation. seeing the fact in the field that the ability of mathematical understanding needs to be optimized early and the development of ict in the area of education to apply the concept of mathematics, the researchers intend to conduct research explicitly aimed at junior high school students (chotimah, et al., 2018). method the method used in this study is an experimental method, two class groups is one group of a control class and one group of experiment class. both groups of courses are given pretest and posttest. as a form of treatment, the control class group acquired the teaching of mathematics in the usual way, while the experimental class obtained the learning of mathematics using a scientific approach with the help of geogebra software. according to ruseffendi (2010), based onthe type of experimental design that is the selection of random class groups (a), the pretest (0) in the control class and experimental class, and the posttes (0) in both classes, called the pretest-posttest control group design, with the research design for each class group as follows: a 0 x 0 a 0 x 0 description: a: random sampling by class 0: prettest = posttest x: mathematical teaching using scientific approach aided geogebra software rohaeti & bernard, the students’ mathematical understanding ability through … 168 results and discussion the results obtained from the students in the form of the comprehension of students' understanding before and after being given learning from both classes, namely the experimental class and the control class. where, before the two classes there was no significant difference in average grade values, this was based on the processing of statistical data in the description that flat the values of the experiment and the experimental class were 4.85 and the control class was 5.15 with the standard deviation of 1.39 and 2.59, respectively. then the results of the two results are compared using normal tests, and the results show a significant value <0.05 which means that the two data are not normal followed by a non parametric test mann whitney, the significant result is 0.80> 0.05 indicating no average differences between the two class at pretest. however, when posttest, in gain, the values of both classes have increased by showing positive values for the experimental gain average 0.54 and 0.445 control class, where the gain of the experimental class is higher than the control class class and based on the description that the average value at posttest for the class experiment 13.00 and for control class 11.71 with each standard deviation 2.702 and 2.59, after being tested normally, the results of significant values 0.000 and 0.015 means that both are significant <0.05, then followed by the non-parametric mann-whitney average difference test with a significant value of 0.016 <0.05 means that the experimental class whose learning uses geogebra software is better than ordinary learning to improve the mathematical comprehension ability of junior high school students, this is in line with senjayawati & bernard (2018) that learning mathematics that uses geogebra is better than learning in the normal way. learning process using the approach this science consists of five general steps: observing, questioning, exploring, associating, and communicating. in applying these five learning steps, in the process of learning researchers assisted by geogebra software with the aim to facilitate researchers in delivering materials to students. the learning process in the experimental class by using-assisted-scientific approach is geogebra software described with the steps of the activity. figure 1. the researcher gives the picture or object using geogebra software aid to start the learning process (observing activity) the learning process of the scientific approach begins with observation activities, where the students are given an object by researchers related to daily life as well as subject matter that will be delivered. because in this study using the help of geogebra software, the researchers provide an overview of the object through geogebra software. the purpose of this activity is to encourage students' curiosity about what images are intended by researchers. volume 7, no. 2, september 2018 pp 165-172 169 the characteristic of the scientific approach that emerges in this activity is to involve potential cognitive processes in stimulating intellectual development, in which students get an external stimulus in the form of an object displayed, intending to encouraging students' cognitive abilities and intellectual development. figure 2. students ask questions by the object that has been observed (activities questioner) after doing activities observing the object that has been submitted by the researcher, then the researchers continue the learning step of the scientific approach that is the activity of the question. questioning activity is a learning process done after students observe an object by the material that has been studied. in the questioning activities, researchers allow students to submit questions by the object that has been observed. characteristics that emerged in this activity is student-centered learning because in the learning process teachers invite students to ask questions by what they observed in the classroom environment as well as on objects that teachers draw in geogebra software. figure 3. students collect information from various sources whether it is their knowledge, or from the source book (activity exploring) student learning process during this exploration, they note on lks (student worksheet) that have been given at the beginning of learning. here students are trained to work together and learn to respect the opinions of others because when they gather information, they find differences of opinion, but this is not a big problem for students completing the lks according to what they find on the information gathering takes place. rohaeti & bernard, the students’ mathematical understanding ability through … 170 figure 4. students process information obtained from observation and exploration activities (association) this activity is a student activity in processing the information they have collected. in this activity, students try and learn in groups to process concrete information in the form of concepts that have been presented by the teacher in observing activities and activities when collecting information from various sources that support learning activities to the more abstract. figure 5. students present the results obtained at the time of processing information (activities communicate) in this activity students in groups communicate/present results obtained at the time of processing information during the learning process. the material presented each group by the student worksheet which has been discussed by students in groups. each student is allowed to respond, refute, or add the explanation of a friend who is presenting. each group takes turns presenting the results of the group's work in front of the class. in this activity, researchers direct students to be able to appreciate friends who are talking and mutual respect in differences of opinion. for the level of difficulty students, posttest about comprehension for the class that learns using geogebra software with classes with ordinary learning can be seen in table 1 volume 7, no. 2, september 2018 pp 165-172 171 table 1. results value of students' mathematical understanding questions for junior high school experiment and control classes number question experiment class control class score score 4 3 2 1 0 4 3 2 1 0 1 19 17 4 1 13 23 3 2 2 6 23 8 4 1 22 10 8 3 10 22 9 4 23 15 4 3 16 17 15 13 11 2 5 4 12 20 5 2 10 15 14 from table 1, pointing out that the executive class is more mastering the material than the control class can show in problem number 1 many students who have the highest score of the experimental class more students, namely 19 students compared to the control class there are 13 students who master the problem. for problem number 2, there were 6 students in the experimental class who controlled more than the students in the control class there was 1 student. likewise, for numbers 3, 4 and 5, the experimental class students have more control than the control class. with results based on the difficulties, students work on the ability to understand that the class that uses geogebra software is more mastering the material than the class that learns the normal way (hendriana et al., 2017; hidayat & sariningsih, 2018). conclusion based on the results of research to determine the improvement of the ability of mathematical understanding of junior high school students using scientific-assisted approach geogebra software can be summed up things as follows achievement of students' understanding of mathematical ability learning using-assisted scientific approach geogebra better than the learning using the usual approach. improvement of students' mathematical understanding ability whose learning using-assisted scientific approach geogebra is better than learning using the usual approach. references bernard, m. (2014). pengaruh pembelajaran dengan menggunakan multimedia macromedia falsh terhadap kemampuan penalaran matematik. in prosiding seminar nasional pendidikan matematika program pasca sarjana stkip siliwangi bandung, 1, 425429). bernard, m., & rohaeti, e. e. (2016). meningkatkan kemampuan penalaran dan disposisi matematik siswa melalui pembelajaran kontekstual berbantuan game adobe flash cs 4.0 (ctl-gaf). edusentris, 3(1), 85-94. chotimah, s., bernard, m., & wulandari, s. m. (2018). contextual approach using vba learning media to improve students’ mathematical displacement and disposition ability. in journal of physics: conference series, 948(1), 012025. fuadi, r., johar, r., & munzir, s. (2016). peningkatkan kemampuan pemahaman dan penalaran matematis melalui pendekatan kontekstual. jurnal didaktik matematika, 3(1), 47-54. rohaeti & bernard, the students’ mathematical understanding ability through … 172 hendriana, h., rohaeti, e. e., & hidayat, w. (2017). metaphorical thinking learning and junior high school teachers' mathematical questioning ability. journal on mathematics education, 8(1), 55-64. hidayat, w., & sariningsih, r. (2018). kemampuan pemecahan masalah matematis dan adversity quotient siswa smp melalui pembelajaran open ended. jnpm (jurnal nasional pendidikan matematika), 2(1), 109-118. ruseffendi, e. t. (2010). dasar-dasar penelitian pendidikan dan bidang non-eksata lainnya. bandung: tarsito. senjayawati, e., & bernard, m. (2018). penerapan model search-solve-create-share untuk mengembangkan kemampuan penalaran matematis berbantuan software geogebra 4.4. maju: jurnal ilmiah pendidikan matematika, 5(1). sugiman, s. (2008). koneksi matematik dalam pembelajaran matematika di sekolah menengah pertama. pythagoras: jurnal pendidikan matematika, 4(1). infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p123-132 123 the students’ mathematical abstraction ability through realistic mathematics education with vba-microsoft excel nelly fitriani 1 , didi suryadi 2 , darhim 3 1,2,3 universitas pendidikan indonesia, jl. setiabudhi no. 229, isola, sukasari, bandung, indonesia 1 nellyfitriani@student.upi.edu, 2 ddsuryadi1@gmail.com, 3 darhim_55@yahoo.com received: june 30, 2018 ; accepted: august 30, 2018 abstract the purpose of this research is to analyze the level of abstraction mathematical ability of the students through learning using realistic mathematics education (rme) with visual basic application (vba) for microsoft excel. this research uses a descriptive qualitative method and the subject of this study is junior high school students of grade 9 th in one school in ngamprah as many as 35 students. one of the instruments in this study is the abstraction test. results show that as many as 65.71% of students are in the first level of perceptual abstraction. as many as 57.14% of students are in the second level of internalization. as many as 31.43% of students are in the third level of interiorization, and as many as 17.14% of students are at the last level, namely second level of interiorization. overall, the sample in this study can be categorized into four levels of abstraction. keywords: mathematical abstraction, realistic mathematics education, visual basic application for microsoft excel. abstrak tujuan penelitian ini adalah untuk menganalisis kemampuan abstraksi matematis siswa berdasarkan level abstraksinya melalui pembelajaran dengan menggunakan pendekatan realistic mathematics education (rme) berbantuan visual basic application (vba). penelitian ini menggunakan metode kualitatif deskriptif dengan subjek penelitiannya adalah siswa smp kelas ix di ngamprah sebanyak 35 siswa. instrumen dalam penelitian ini diantaranya adalah soal tes abstraksi. hasil yang diperoleh menunjukkan bahwa sebanyak 65,71% siswa sudah ada pada level pertama yaitu perceptual abstraction. sebanyak 57,14% siswa sudah ada pada level kedua yaitu internalization. sebanyak 31,43% siswa sudah berada pada level ketiga yaitu interiorization dan sebanyak 17,14% siswa sudah berada pada level terakhir yaitu second level of interiorization. secara keseluruhan, sampel dalam penelitian ini dapat dikategorikan ke dalam empat level abstraksi. kata kunci: abstraksi matematis, realistic mathematics education, visual basic application for microsoft excel. how to cite: fitriani, n., suryadi, d., & darhim, d. (2018). the students’ mathematical abstraction ability through realistic mathematics education with vba-microsoft excel. infinity, 7(2), 123-132. doi:10.22460/infinity.v7i2.p123-132. mailto:ddsuryadi1@gmail.com fitriani, suryadi, & darhim, the students’ mathematical abstraction ability through … 124 introduction concepts in mathematics are abstract (including geometry). students will have difficulty if they are emphasized to memorize. concepts should be constructed in the minds of students and not transferred by the teacher to students directly. the concept construction process that occurs in the minds of students by utilizing their initial experience or knowledge is called the process of mathematical abstraction (nurhasanah, kusumah, sabandar, & suryadi, 2017). in line with this, process of abstraction occurs when a person realizes the similarity of characteristics between objects based on experience that already happens (skemp, 2012). these similarities are used as a basis for classification so that one can recognize a new experience by comparing it with experiences already established in the previous thought. this process is called the process of abstraction. the result of the abstraction process is a concept. based on the above definition, the process of abstraction will exist through one's experiences, i.e. the student's experience in constructing an initial mathematical knowledge and the concept. here, the concept in mathematics feels very meaningful, because the concepts are interrelated and mutually required. so, it is essential for students to master the process of abstraction. some experts who have reviewed the issue and in general, previous studies were still on topics outside of geometry (ferrari, 2003). the process of abstraction has a vital role in geometry learning (mitchelmore & white, 2007). however, from a cognitive point of view, abstraction is one of the reasons for failure in mathematics, including geometry (ferrari, 2003). in indonesia, geometry becomes one part of the mathematics material of the school where students experience many problems. it is in line with the results of trends in international mathematics and science study (timss) in 2011 which conveyed that the dimension of indonesian students' lowest content is geometry. the failure is allegedly related to the way to form abstract mathematical objects. the formation cannot be done only through the delivery of information directly but requires an object forming process through a series of experiences directly by students. so it becomes something interesting to examine students' abstraction in understanding geometry. there are several ways that abstractions may arise in learning, namely: familiarize students to find relevant contexts; direct students to recognize commonalities across contexts; make students feel the same so they can form a universal concept; direct students to apply the concept in new situations (mitchelmore & white, 2007). based on the above, a learning design that can facilitate the abstraction process is using vba-assisted rme approach. the rme approach designed to direct students to find relevant contexts early in the learning process (fitriani, 2015). then with the help of vba, students are directed to recognize the similarities of the properties contained in the concept of tubes and circles. the programming language in vba produces a dynamic view of the system. the volume of the constructed tube is the sum of the area of the circle multiplied by its thickness. students will form a universal concept in a new situation. besides, rme is applied because it has the main characteristics of a self-developed model. it greatly facilitates the occurrence of mathematical abstraction processes. self-developed models can bridge the gulf between informal and formal knowledge in mathematics. students build their knowledge gradually from the knowledge volume 7, no. 2, september 2018 pp 123-132 125 they possess as a result of their interaction with the environment, enhanced toward a semiconcrete form, then move into semi-abstract and abstract. it greatly facilitates the process of abstraction. thus, it is possible that the rme approach is applied to train students to produce a mathematical abstraction process. mathematical abstraction piaget (gray, 2007) distinguishes three kinds of abstractions, namely, empirical abstraction, pseudo-empirical abstraction, and abstraction reflective. empirical abstraction states that knowledge comes from experiences. pseudo-empirical abstraction is between empirical and reflective abstraction. it occurs when the subject is confronted with an object and then finds the properties of the object through the process of imagining an action on the object. piaget describes abstraction reflective as the general coordination of action in such a way that the source is a subject equipped with a full internal nature. abstractions divided into empirical and theoretical abstractions (mitchelmore & white, 2007). theoretical abstraction consists of forming concepts that correspond to several theories. a clear example of the difference between empirical and theoretical abstractions is: when studying the concept of a tube, according to an empirical abstraction the process is that students recognize the various forms of representation of the tube first, as are examples of the forms of the tubes in everyday life. students will recognize the same characteristics based on experiences with real objects. from various contexts, it will introduce a concept. in a theoretical abstraction, the teacher introduces the students to the concept and definition of a prism. after that, students are led to conclude that the tube is a unique prism. it is useful for students to come up with a generalization process based on the relationship between the two concepts. from these examples, there is a difference between the two abstractions. in empirical abstraction, individuals form new concepts based on observation and experience. while in theoretical abstraction, new concepts will emerge by matching existing concepts with experiences that have already been formed and stored first in individual thought. piaget's theory of reflective abstraction is in the category of theoretical abstraction. in this study, the focus is on empirical and reflective abstractions. based on the characteristics of the two types of abstraction, the empirical abstractions appear before a reflective abstraction. to coordinate and reorganize (collect, arrange, develop) mental actions, one must first recognize the same characteristics of an object, then analyze its association with the existing knowledge (nurhasanah, 2018). visual basic application for microsoft excel visual basic application (vba) for microsoft excel is a data processing and numeric software that automatically utilizes mathematical functions using the help of visual basic codes. the advantages of microsoft excel are the use of mathematical functions associated with daily life such as financial, statistics, engineering, information and the web. besides, the software has many different shapes and images of type and size. however, the relationship between mathematical functions and images, cannot run correctly when not using vba for microsoft excel. with the visual basic, the images in microsoft excel become interactive by connecting the mathematical functions in the form of the code language. vba for excel can create automated commands to run programs on mathematical arithmetic operations associated with images as a mathematical medium (chotimah, bernard, & wulandari, 2018). utilization of ict-based media can provide students with a more useful understanding of audio-visual by utilizing image features in mathematics learning (bernard, 2015). the use of fitriani, suryadi, & darhim, the students’ mathematical abstraction ability through … 126 vba for excel in mathematics learning media is an effort to improve the ability of mathematical thinking and student activeness because of the interactive images. method this research uses a descriptive qualitative method and the subject of this study is junior high school students of grade 9 th in one school in ngamprah as many as 35 students. the main instruments in this research are mathematics abstraction ability test. this is an example of an instrument to measure the level of students' mathematical abstraction ability in level 2, figure 1. sample abstraction test questions researchers want to analyze in depth the abstraction abilities of mathematical students based on abstraction level through learning by using rme approach through vba. the research question is: what is the level abilities of students' mathematical abstraction in the class using realistic mathematics education (rme) approach with visual basic application (vba) for microsoft excel. results and discussion the mathematical thinking process based on the rme approach assisted by vba excel helps students bridge the horizontal mathematization process towards vertical mathematization mathematical formations. through this, the process of abstraction do well, and learning becomes more meaningful. furthermore, to detect abstractions that occur in learning, then we must arrange the indicator. this study modifies indicators for abstraction level abilities. this indicator is for analyzing abstraction levels in junior high school students (hong & kim, 2016; nurhasanah, 2018), tabel 1. indicator of mathematical abstraction level mathematical abstraction level indicator perceptual abstraction a. know the properties of mathematical objects based on the utilization of physical objects. b. recognise previous experience related to the problem at hand. internalization a. represents the results of thought in the form of mathematical symbols, words or diagrams. b. able to resolve/manipulate the problem. interiorization reorganize (collect, organise, develop, coordinate) concepts into new understandings or new knowledge. second level of interiorization generalize new knowledge in different contexts. volume 7, no. 2, september 2018 pp 123-132 127 the student's abstraction level in solving problems related to the concept of the curved -face three-dimensional objects will be described in table 2 below, table 2. results of data analysis on abstraction level answer characteristics level of abstraction number of participants percentage % students recognize the properties of the tube by utilizing the image of its nets. based on the image, students can imagine the surface area of the tube. students recall the previous experience of the circumference of the circle with the size of the radius of 14 cm and the definition of the rectangle (the dc length is equal to ab and parallel, i.e., 88 cm), so the student can deduce that the nets can form a tube or not. students must conclude that abcd is a square. then students can also conclude that the circumference of a circle with dc or ab length, is the same length. thus, the nets can form a tube. perceptual abstraction 23 65.71 students can represent the results of their thinking by sketching the image of a quadrilateral prism, pentagon, hexagon, and n sides. with the image, the students can construct the concept of a tube (a tube is a unique prism with a base and roof is n sides or a circle). students will realize that the tube is a unique prism, so to determine its volume, they use a similar formula, only adapted to the base form. the student can solve the problem because he can deduce the formula of the tube. internalization 20 57.14 students can gather information on the issue. he was able to conclude that the object contained in the problem is a truncated cone. on the problem, it is not required to sketch, but the student can sketch it. then, he was able to coordinate the concept of value comparison and phytagoras concept in solving the problem. a direct proportion concept is used to determine the length of the painter's line (s) on the lower cone. as a result, s = 10 cm. the pythagoras interiorization 11 31.43 fitriani, suryadi, & darhim, the students’ mathematical abstraction ability through … 128 answer characteristics level of abstraction number of participants percentage % concept is used to determine the height of a small cone, which is 8 cm, and the height of a large cone, which is 18 cm. after all the data is complete, then he can determine the truncated cone volume, by calculating the difference between the full cone and the small cone at the bottom. students can recall the concept of pyramid volume. they can sketch the pyramid even its not asked in the problem. they generalized that a tiny collection of pyramids would form a solid ball through proof of the sum of pyramid volumes of n pyramid. if the formula is simplified, it will form a formula . after generalizing the ball volume formula based on the number of pyramid volumes of n pieces, the student can solve the problem that the ball volume is 36  cm 2 . second level of interiorization 6 17.14 based on the theory of abstraction levels, the results of the student work analysis show that some students have met all types of abstraction levels (hong & kim, 2016; nurhasanah, 2018). as many as 65.71% of students already in the first level, namely perceptual abstraction. student errors occur when they can not recognize the concept of the circumference of the circle. students cannot argue that the circumference of the circle should be as large as the length of ab or cd. as many as 57.14% of students already exist in the second level, namely internalization. student errors occur when they can not sketch the prisms. they are unable to generalize the tube which is a unique prism and cannot solve the problem. a total of 31.43% of students are already in the third level, namely interiorization. student errors occur when they can not conclude that the object is a cone cut off, they conclude that the object is a tube, so the solution to the problem was not appropriate. a total of 17.14% of students are already in the fourth level, namely second level of interiorization. student errors occur when they can not generalize new knowledge to the concept of the pyramid in the context of a sphere. the following will describe the results of interviews and student work results for the given abstraction test. g : can images of these nets form a tube build? s1 : yes i can, because based on the information given on the problem, it is known that the dc length is the same length and parallel to the ab, and the circumference of the base is the same as the length of the dc or ab. s2 : maybe volume 7, no. 2, september 2018 pp 123-132 129 g : why is it possible? s2 : because the picture is indeed tube nets g : should there be similarities between the circumference of the circle with the length of dc? s2 : not always, depending on what we will make interviews were conducted on two sample students, where s1 was a student who was classified as having high ability, he was at the perceptual abstraction level. s2 is classified as a student with low ability, he is not at the perceptual abstraction level, it is proven that he cannot put forward his argument where the circumference of the circle must be as large as the length of the ab or cd. the disadvantages that occur to s2 students, he does not understand the characteristics that must be fulfilled so that the nets can form a tube, and only utilize it from direct observation. the picture below is an example of student work, figure 1. s1 work results based on the results of the work in figure 1., students are able to recall experiences of previous concepts, such as the concept of alignment, segment length, and circumference of the circle that is needed to solve the given problem, remembering that all of this is a mutually successive scheme forming a concept (van oers & poland, 2007). based on his experience of these concepts, he was able to reorganize his knowledge into a complete concept, and he was able to conclude that these can be formed in a whole tube. clinical interviews continued with regard to problem number 2, with samples of students who were still the same. clinical interviews conducted are of an unstructured type the main purpose is to obtain accuracy relevant information (jones, 2010), g : can you make a sketch of the prisms? s1 : i can s2 : (silent) g : from the sketches you made, according to your prediction, the n-aspect prism is what space is it built? s1 : tube ma’am g : how about you? s2 : i can only make sketches of triangle and rectangular prisms, ma’am g : can you guess or imagine what the n-shape prism looks like? s2 : sorry ma'am, i can't imagine it yet... g : from the pictures that you sketched, are there similarities between them? s1 : they have the same large and parallel base and roof fitriani, suryadi, & darhim, the students’ mathematical abstraction ability through … 130 s2 : same with you, i'm also the answer g : to determine the area of the base and height of the prisms, what concept do you use? s1 : prism volume g : can you determine the volume of a rectangular prism? s1 : yes i can ma’am g : prism in terms of n? s1 : (silent), because it has the same properties as all, the n-prism of the volume must be the same g : what distinguishes it? s1 : depending on the shape of the base or roof based on clinical interviews conducted, it appears that s1 is already on the second level, internalization, while the s2 has not yet reached this level. s2 errors occur when they cannot make sketches of the prisms, so he is unable to make a generalization about the tube which is a special prism and so in solving the problem. for the level of interiorization, only 11 students experienced it. in general, students are not able to reorganize (collect, compile, develop, coordinate) concepts into new understanding or new knowledge (nurhasanah, 2018). students cannot conclude that the building presented is a building from a hollow cone, they are more of a view that the building is a tube so that the solution becomes very inappropriate. likewise for the second level of interiorization, only a few have achieved it. students still find it difficult to generalize new knowledge in different contexts, students tend to be able to solve problems for similar contexts. overall, based on the action taken has made the student abstraction process appear, and the sample in this study can be categorized into four levels of abstraction. conclusion based on the learning that has been done to 35 sample students, some students are appropriate for all types of abstraction levels. as many as 65.71% of students are in the first level (perceptual abstraction). as many as 57.14% of students are in the second level (internalization). as many as 31.43% of students are in the third level (interiorization), and 17.14% of students are at the last level (second level of interiorization). overall, the sample in this study can be categorized into four levels of abstraction. acknowledgments thanks to technology research and high education ministry for providing doctoral dissertation research grant, so the researcher can conduct this study to accelerate the completion of her study of the doctorate program and thanks to many people who have helped in completing this research, especially to mrs. farida nurhasanah, because she is very helpful in directing writers when experiencing a deadlock. references bernard, m. 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(2016). mathematical abstraction in the solving of ill-structured problems by elementary school students in korea. eurasia journal of mathematics, science and technology education, 12(2), 267–281. https://doi.org/10.12973/eurasia.2016.1204a jones, k. d. (2010). the unstructured clinical interview. journal of counseling & development development, 88, 220–227. https://doi.org/doi:10.1002/j.15566678.2010.tb00013.x mitchelmore, m. c., & white, p. (2007). abstraction in mathematics learning. mathematics education research journal, 19(2), 1–9. https://doi.org/10.1007/bf03217452 nurhasanah, f. (2018). mathematical abstraction of pre-service mathematics teacher in learning nonconventional mathematics consepts. universitas pendidikan indonesia. nurhasanah, f., kusumah, y. s., sabandar, j., & suryadi, d. (2017). mathematical abstraction: constructing concept of parallel coordinates. journal of physics: conference series, 895(1), 012076. https://doi.org/10.1088/1742-6596/895/1/012076 skemp, r. r. (2012). the psychology of learning mathematics: expanded american edition. new york: routledge. van oers, b., & poland, m. (2007). schematising activities as a means for encouraging young children to think abstractly. mathematics education research journal, 19(2), 10–22. https://doi.org/10.1007/bf03217453 fitriani, suryadi, & darhim, the students’ mathematical abstraction ability through … 132 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p173-182 173 attitudes toward statistics and achievement: between students of science and social fields dian cahyawati 1 , wahyudin 2 , sufyani prabawanto 3 1 universitas sriwijaya, jl. palembang prabumulih km 32 ogan ilir, south sumatera, indonesia 2,3 universitas pendidikan indonesia, jl. dr. setiabudi no. 229, bandung, west java, indonesia 1 dianc_mipa@unsri.ac.id, 2 wahyudin.mat@upi.edu, 3 sufyani@upi.edu received: june 29, 2018 ; accepted: september 08, 2018 abstract the purpose of this study was to explore students' attitudes toward statistics (ats) based on the beginning and the end of learning, based on differences in fields of study, and examine its relationship with statistical acheivement. ats was measured by an attitude instrument, namely the survey of attitudes toward statistics (sats) which consists of six dimensions (affective, cognitive competence, difficulty, value, interest, effort). research respondents were undergraduate students who took lectures on statistical recognition that came from at one of the universities in south sumatra. the results of the descriptive analysis showed the variation of ats in each dimension of attitudes classified into positive, neutral, or negative attitudes. although there are variations in the response of student attitudes in each dimension, but the results of statistical tests have not been able to show differences in attitudes between the beginning and the end of learning in each dimension. the differences in attitudes between the beginning and the end of learning that are statistically significant are in the dimensions of affective, value, and effort. the difference in the field of student science shows the difference in ats, but only in the dimension of value. this study does not have enough evidence to state that there is a significant relationship between student attitudes to statistics and the results of learning statistics. keywords: attitudes toward statistics, field of study, statistics achievement. abstrak tujuan penelitian ini adalah untuk mengeksplorasi sikap mahasiswa terhadap statistika (ats) berdasarkan awal dan akhir pembelajaran, berdasarkan perbedaan bidang studi, dan menguji hubungannya dengan hasil belajar statistika. ats diukur dengan instrumen sikap yaitu the survey of attitudes toward statistic (sats) yang terdiri dari enam dimensi (affective, competency cognitive, difficulty, value, interest, effort). responden penelitian adalah mahasiswa tingkat sarjana yang mengikuti perkuliahan pengenalan statistika yang berasal dari salah satu perguruan tinggi di sumatera selatan. hasil analisis deskripsi menunjukkan adanya variasi ats pada setiap dimensi sikap yang diklasifikasikan menjadi sikap positif, netral, atau negatif. meskipun terdapat variasi respon sikap mahasiswa pada setiap dimensi, tetapi hasil pengujian secara statistika belum dapat menunjukkan adanya perbedaan sikap antara awal dan akhir pembelajaran pada setiap dimensi. perbedaan sikap antara awal dan akhir pembelajaran yang signifikan secara statistik ada pada dimensi affective, value, and effort. perbedaan bidang ilmu mahasiswa menunjukkan adanya perbedaan ats, tetapi hanya pada dimensi value. penelitian ini belum memiliki cukup bukti untuk menyatakan bahwa ada hubungan yang signifikan antara sikap mahasiswa terhadap statistika dengan hasil belajar statistikanya. kata kunci: sikap terhadap statistika, bidang ilmu, hasil belajar statistika. how to cite: cahyawati, d., wahyudin, w., & prabawanto, s. (2018). attitudes toward statistics and achievement: between students of science and social fields. infinity, 7(2), 173-182. doi:10.22460/infinity.v7i2.p173-182. mailto:dianc_mipa@unsri.ac.id mailto:wahyudin.mat@upi.edu mailto:sufyani@upi.edu cahyawati, wahyudin & prabawanto, attitudes toward statistics and achievement … 174 introduction statistics is a necessary and important topic in all education area. it is used by various disciplines and all levels of education (andrews, 2010) from primary to university level (chan & ismail, 2013). the importance of statistics makes this topic a compulsory course in higher education institutions (ashaari, judi, mohamed, & tengku wook, 2011). similarly, in indonesia, every program in higher education is required for including either mathematics or statistics or both of it (menkumham-ri, 2005). that requirement aims to provide students’ ability in basic understanding and applying quantitative methods. especially for a statistics course, this topic is focused on statistical literacy, statistical reasoning, and statistical thinking those are essential abilities in the 21st century. the usefulness of statistics as a method for processing data in the various field was written by david moore (biehler, ben-zvi, bakker, & makar, 2013; garfield & ben-zvi, 2008). the importance of statistics as one of the scientific tools for suggesting to determine a decision in various experiment fields is the educators challenge to deliver the best learning for their students in improving statistics achievement. some observation to the statistics achievement from the first year undergraduate students in statistics introductory class in one of the universities at south sumatera from 2016/2017 academic year found that there were only 49% of the 155 students who reached statistics achievement in top three (from 5-grades of quality) and the rest as much as 51% they reached in lower grades. it happened also in the previous academic years because of their difficulties in learning statistics, therefore some students tended to remedy statistics introductory class. the difficulties in learning statistics can be related to cognitive and non-cognitive factors, but according to ashaari et al. (2011) the dominant factor was non-cognitive factors. learning motivation, learning interests, learning habits, self-concept, and attitudes was an example of a non-cognitive factor or an affective domain (djaali, 2014) that can lead to difficulties in learning statistics (ashaari et al., 2011). the attitudes toward statistics influence academic achievement (hilton, schau, & olsen, 2004). it can be a determinant factor in learning activities and a risk factor in academic achievement including statistics achievement. the attitude has an important role in determining the students’ learning atmosphere. a negative attitude can disturb in learning atmosphere both in the class (liau, kiat, & nie, 2014) and outside. therefore, students’ attitudes toward statistics are important to be studied and assessed (hilton et al., 2004). the research on the attitudes toward statistics has been done for various purposes, among others aimed to reform the used of learning approaches, to improve academic achievement in statistics, as well as to improve the attitudes toward statistics itself. there were out of 17 articles which conducted research about the attitudes toward statistics, but those articles were not yielded a similar result in conclusion (ramirez, schau, & emmioǧlu, 2012). a total of 15 articles showed there was statistically relation between attitudes toward statistics and statistics achievement, but two other articles showed the different things. jahan, al-saigul, & suliman (2016) examined the association between attitudes toward statistics and statistical ability and they expressed that someone who has positive attitudes toward statistics still indicated a difference in terms of statistical ability or achievement. based on the major of studies, ashaari et al. (2011) stated that there was no significant difference in attitudes toward statistics. these results showed that attitudes toward volume 7, no. 2, september 2018 pp 173-182 175 statistics and statistics achievement have not been consistent in its relationship. the meaning is the relationship of attitudes toward statistics and statistics achievement could be either statistically significant or statistically no significant. this study aimed to explore students' attitudes toward statistics based on the beginning and the end of learning, based on differences in fields of study, and to examine the association between attitudes toward statistics and statistics achievement. method the research method used the quantitative method. the population was the first year undergraduate students who joint statistics introduction class during the even semester in 2016/2017 academic year in one of the universities at south sumatera. the sample research was chosen purposively from science fields and another from social field. those fields were known respectively as a group of sci-tech and soc-hum as described by (cahyawati, 2015). the research variables were the attitude towards statistics, statistics achievement, academic achievement, and field of study. the attitude was defined psychologically (ashaari et al., 2011; chiesi & primi, 2009; gal, ginsburg, & schau, 1997; hannula et al., 2016; hilton et al., 2004; jahan et al., 2016). based on their definitions, operational variable of attitudes toward statistics (ats) can be written as a tendency of students’ respond on statistics topic. the research instrument for measuring the ats instrument was the survey of attitudes toward statistics (sats) that was adopted from (schau, 2004; 2008). the sats scale uses 7-points likert that is consisted of six dimensions and 36 items that can be descripted in table 1. table 1. the description of attitude dimensions dimension description number of item(s) affective students’ positive or negative feeling about statistics 6 cognitive students’ intelectual knowledge about statistics 6 difficulty students’ attitude about the difficulty of statistics 7 value students’ attitude about the usefulness, relevance, and worth of statistics 9 interest students’ level of individual interest in statistics 4 effort students’ expends to learn and work in statistics 4 all of the sats items in this study produced statistical high validity and reliability the same as other researchers who used the same instrument. the cronbach's alpha coefficient range was 0.71 to 0.85 except for the difficulty dimension was 0.56. nevertheless, that value is still acceptable for its reliability because all of the items in its dimension had the validity measure (hair, black, babin, & anderson, 2014). the difficulty dimension had the lowest reliability coefficient that was in line with (garcía-santillán, escalera-chávez, rojas-kramer, & pozostexon, 2014; hilton et al., 2004; schau, 2008; schau, dauphinee, del vecchio, & stevens, 1999; tempelaar, schim van der loeff, & guselaers, 2007) in their research. the other variable was academic achievement. it was a result of learning effort that related to aspects of knowledge (arifin, 2016) and that was an outcome of education usually measured by a test or assessment (kumari, 2014). academic achievement was measured by a grade cahyawati, wahyudin & prabawanto, attitudes toward statistics and achievement … 176 point average (gpa). statistics achievement or learning achievement (yuwono, 2018) was measured by a final score on the statistics topic aslike (liau et al., 2014) then it was transformed to 5-grades of quality value, namely a, b, c, d, and e. the data was collected by asking students to give their respond about the sats instrument. students gave the respond twice, the first was at the beginning of the semester (pre-test data) and the second was at the end of the semester (post-test data). there were 118 respondents from 155 students who were participated in data collecting both of pre-test and post-test. the post-test data was analyzed by statistics descriptive. the age was ranging from 17 to 22 years. the most respondent about 70% was women. then some statistical inferential test was used to answer the questions research. results and discussion results attitudes toward statistics attitudes toward statistics can be expressed as a tendency of students’ respond on statistics topic. their respond can be a positive, neutral, or negative. table 1 shows the descriptevely result of attitudes toward statistics from 118 students. table 1. frequency and percentage of each attitudes dimension dimension attitude affective cognitive difficulty value interest effort f % f % f % f % f % f % positive 18 15.2 18 15.2 11 9.3 37 31.4 36 30.5 102 86.4 neutral 55 46.7 71 60.2 89 75.4 60 50.8 66 55.9 12 10.2 negative 45 38.1 29 24.6 18 15.3 21 17.8 16 13.6 4 3.4 for knowing whether there were any differences attitude towards statistics between the beginning and the end of learning the t-test was done. table 2 shows the result of statistical test to each dimensions of attitude. table 2. the results of statistical test dimension pretest posttest n-gain t p-value effect size cohen’s d mean sd mean sd affective 4.95 1.17 4.50 0.98 -0.22 -4.01 0.000 -0.41 cognitive 4.36 0.95 4.31 0.75 -0.02 -0.61 0.542 -0.06 difficulty 3.27 0.81 3.32 0.78 0.01 0.62 0.539 0.07 value 5.49 0.83 5.20 0.86 -0.18 -3.08 0.003 -0.33 interest 5.25 1.15 5.20 0.86 -0.02 -0.83 0.696 -0.04 effort 6.25 1.09 5.93 1.03 -0.41 -2.47 0.015 -0.29 attitudes toward statistics based on the fields of study figure 1 shows the mean score of students’ response to each attitude dimensions based on the fields of study namely science and social fields. volume 7, no. 2, september 2018 pp 173-182 177 figure 1. the attitude dimension mean scores based on the field of study a t-test was carried out to find whether there was any difference in students’ attitudes between science and social fields. the result of the test reached the sig. value (p-value = 0.01) for the value dimension but p-value > 0.05 for other dimensions. academic achievement based on the field of study statistics achievement was measured using 5-grades of quality value (a, b, c, d, and e) and academic achievement was measured using gpa. the students’ statistics and academic achivements based on the field of study is shown in figure 2 then, a t-test conducted to see whether academic achievement between students of science field and students of social fields were different. the results was a significant difference in academic achievement between them (t = -13.082, p < 0.01). figure 2. statistics achievement and the gpa based on the fields of study the association between attitudes toward statistics and statistics achievement the chi-square test was carried out to find the association between students’ attitude respond (positive, neutral, negative) and their statistics achievement (a, b, c, d, e). table 3. pearson chi-square test for attitude dimensions affective cognitive difficulty value interest effort chi-square 8.494 8.042 2.339 16.159 3.868 5.707 df 6 6 6 6 6 6 sig. 0.204 0.235 0.886 0.013 0.695 0.457 0 1 2 3 4 5 6 7 affective cognitive difficulty value interest effort science social cahyawati, wahyudin & prabawanto, attitudes toward statistics and achievement … 178 the outcome of correlation test among the attitude dimensions is shown on table 4. table 4. correlation coefficient and significant value among attitude dimensions affective cognitive competence difficulty value interest effort affective cognitive competence 0.619 difficulty 0.285 0.370 value 0.463 0.323 -0.105 (0.257) interest 0.427 0.293 -0.061 (0.515) 0.603 effort 0.386 0.098 (0.290) -0.291 0.579 0.561 discussion attitudes toward statistics students responded the sats instrument at the beginning and the end of the semester. their response could be classified to the positive, neutral, or negative attitude. the data of the end of the semester, namely post-test data was chosen to classify students’ response. the reason was the attitude at the end of the semester was a good predictor of statistics achievement as stated by wisenbaker (chiesi & primi, 2010). the post-test data descriptively was shown on the table 1, there was a variation enough of students’ response in each attitude dimensions the effort dimension had the most percentage. all the items in this dimension were responded positively by students. this shows that students have had a good positive attitude in the effort to learn statistics. this result was in line with ghulami, hamid, & zakaria (2015) that found the students have given the great effort to learn statistics. the greater effort to learn statistics should reward a higher-quality grade in statistical achievement. unfortunately, that was not shown in the next figure 2. in that figure, students who got the low-quality grade in statistics achievement were quite a lot. the only one dimension that was responded neutrally in all items was the interest dimension. this can be an indication that students have no good interest yet in statistics topic. they attended statistics course as a necessary one because the statistics course is one of the compulsory courses. students had not interested yet in the statistics topic and that could be caused by many factors. one of them is historically the topic of statistics was a difficult and unpleasant discipline to study (garfield & ben-zvi, 2008). however, they responded positively to the item "i can study statistics" in the instrument and believed in being able to learn statistics. they believed that statistics is a valuable topic but it was strangely enough, that other items about the value of statistics in daily life and occupation were still responding neutrally. this can happen because they were new students for the first year in higher education. they have not thought yet about the future field of work which they will be employed. according to the table 1, the largest percentage of positive attitudes was in the effort dimensions. the largest percentage of neutral attitudes exists in the difficulty dimension that volume 7, no. 2, september 2018 pp 173-182 179 was rather different with ashaari et al. (2011) who classified the difficulty dimension into the positive attitude. the greatest percentage of negative attitudes was in the affective dimension that was in line with (ashaari et al., 2011). the statistical test was conducted to the pre-test and the post-test data for testing whether there were any differences of attitude between the beginning and the end of the learning process. the results of statistical test was shown on table 2. all of the n-gain scores in table 2 is less than 0.3 that means the changing in attitudes is in a low category (hake, 1999). the p-value for t-test shows a significant negative change in the affective, value, and effort dimensions, while other dimensions were not. some of the results are in contrast to liau et al. (2014) which showed a positive change in both affective and cognitive competence dimensions, and no change in the value dimension as well. some of the results are in line with liau et al. (2014) whereas the effort dimension showed a negative change and both of the difficulty and interest dimensions did not show any change. the negative changes in the attitude dimensions would be a problem because it indicates a decrease in attitudes responds to statistics topics. it would effect understanding, comforting, and enjoying in learning statistics and learning environment in the classroom as written by gal et al. (liau et al., 2014). furthermore, the negative attitude has influence factor to their statistics achievement. the score of cohen’s d in table 2 ranges from 0.04 to 0.33 which indicates the small effect criterion (cohen, 1992; thalheimer & cook, 2002). this shows that statistics introductory learning just carried out a very little effect on students’ attitude changing. that means that pretest and posttest data were not significantly different, but for the next analysis, the postest data would be chosen to be used. the reason was that wisenbaker (chiesi & primi, 2010) stated that the attitude at the end of the semester was a good predictor of statistics achievement. attitudes toward statistics based on the fields of study the t-test result showed that there were no significant different attitude in all attitude dimensions between the science and the social fields of study (p > 0.05) except for the value dimension (p = 0.01) that indicated the existence of the difference students’ attitude in the value dimension based on the fields of study. the result was in line with griffith, adams, gu, hart, & nichols-whitehead (2012) who stated that majors differ with regard to general attitudes toward statistics. the resulting contrast to ashaari et al. (2011) who stated the attitudes toward statistics was no significant difference based on the fields of study. the association between attitudes toward statistics and statistics achievement based on the result of chi-square test on table 4, only the affective dimension shows a significant association with statistics achievement.this result was in line with hilton et al. (2004) that stated a significant association between several components of attitudes toward statistics and course achievement (measured by test scores or course grades) although the association in small to moderate relations. however, there were some significant associations among attitude dimensions in small to moderate correlation coefficient as shows on table 4. this means that the attitude towards statistics could be improved by paying attention to all dimension simultanously not only one by one dimensions. cahyawati, wahyudin & prabawanto, attitudes toward statistics and achievement … 180 conclusion attitudes toward statistics between students of science and students of social fields were different in the value dimension only. the value dimension means the students believed in a usefulness of statistics in their education, their life, and their future work. the mean score of science field students was higher than social field students. this suggests that the value of statistics need to be improved for students of the social field in order to reach their positive attitudes about statistics topic. this study has not been a success to show the association between attitudes toward statistics and statistics achievement. the characteristics of sample unit and sample size may be the reason. this has to be explored carefully for the next study. references andrews, s. 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(2018). the correlation between cognitive style and students’ learning achievement on geometry subject. infinity journal of mathematics education, 7(1), 35–44. https://doi.org/10.22460/infinity.v7i1.p35-44 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p147-154 147 students’ representation in solving word problem ardhi sanwidi universitas nahdlatul ulama blitar, jln. masjid no. 22, blitar, east java, indonesia ardhisanwidi@unublitar.ac.id received: july 03, 2018 ; accepted: august 31, 2018 abstract the purpose of this research is to describe the representation of sixth grade students in solving mathematics word problems. the focus of the representation of this research is an external representation which is viewed from students with high mathematical abilities. the method used in this research is task-based interview, by giving a problem test of word problems. students who have a high level of abilities, he makes pictures of all problems and successfully solve the problems. students whose level of abilities is lacking, he only makes incomplete symbol / verbal representations, he has wrong when solving the problems. various kinds of representations and increasing abilities in many problems such as multiplying exercises and solve the word pronlem. applying various representations to students are very important to be improved by students in order to succeed in solving various mathematical word problems. keywords: representation, solving problem, word problem. abstrak tujuan dari penelitian ini adalah mendeskripsikan reprsentasi siswa kelas vi dalam menyelesaikan masalah cerita matematika. fokus representasi penelitian ini adalah representasi eksternal yang ditinjau dari siswa berkemampuan matematika yang tinggi. metode yang dilakukan dalam penelitian ini wawancara berbasis tugas, dengan memberikan tes masalah soal cerita matematika. siswa yang memiliki tingkat pemahaman yang tinggi membuat gambar dari semua masalah dan berhasil menyelesaikan masalah. siswa yang tingkat pemahamannya kurang, dia hanya membuat representasi symbol/verbal tidak lengkap, dia tidak tepat dalam menyelesaikan masalah. berbagai macam representasi dan meningkatkan pemahaman dari berbagai masalah yang ada seperti memperbanyak latihan soal dan menerapkan berbagai representasi kepada siswa memang sangat penting untuk ditingkatkan oleh siswa supaya berhasil dalam menyelesaikan berbagai masalah cerita matematika. kata kunci: representasi, menyelesaikan masalah, masalah cerita. how to cite: sanwidi, a. (2018). students’ representation in solving word problem. infinity, 7(2), 147-154. doi:10.22460/infinity.v7i2.p147-154. sanwidi, students’ representation in solving word problem 148 introduction representation is very important in learning mathematics, it is stated in nctm (2000) which states that representation is the core of mathematics study. students can develop and deepen their understanding of concepts and mathematical relationships as they create, compare, and use multiple representations. representations such as physical objects, images, and symbols can help students communicate their thoughts. representations can represent the students' minds of representations captured by the brain, and are reproduced back into another form (markmann, 1999). salkind (2017) reveals that representations are useful, in mathematical learning. objectives for student learning include the development of internal representation systems, understanding traditional external representation systems, creating and using representations as a tool for communication and problem solving. one emerging theme is that student learning involves establishing relationships between different types of representation: pictorial and symbolic; verbal and visual; internal and external. external representation systems are usually symbolic. the system of internal representation is made in one's mind and is used to define mathematical meanings. numbering systems, mathematical equations, algebraic expressions, graphs, geometric figures, and numerical lines are examples of external representations (goldin & shteingold, 2001). external representation has been developed and widely used in various learning. external representation is a written and spoken language. resolving a problem is an attempt to find a way out of a difficulty, reaching a goal that can not be achieved immediately (polya, 1981). stages in settling by polya are understanding the problem, planning the settlement, executing the settlement plan and re-examining the result of the settlement. resolving word problems is one of the important components of mathematical problem solving that combines problems and everyday life, but many studies reveal that students express great difficulty in dealing with word or matter of story (ahmad, tarmizi, & nawawi, 2010; boonen et al., 2014; van der schoot et al., 2009). the term word problem is used to refer to mathematical exercises where significant background information on the problem is presented as text rather than in mathematical notation, since word problems often involve such narratives, they are sometimes also referred to as word problems (boonen et al., 2013). as gagne (ahmad et al., 2010) suggests, in the process of solving the problem of mathematical stories, students must be able to translate concretely into abstract or abstract to concrete. hence the problem solving of the story is more unique and a challenging task than the usual mathematical task. students usually find difficulty in solving the initial word problem from translating word representation into mathematical representations. particularly in science, visual images are preferred to display many relationships and processes that are difficult to describe. research studies show that this type of visual representation can determine how strongly illustrations will be a learning aid and facilitate in solving a problem. in psychology, mathematical representation means the description of the relationship between the object and the symbol. there are two tipes of representation in this research, namely external representation (real world) and internal representation (in mind). lesh, post & behr (1987) shows the five external representations used in mathematics education including real-world representation of objects, concrete representations, arithmetic symbol representations, oral language representations, and visual images or representations. the representation of arithmetic symbols, oral representations of language, and images or visual representations, is a more abstract representation and a higher level of representation for mathematical problem solving (hwang et al., 2007). the ability to represent the spoken symbols is the ability to translate the properties examined and their relation to mathematical problems into verbal representations. volume 7, no. 2, september 2018 pp 147-154 149 the ability to represent written symbols is the ability to translate mathematical problems into the representation of mathematical models. the ability to represent images is the ability to translate math problems into images, diagrams, or graphs. abstract mathematical ideas or concepts can be concrete and more easily understood if planned or deliberately planned by the teacher in multi-representation, so that the learning can run smoothly, and the goal is optimal (hwang et al., 2007). method the type of research used in this study is descriptive qualitative research with the purpose of this study describes the representation of elementary students in solving word problem. the problem is given as follows: 1. dalam perlombaan atletik, jim berada empat meter di depan tom dan peter berjarak 3 m di belakang jim. seberapa jauh peter di depan tom? 2. nindi berjalan kearah utara dari rumahnya menuju kekota a dengan jarak 66 km, lalu dia pergi kekota b dengan jarak 13 km kearah timur. beberapa saat kemudian dia disuruh ibunya pergi kekota c sejauh 66 km kearah selatan. oleh ibunya dia disuruh untuk segera pulang kerumah , berapa jarak rumah si nindi dari kota c? the focus of representation used in this research is external representation. the method used in this research is task-based interview, by giving math word problems. the subjects of this study were the sixth grade elementary school students in blitar east java with the subject of 1 male student and 1 female student with the category of high math ability based on the results of the average of the previous test scores. the researcher acts as the main instrument that is assisted by supporting instruments for giving tests and confirmed by interviews. early stdp conducted in making instrument summary and designing research instrument. all of instrument was validated by validator. the test is given at the time of the lesson outside, with 10 minutes working on the problem. after completing each question, a short interview was conducted to find out what representation the students used to solve the word problem. data in this research is qualitative data. qualitative data gained from validation result of expert to research instrumen, observation result in student activities, interview result of student. when students create visual representations, three categories are distinguished: accurate visual-schematic representation, inaccurate visual schematic representation, or pictorial representation (boonen et al., 2014). results and discussion results the following research results obtained from 2 subjects named (pseudonyms) ihsan and excelina are described as follows: subject of ihsan at the time of understanding the problem, ihsan read over and over to understand the problem, ihsan understood the problem (no. 2) that nindy walked from her house and strolled through to the end of town c. nindi walked in different directions until nindy ended in a city c. ihsan answered the question degan looking for problems that exist on the matter, which is calculate the distance between the city of c and nindy's house or the starting place nindy sanwidi, students’ representation in solving word problem 150 started walking. ihsan did the same in the no. problem. 1 but only briefly explained, because in the interview mentioned that those who ran a distance of only 1 meter only. ihsan only imagined in his mind and managed to find the answer. ihsan drew up a settlement plan (problem no. 2) by drawing the nindy walking from his house to a town, then walking back east toward kora b, then heading south toward town c and ending in town c anyway. in implementing the ihsan settlement plan begin drawing and writing out the results of the answers. ihsan draws the boxes which he thinks is a city that the nindy will pass. in doing this internal representation the subject of ihsan does a visual representation of the problem internally in his mind (hwang et al., 2007; boonen et al., 2014), then he wrote externally in the form of drawings representing everything ihsan know of about story (boonen et al., 2014). in re-examining the answer, the subject of ihsan only glance at it without counting or re-checking carefully, because the subject assumes the question of this story makes him confused and thinks he can not solve it. the same is done for problem no. 1 where ihsan drew the shape of the person as a description of jim, peter, and tom contained in the question, but only two people were drawn, when asked why only 2 people, ihsan replied that the other one was fronted. internal representations of ihsan are seen when interviewed based on the answers he expressed. figure 1. ihsan work results at the time of solving the problem, the subject of ihsan performs abstraction in the form of pictorial and symbolic representation of the word problem containing any information contained in the matter of the story (internal representation), such as drawing his city, the roads through which nindy is in question number 2 (external representation). the internal representation of ihsan appeared at the interview, he mentioned that 3 people were in the matter, but only 2 were drawn by ihsan. ihsan think of the three people in the same way but with different distances. in terms of understanding the problem, ihsan subjects do a repetition in reading. when planning and executing a plan of completion, the subject of inspiration makes an accurate representation of the schema that describes the situation in the matter or represents again in the form of an image, any variables derived from matter into image form (boonen et al., 2014). at the time of completing the question, ihsan connect the knowledge of the sum of numbers and recall the form of the wake up in solving problem number 2. excel subject excel understands the problem given by reading it for a long time and mumbling. excel says that solving the problem of number 1 only adds it will find the answer, because it is a matter of addition and subtraction. excel understands that question number 2 is to calculate the volume 7, no. 2, september 2018 pp 147-154 151 distance between nindi's house and city c which previously runs into town a, then town b, and ends in town c, which is asked is the distance from nindy's house to town c. in executing the settlement, the excel subject adds all the distance between nindy's house and every town that is till ends in town c as shown below figure 2. excel work results excel subjects understand the problem of this story by starting and reading for a long time. in planning and executing the plan, the excel subject does not represent visually as the previous subject (ihsan) it represents into the algebraic symbol, but is still wrong. it represents the city of a = 66 km, whereas 66 km is the distance between nindy's house and city a. whereas in solving this word problem it will be easier to visualize all the variables in question number 2 (boonen, 2014). excel solves this problem by adding all the distance that is in the matter, because the understanding of excel in question is the distance of the house and the last city distance passed by nindy. in re-examining the results of his work, the subject of this excel is less visible, because the subject of excel only see it without counted back or read carefully the answer. in question number 1, excel responded by drawing a line on the answer. excel in his interview replied that the problem is a trick problem, because according to excel the answer on the matter is 4 meters the same as in the matter. without thinking long, excel justify for the result of answer about the number 1 is true without checking again. discussion this research examines the importance of students' visual representation by using test items for word poblem and interview questions to explore the representations made by students in external or internal representation (salkind, 2017). from the written test results strengthen the evidence that in solving word problems, students who draw, the answer to the problem is correct. conversely, students who do not draw or just write, the answer to the problem is wrong. furthermore, to see more about the representation being built, an interview was conducted. based on the results of the interview, it strengthens the evidence that the (external) representation of writing or images written by students influences students' understanding of word problems. it was proven that after interviewing on subject 2, the level of understanding of the problem was low, using the schema representation was not accurate, the answer was wrong. conversely on subject 1, when the interview is done based on the results of the test or the writing looks high level of understanding, using an accurate pictorial scheme representation, the correct answer. of the 2 research subjects, it can be seen that students who solve the problem of word problems by using better pictures and answers to solutions are more appropriate than students sanwidi, students’ representation in solving word problem 152 who solve the problem only by making a symbolic or variable representation such as the 2nd subject (ahmad et al., 2010; boonen et al., 2013; boonen et al., 2014). in solving a problem, each student does it in different ways but the stages are sequentially understanding the problem, planning a solution, completing the process and seeing the work again (polya, 1981) conclusion judging from the truth of subject 1 and subject 2, it can be seen that the students who represent the solution of the story with the shape of the picture (accurate schema representation), make it easier for students to finish it and the result of the completion is also appropriate (ahmad et al., 2010; boonen et al., 2014; hwang et al., 2007). unlike the subject 2, that only represent the symbol and verbal (inaccurate schema representation) that resulted from the completion of students is less precise. in terms of understanding the problem, subject 2 looks very less. subject 2 only describes the straight line when solving the problem, whereas much of the information contained in the subject is less noticeable by subject 2. it is seen to be able to represent students of various kinds either verbally or visually representation (lesh, post & behr, 1987). in this research, like subject two which changes (translates) city a with 66 km, while 66 km is the distance between nindy's house and city a. it can be seen that students' understanding is still weak, for that the teacher needs to provide experience to improve understanding and word problem solving to student. as well as providing more knowledge to represent mathematical problems, because representation is an important part of solving mathematical problems (salkind, 2017). in this study, students who represent the word problem by drawing what is understood in the problem (accurate schema representation) can solve the problem correctly. students who represent with symbols or writing are not appropriate for solving problems (inaccurate schema representations). internal representation in this research is still a bit unearthed, because this research focuses only on students' external representation. the potential internal representation that can be explored more deeply is what students say during the interview, but in describing it in different writing forms and the students respond to it to be difficult to express in written form and in their minds. in other words is to discuss the written or written completion of the students either in the form of writing, symbols, pictures, graphics and other of the given problem. for further to further research discuss the imagination or what the student thinks about first (internal representation) in understanding the given problem as well as the process that the student does to shape, use, read, change the representation to solve the problem as well as further developed about the visual representation against the subject of teachers or other students who will later students will easily solve math problems and mewujutkan what is in the imagination of students in the form of images and symbols so easy for students to solve problems. references ahmad, a., tarmizi, r. a., & nawawi, m. (2010). visual representations in mathematical word problem solving among form four students in malacca. procedia-social and behavioral sciences, 8, 356-361. volume 7, no. 2, september 2018 pp 147-154 153 boonen, a. j., van der schoot, m., van wesel, f., de vries, m. h., & jolles, j. (2013). what underlies successful word problem solving? a path analysis in sixth grade students. contemporary educational psychology, 38(3), 271-279. boonen, a. j., van wesel, f., jolles, j., & van der schoot, m. (2014). the role of visual representation type, spatial ability, and reading comprehension in word problem solving: an item-level analysis in elementary school children. international journal of educational research, 68, 15-26. goldin, g., & shteingold, n. (2001). systems of representations and the development of mathematical concepts. the roles of representation in school mathematics, 2001, 1-23. hwang, w. y., chen, n. s., dung, j. j., & yang, y. l. (2007). multiple representation skills and creativity effects on mathematical problem solving using a multimedia whiteboard system. journal of educational technology & society, 10(2). lesh, r., post, t. r., & behr, m. (1987). representations and translations among representations in mathematics learning and problem solving. in problems of representations in the teaching and learning of mathematics. lawrence erlbaum. markmann, a. b. (1999). knowledge representation. mahwah, nj: erlbaum nctm (2000). principles and standards for school mathematics. reston/va: national council of teachers of mathematics polya, g. (1981). mathematical discovery on understanding, learning, and teaching problem solving. united states of america salkind, g., m. (2017). mathematical representations. edci 857 preparation and professional development of mathematics teachers van der schoot, m., arkema, a. h. b., horsley, t. m., & van lieshout, e. c. (2009). the consistency effect depends on markedness in less successful but not successful problem solvers: an eye movement study in primary school children. contemporary educational psychology, 34(1), 58-66. sanwidi, students’ representation in solving word problem 154 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p31-42 31 learning motivation and mathematical understanding of students of islamic junior high school through active knowledge sharing strategy rahayu kariadinata 1 , r poppy yaniawati 2 , hamdan sugilar* 3 , dede riyandani 4 1,3 universitas islam negeri sunan gunung djati 2,4 universitas pasundan article info abstract article history: received aug 7, 2018 revised sept 6, 2018 accepted jan 4, 2019 this study aims to analyze and describe learning motivation, mathematical understanding and activities of students' and teachers at islamic junior high school (madrasah tsanawiyah/mts) through active knowledge sharing strategy. the research is conducted in mts. al-mukhlisin bandung indonesia. the method of this research is a mixed method with classroom action research (car). the instruments used were motivation questionnaires, observation sheets, mathematical understanding tests and observation sheets. the results of the research indicate that learning motivation, mathematical understanding and activities of teachers and students through the active knowledge sharing strategy increase in each cycle. students' learning motivation is shown from the result of the questionnaire, states that the average percentage of motivation indicator increase in each cycle, those are 70,35% (cycle i), 71,17% (cycle ii ) and 72.15% (cycles iii), student mathematical understanding indicated from the test results states that the average value increases in each cycle and the end of the cycle has reached the classical completeness, that is 79.41%, the activities of students and teachers during teachinglearning increased in each cycle of 60.95% (cycle i), 73.33% (cycle ii) and 86.67% (cycles iii). keywords: active knowledge sharing strategy learning motivation mathematical understanding copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: hamdan sugilar, departement of mathematics education, universitas islam negeri sunan gunung djati, jl. a.h. nasution no. 105 bandung, indonesia. email: hamdansugilar@uinsgd.ac.id how to cite: kariadinata, r., yaniawati, r. p., sugilar, h., & riyandani, d. (2019). learning motivation and mathematical understanding of students of islamic junior high school through active knowledge sharing strategy. infinity, 8(1), 31-42. 1. introduction the existence of islamic education in indonesia is instrumental for shaping the character of students. islamic values applied in the institution are expected to shape the character of students which are moral, high spirits, respect for spiritual values and humanity, teaches attitude and honest behavior, and simple life. the development of education in indonesia is marked by the emergence of various forms of various institutions, mailto:hamdansugilar@uinsgd.ac.id kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 32 such as boarding schools, madrasah, surau, and meunasah (akhiruddin, 2015). madrasah is one form of islamic education in indonesia, initially only teach a variety of religious knowledge, train experience in islamic teachings, including worship practices, muamalah and morals (nurhasnawati, 2015). madrasah is a container or place of learning islamic knowledge and other skills that developed in his time. thus it can be concluded that the term madrasah comes from islam itself (akhiruddin, 2015). the word "madrasah" comes from arabic meaning 'place of learning' equated with the word 'school'. however, within the framework of the national education system they are different. the school is known as a primary and secondary educational institution whose curriculum focuses on general subjects, and its management is under the auspices of the ministry of national education. while madrasah are known as basic and intermediate religious education institutions which, therefore, focus more on religious subjects, and their management is the responsibility of the ministry of religious affairs (kosim, 2007). in 1994, a typical islamic madrasah concept was formed with 70% curriculum content of general knowledge and 30% religious knowledge. that is, this curriculum modification equates the substance and content of the madrasah curriculum with public schools according to the national education system (nurhasnawati, 2015). for example, for islamic junior high school (mts) curriculum is the same as the junior high school curriculum, only mts has more portion of islamic education. such as: quran and hadith, aqidah akhlak, fiqh, history of islamic culture, and arabic. the combination of common and religious subjects into a single curriculum under the auspices of the ministry of religious affairs. however, the impact of the curriculum content composition is the increasing burden on the madrasah. on the one hand, it has to improve the quality of education generally in accordance to the standards applicable in schools. on the other hand, however, the madrasah, as an islamic educational institution, must keep the quality of religious education well. their burdens to receive so many lessons has an impact on learning motivation and the ability to understand common subjects, especially subjects that are students have difficulty in mathematics. based on the results of preliminary studies conducted by researchers on students of islamic junior high school al mukhlisin bandung indonesia, when learning activities of mathematics, students are less passionate in learning, students tend to be passive in doing tasks given by teachers, students do not timely in collecting tasks. there are even some students who do not work at all with difficulties solve the problem. teachers have tried to provide various motivations in the learning process, such as giving comments on each students' answers, giving quizzes with additional value bonus, giving praise to the students who got good grades, giving some gift to the students who scored high on every daily test. but the motivation of students to actively learn math remains low and attitude. attitude towards mathematics is factors that influence learning achievementmathematics (rusnilawati, 2016). such low motivation has an impact on understanding the concept of mathematics. this is obtained based on the results of student work in completing the mathematics that the researchers gave during the preliminary study. some problems that require understanding of mathematical concepts have not been optimally mastered by students, such as the weak understanding of one concept with other concepts needed to solve of mathematical problems. learning motivation is very important as a determinant of the success of student learning outcomes. learning outcomes will be optimal if there is motivation. the higher the motivation the better the learning outcomes. so motivation will always determine the intensity of the learning effort for the students. learning motivation is defined as the impulse arising from within due to the influence of within itself or outside influences that aims to achieve learning success. through the motivation found in students, volume 8, no 1, february 2019, pp. 31-42 33 something new can be learned. in this case, it is needed teachers’effort to generate students’motivation to strengthen the response that has been learned. teachers should be able to innovate learning and motivate students to learn more actively, creatively and systematically to find mathematical knowledge independently. students' creative thinking ability cannot develop well if the teacher doesn’t involve the students actively in learning process (sugilar, 2013). learning strategy is needed to overcome it, that can encourage students to be more active. active learning is one of the alternatives that can be applied by engaging intellectual and emotional students so that they play an active role and participate in learning activities. students are expected also to make improvement of their performance in learning through positive behavior as part of the soft skills (rosyana, afrilianto, & senjayawati, 2018). learning process is not on the delivery of information (students as listeners) but the development of higher-order thinking such as analysis, synthesis and evaluation, 2) emphasis on values and attitudes related to the material, 3) students are actively involved in reading, writing and discussion (bonwell & eison, 1991). active knowledge sharing is a strategy requires students to be ready to learn quickly so that his or her ability in team work can be observed (zaini, 2007). siberman (2007) suggests some things to be careful of when teachers will initially apply learning using the active knowledge sharing strategy to achieve the objectives of: 1) establishing a discussion team so students know each other and create a spirit of cooperation, 2) assess attitudes, motivation, knowledge and student experience, 3) motivate students’ interest at the beginning of the lesson. the learning that uses active knowledge sharing strategy in this research consists of several stages, namely the stage of questioning, discussion, knowledge sharing and discussion. the role of teachers in learning as a motivator and facilitator who can invite students to construct their own knowledge and find their own way of learning. the purpose of this research is to know the students' learning motivation, mathematical understanding and student and teacher activity through active knowledge sharing strategy. 2. method this type of research is a mixed-method with classroom action research (car), a method that focuses on collecting, analyzing, and mixing quantitative and qualitative data in one study or series of studies (indrawan & yaniawati, 2014). this research is conducted at mts.al-mukhlisin bandung, indonesia to the eighth grade students. activity of car by implementing active knowledge sharing follow the main component in action research as follows: 1) planning, 2) acting, and observing and 3) reflection is done in three cycles which each cycle consist of three meeting. the instruments used are mathematical understanding test, learning motivation questionnaire, student and teacher activity observation sheets in active knowledge sharing learning and interview guide. the data to be obtained in this study consists of quantitative data and qualitative data. quantitative data obtained through mathematical understanding test, filling questionnaire student learning motivation, observation sheet while qualitative data obtained through interview. in the questionnaire of learning motivation, students are asked to fill out by choosing answers in the form of: ss (strongly agree), s (agree), ts (disagree), n (neutral), and sts (strongly disagree) towards learning with active knowledge sharing strategy. kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 34 3. results and discussion 3.1. results 3.1.1. description of student learning motivation data of students’ learning motivation in teaching-learning mathematics through active knowledge sharing strategy, obtained from the motivation questionnaire given to students’at each end of the cycle. this motivation questionnaire uses a likert scale. there are 8 indicators of learning motivation: 1). duration of learning activities, 2) frequency of learning activities, 3) persistence in learning, 4) firmness, tenacity and ability to face obstacles and difficulty in achieving goals, firmness, tenacity and ability to face obstacles and difficulties in achieving goals, 5) level of students’ aspiration in learning , 6) loyalty and sacrifice for achievement, 7) the level of qualification and learning achievement, and 8) the direction of students' attitudes in learning (syamsudin, 2003). the result of questionnaire analysis of students' learning motivation can be seen in table 1. table 1. analysis results of students’ learning motivation at any cycle based on table 1, students’ learning motivation in each cycle is mostly in the high category, that is, the average of the percentage of cycle i is 70.35% cycles ii (71.17%) and cycles iii (72.15%). 3.1.2. description of students’ mathematical understanding on every cycle based on the results of research in cycle i, cycles ii and cycle iii, each cycle consists of 3 meetings. any average grade that a students’ has achieved in each cycle is compared with the minimum completeness criteria (mcc). students’have understood the no indicator descriptor percentage (%) in : cycle i cycle ii cycle iii 1. duration of learning activities a. duration of use of study time 78.50 78.25 80.50 b. the duration of concentration at the time of study 75.33 76.00 77.33 2. frequency of learning activities a. frequently reading textbooks 67.67 70.67 71.33 b. often do learn 65.00 66.50 67.00 3. persistence in learning a. provision in achieving goals 71.33 74.67 71.67 b. stickiness in achieving learning goals 68.33 69.33 68.00 4. firmness, tenacity and ability to face obstacles and difficulty in achieving goals , a. steady in the face of obstacles 65.50 65.75 67.00 b. tenacious in achieving goals 65.00 66.50 70.00 5. level of students’ aspiration in learning a. have high ideals 80.50 82.00 84.00 volume 8, no 1, february 2019, pp. 31-42 35 concept of mathematics when it reaches mcc. each school has defined a mcc for each subject adjusted to the students’ average ability level, indicator and condition of the education unit (depdiknas, 2008). in mts.al-mukhlisin bandung indonesia which is the location of research, mcc for mathematics subject is 75. so students’ are said to be solved individually if it has reached the value of  75. the description of students’ mathematical understanding in each cycle is presented in table 2. table 2. average of students’ mathematical understanding of every cycle based on the results listed in table 2, it can be seen that the average score of students’ at the end of the third cycle has not reached the mcc, but after being given a final test (postest) the average score is above the mcc value and has reached completeness in a classical manner, average and number of students’ who have achieved learning mastery. 3.1.3. description of students’ and teacher activity through active knowledge sharing strategy observational data during learning activities were obtained from observation sheets observed by observers. observations of students’ and teacher activity in each cycle are presented in table 3 and table 4. table 3. students’ activity observation results on each cycle no learning activity avarage mcc number of completed students’ percentage of classical completed category 1. cyrcle initial test (pretest) 14.95 75 75 75 75 75 0% not completed 2. cycle i 61. 29 9 26.47% not completed 3. cycle ii 66. 06 14 41.17% not completed 4. cycle iii 70. 21 18 52.94% not completed 5. cycle end test (postest) 78. 67 27 79.41% completed no students’ activity observed achievement (%) in cycle i cycle ii cycle iii 1 students’ ask the questions about course material when discussing the results of the discussion 53.33 80,00 86,67 2 students’ work on the worksheets given by teachers 80.00 93,33 93,33 3 students’ make the conclusions on the subject matter at the end of the lesson 66.67 80,00 80,00 4 students’ express their opinions when discussing the results of the discussion 53.33 73,33 80,00 5 students’ give advice when discussing the results of the discussion 46.67 60,00 73,33 6 students’ respond to questions when discussing the results of the discussion 53.33 73,33 86,67 7 students’ prepare to learn by making assigned home tasks 73.33 80,00 80,00 average 60. 95 73,33 86,67 average overall = 73.65 kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 36 based on table 3, there is an increase in the percentage of students’ activity in each cycle, and the average students’ activity throughout the cycle is 73.65% (good). this shows that active knowledge sharing strategy is effectively applied in mathematics learning. furthermore, observation result of teacher activity in each cycle presented in table 4. table 4. teachers’ activity observation results on each cycle the result of observation showed that all of teachers’ activity on observation sheet majority in good category, and the average activity on whole cycle equal to 81.97% (good). this shows that teachers can implement active knowledge sharing strategies well. 3.1.4. reflection on every cycle in each cycle of teachers and observers analyze learning outcomes that have been implemented. various obstacles are found when teachers apply the active knowledge sharing strategy. details of learning reflections on each cycle are presented in table 5, table 6 and table 7. no active knowledge sharing stages students’ activity observed achievement (%) in cycle i cycle ii cycle iii 1. questioning stages begin the lesson by giving positive suggestions to condition the students’ 80.00 80.00 80.00 revisiting the previous material and giving the students’ an opportunity to ask and respond 86.66 86.66 93.33 guiding students’ in filling students’ worksheets (sw) 86.66 86.66 80.00 2. group discussion stages guiding students’ on group activities 80.00 86.66 80.00 directs students’ to find various information that can support problem solving 80.00 80.00 80.00 3. knowledgesharing stages observe and direct students’ work 80.00 80.00 80.00 guiding students’ to communicate the results of the discussion 80.00 80.00 80.00 control the course of group presentations 80.00 80.00 86.66 4. disscussion stages guiding students’ to conclude material at the end of learning 80.00 80.00 80.00 average 81.48 82.22 82.22 average overall = 81.97 volume 8, no 1, february 2019, pp. 31-42 37 table 5. learning reflectionsons on cycle i table 6. learning reflections on cycle ii table 7. learning reflections on cycle iii active knowledge sharing stages progress suggested remedies questioning stages students’ are active to answer questions on the worksheet group discussion stages students’ are active in group discussions active knowledge sharing stages obstacles suggested remedies questioning stages students’ are still confused with active knowledge sharing learning strategy the description of the active knowledge sharing strategy steps needs to be clarified group discussion stages students’ are still not active in asking questions during the discussion 1. teachers need to guide and direct the students’ during the discussion 2. teachers need to explain the difference between the diagonal plane and the diagonal of space knowledge sharing stages students’ are not yet optimal in sharing knowledge disscussion stages some students’ have not been able to distinguish mathematical concepts such as differences between the diagonal plane and the diagonal of space active knowledge sharing stages obstacles suggested remedies questioning stages students’ have started actively answering questions on the worksheet, but still need to be guided group discussion stages students’ have not shown their activity in group discussions teachers need to motivate students’ to be active in the discussion knowledge sharing stages students’ have started to dare to share knowledge disscussion stages some students’ have not grasped the concept of formulas in mathematics, for example the formula of the surface area of a flat side room teacher directs the way of understanding the concept of formulas in mathematics, for example the surface area of a flat side room kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 38 active knowledge sharing stages progress suggested remedies knowledge sharing stages students’ are dare to share knowledge disscussion stages students’ understand the material they have learned 3.1.5. reflection on every cycle to get an overview of the implementation of the active knowledge sharing strategy, interviews were made of a number of students’. the results of interviews from 9 students’ consisting of 3 highly skilled students’, 3 medium-skilled students’, and 3 students’ of low-ability students’ were presented in table 8. table 8. interview results with students’ 3.1.6. analysis students’ motivation through active knowledge sharing strategy based on the results of the analysis, students’learning motivation in each cycle is in medium and high category, it means that students’are motivated in learning mathematics through active knowledge sharing strategy. increased motivation in each cycle is seen during group discussions at the knowledge sharing stage. through the sharing of knowledge students’ will experience increased understanding because of the mutual between students. the exchange of knowledge can be through the internet, peers and other no interview questions respondents answer 1. how do you feel about learning using the active knowledge sharing strategy? active sharing knowledge learning becomes more interesting and fun because we are helped by working groups and sharing knowledge so that worksheets can be filled properly. 2. what are the obstacles in mathematics learning using the active knowledge sharing strategy? during the knowledge sharing stage, the classroom atmosphere became crowded, so we did not focus on learning the material, the timing of the problem was too short so it was difficult to understand the material taught by the group representatives. 3. what are the benefits of learning using active knowledge sharing strategy? through active knowledge sharing learning, mathematical material is easy to understand and the atmosphere of togetherness is getting closer. 4. what are your expectations after following the mathematics learning using active knowledge sharing? we hope to get good grades; can solve problems correctly. 5. what is the general group response to the active knowledge sharing strategy? we hope the active knowledge sharing strategy is always applied in mathematics learning. volume 8, no 1, february 2019, pp. 31-42 39 sources. the principle of mutual exchange of knowledge is transferring knowledge to others. between one person and another can exchange knowledge derived from their own experience (bechina & bommen, 2006). students’ motivation on learning activity duration indicator is high in every cycle, it is indicated from percentage score reaching 66.75%, meanwhile students’motivation on learning activity frequency indicator is increase from medium level in cycle i (66.34%) to high category in cycle ii (68.59%) and cycle iii (69.17%). this aspect refers to the frequent or not of students’ learning activities. this motivation is needed when students’ are studying in the classroom or outside the classroom. while learning in the classroom with the active knowledge sharing strategy there is a questioning stage, they must set the time for each step to pass well. providing questions (tasks) is one effort to keep students’ motivation so that knowledge will be obtained. motivation is a determinant factor in learning, but students’ motivation can change, when exposed to the environment that stimulate their attention. to keep students’ engaged, students’ need to maintain a taskoriented outlook on learning, which is associated with deep-level learning and learning for the sole gratification of acquiring knowledge. in the indicators of diligence in learning, students’ learning motivation shows a high percentage score in each cycle ( 65%). perseverance in learning can be interpreted as violence of determination and sincerity in learning. in addition based on the questionnaire results this is also reflected when they do learning activities, they are diligent in undergoing all the stages in learning. while the indicators of fortitude, tenacity and ability to face percentage difficulties in the cycle i and cycle ii in the category of medium (≤ 65%) and increased to high category in cycle iii (68.50%). this indicates that students’have tried as much possible to overcome problems in learning. the students’ effort through searching information related to mathematics learning that is internet, teacher, and classmates. high learning motivation is reflected by the non-breaking persistence to achieve success despite being confronted by difficulties. some of the characteristics of good learning motivation such as facing duties, tenacious to face difficulties and not quickly satisfied with the achievements that have been achieved. score percentage of students’ aspiration level in teaching-learning activity in high category at each cycle ( 65%). similarly, the level of loyalty and sacrifice to achieve learning achievement in the high category ( 65%). the knowledge sharing stage in active knowledge sharing shows that the strength of the students’ learning and the students’ motivation to gain knowledge will be shared with their friends in the discussion. through this stage each students’ has valuable knowledge in the group. furthermore, the qualification level indicator shows the percentage score in the medium category (≤ 65%) in each cycle, it indicates that the skills that students’ have in performing the stages in active knowledge sharing are still not optimal, because the students’ are not yet familiar with a learning that requires liveliness in thinking to share knowledge, while the indicator of students’ attitude toward learning achieves high percentage score in each cycle ( 65%). the characteristics of the active knowledge sharing strategy, including greater emphasis on the exploration of values and attitudes of students’ (bonwell & eison, 1991). 3.1.7. students’ and teacher activity in mathematics learning with active knowledge sharing strategies on classroom action implementation implementation of learning carried out in accordance with the steps of classroom action research starts from planning (planning), action (acting) and observation (observing) and ends with reflection (reflecting). at the planning stage the teacher prepares various kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 40 tools that support the implementation of mathematics learning using active knowledge sharing strategies such as making the learning plan) students’ worksheet, determining the time required for each cycle that is 6 x 40 minutes (3 x meetings). implementation of action in each cycle begins with apperception and motivation. at this stage the teacher explains the learning objectives and the relevance between the subject matter and the daily life. in addition, the teacher informs the lesson that will be implemented using the active knowledge sharing strategy and the steps. the first stage of the active knowledge sharing strategy, which is the questioning stage, teachers first group students’ heterogeneously based on academic ability. the teacher divided the group into 7 groups, each group consisting of 4 and 5 people. furthermore, the teacher distributes the students’ worksheet containing a list of questions related to the subject matter to be taught. during the discussion phase, the teacher invites the students’ to discuss answers on the worksheets with their respective group members within the stipulated time. the teacher provides guidance and supervises the course of the discussion. in the next stage of knowledge sharing, each group assigns one students’ as a group representative to ask another group within a predetermined time, where the group's representatives are different at each meeting. groups that know the answer are obliged to share knowledge actively to other students’ who do not understand. representatives of groups who have completed the worksheets sit back in their respective groups. group representatives are responsible for sharing the knowledge or information they get from other groups to their respective group members. after that the students’ work on the mathematical understanding problems that are on their worksheets individually. during the discussion strage, the teacher selects one of the group representatives at random to present the part of the completed worksheet and provide new information obtained from other group index cards. the information should not be the same as the information on the index cards in the group. members of other groups are given the opportunity to give opinions and responses about what has been presented. then the teacher and the students’ discuss the results of the group work and provide reinforcement of the students’ work. teacher give rewards the best group. each action requires students’ and teachers to take an active role in performing the stages of active knowledge sharing. communication and interpersonal skills and the expression of students’ ideas will arise during the learning process. this is in accordance to the opinion of (majid & chitra, 2013) stating that: “…active knowledge sharing, brings many benefits to students’ such as better academic achievements, improved communication and interpersonal skills, appreciation for diverse ideas and viewpoints, positive inter-dependence, and a sense of satisfaction for contributing towards learning of others.” observation of the actions performed on students’ and teachers obtained good results with the average percentage of each of 73.65% and 81.97%. it indicates that students’ and teachers succeed in learning mathematics with active knowledge sharing strategy. reflection activities are conducted between the teacher and the observer to discuss the constraints at each stage of active knowledge sharing. these constraints are discussed together and look for remedial solutions for planning and action at the next cycle meeting. based on the results of reflection analysis in each cycle it appears that there is a reduction of constraints for students’ and teachers at each stage. by referring to reflection, students’ and teachers demonstrate good activity in the following cycles. volume 8, no 1, february 2019, pp. 31-42 41 3.2. discussion students’ mathematical understanding through active knowledge sharing strategy improves in every cycle. through the stages of discussion, sharing knowledge and discussion, students’ can solve mathematical solutions by linking a concept with other concepts. based on the analysis and discussion, overall students’ learning motivation through active knowledge sharing strategy in each cycle is high. thus, active knowledge sharing strategy can improve students’ motivation. learning motivation is an internal and external impulse found in a person while learning to change behavior, and learning achievement. interaction between students’ which becomes an important stage in active knowledge sharing has an impact on students’ learning motivation. the use of this strategy can motivate the students’ so interested to follow the learning, because at the beginning of learning students’ have been motivated by giving questions. based on the results of interviews with students’, the students’ felt attracted and helped by the sharing of knowledge among members of the discussion group, the mathematical material that had been considered difficult, became easy to understand, and the students’ felt a close togetherness learn mathematics. obstacles of students perceived when sharing knowledge, it is necessary for information from the teacher about the strategy of active knowledge sharing, so that the learning process runs without noise.the proposed description concludes that mathematics learning through active knowledge sharing strategy with classroom action setting is very effective to solve problems related to learning motivation, concept comprehension, and students’ activeness. classroom action research that has been done can answer the problems that have been appearing in class. action research is an effort to alleviate real problems, to increase effectiveness (hopkins, 2014). 4. conclusion students’learning motivation in mts. al mukhlisin bandung indonesia through active knowledge sharing strategy shows the percentage increase in every cycle, that is 70.35% (cycle i), 71.17% (cycle ii) and 72.15% (cycles iii). the increase of learning motivation indicates that active knowledge sharing strategy is very effective to be applied to mathematics learning. stages in active knowledge sharing (giving questions, group discussions, sharing knowledge and discussion) provide opportunities for students’ to display self-actualization to achieve good mathematics learning outcomes. students’mathematical understanding in mts. al mukhlisin bandung indonesia through active knowledge sharing strategy that increases in each cycle and the end of the cycle has reached classical mastery, that is 79.41%. completeness of students’ learning achieved during the last cycle, this indicates to improve understanding of mathematical concepts need to be carried out several cycles. students’ and teacher activity of mts.al mukhlisin bandung indonesia during learning with active knowledge sharing increase in every cycle are 60.95% (cycle i), 73.33% (cycle ii) and 86.67% (cycle iii). increased activity indicates that students’ and teachers have mastered the stages of active knowledge sharing strategy. reflection on each action becomes a reference for improvement of the next meeting. references akhiruddin, k. m. (2015). lembaga pendidikan islam di nusantara. jurnal tarbiya, 1(1), 195-219. kariadinata, yaniawati, sugilar, & riyandani, learning motivation and mathematical … 42 bechina, a. a., & bommen, t. (2006). knowledge sharing practices: analysis of a global scandinavian consulting company. the electronic journal of knowledge management, 4(2), 109-116. bonwell, c. c., & eison, j. a. (1991). active learning: creating excitement in the classroom. 1991 ashe-eric higher education reports. eric clearinghouse on higher education, the george washington university, one dupont circle, suite 630, washington, dc 20036-1183. depdiknas (2008). rancangan hasil belajar. jakarta: direktorat pembinaan sekolah menengah atas-direktorat jendral manajemen pendidikan dasar dan menengahdepartemen pendidikan nasional. hopkins, d. (2014). a teacher's guide to classroom research. mcgraw-hill education (uk). indrawan, r., & yaniawati, p. (2014). metodologi penelitian kuantitatif, kualitatif, dan campuran untuk manajemen, pembangunan, dan pendidikan. bandung: refika aditama. kosim, m. (2007). madrasah di indonesia (pertumbuhan dan perkembangan). tadris, 2(1), 41–57. majid, s., & chitra, p. k. (2013). role of knowledge sharing in the learning process. literacy information and computer education journal (licej), 2(1), 1201-1207. nurhasnawati, n. (2015). pendidikan madrasah dan prospeknya dalam pendidikan nasional. potensia: jurnal kependidikan islam, 1(1), 85-98. rosyana, t., afrilianto, m., & senjayawati, e. (2018). the strategy of formulate-sharelisten-create to improve vocational high school students’mathematical problem posing ability and mathematical disposition on probability concept. infinity journal, 7(1), 1-6. rusnilawati, r. (2016). pengembangan perangkat pembelajaran matematika bercirikan active knowledge sharing dengan pendekatan saintifik kelas viii. jurnal riset pendidikan matematika, 3(2), 245-258. siberman, l. m. (2007). active learning strategi pembelajaran aktif. yogyakarta: pustaka insan madani. sugilar, h. (2013). meningkatkan kemampuan berpikir kreatif dan disposisi matematik siswa madrasah tsanawiyah melalui pembelajaran generatif. infinity journal, 2(2), 156-168. syamsudin, a. (2003). psikologi pendidikan perangkat sistem pengajaran modul. bandung: pt. remaja rosdakarya. zaini, h. (2007). strategi pembelajaran aktif. yogyakarta: insan madani. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 2, september 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i2.p157-166 157 developing a local instruction theory for learning combinations ika meika *1 , didi suryadi 2 , darhim 3 1 universitas mathla’ul anwar banten 2,3 universtas pendidikan indonesia article info abstract article history: received july 9, 2018 revised august 29, 2019 accepted sept 2, 2019 this research aims at developing local instruction theory for learning combinations by using realistic mathematic education (rme). the lit flow developed in this study is to find an easy learning path to help students build basic concepts in combination material. to achieve the objectives of this study, researchers used design research. the hypothetical learning trajectory (hlt) was developed from a series of activities to get a better understanding about combinations at senior high school (shs) students. theoretical development is supported by interactive process in designing learning activities, conducting teaching experiment and conducting retrospective analysis to contribute to the development of lit combinations. an understanding of combinations emerges and develops during classroom learning activities. qualitative analysis of teaching experiments shows that by using lit teaching materials that contain the characteristics of pmr, students can build basic concepts and develop their understanding about combination. keywords: combinations, hlt, local instruction theory, rme copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: ika meika, departement of mathematics education, universitas mathla’ul anwar banten, jl. raya labuan km 23, saketi, pandeglang, banten 42273, indonesia email: ikameika@unmabanten.co.id how to cite: meika, i., suryadi, d., & darhim, d. (2019). developing a local instruction theory for learning combinations. infinity, 8(2), 157-166. 1. introduction permutations and combinations are the basic materials in studying probability. probability is part of mathematics that needs to be mastered by high school students as a prerequisite of statistical material that is very widely used in designing research and processing data of research results from various branches of science (azhar & kusumah, 2011). garfield & ahlgren (1988) revealed that permutations and combinations are the important parts of introduction to statistics at some universities. according to van de walle (2008) students' ideas about probability must evolve from experience (exploration). further, busadee & laosinchai (2013) stated that in studying permutations and combinations, students need real relevant issues to stimulate their learning and knowledge. not a few students are mistaken in solving combinatorics problems, especially on sub-topic of permutations and combinations. a common mistake is when using mailto:ikameika@unmabanten.co.id meika, suryadi, & darhim, developing a local instruction theory … 158 permutations and combinations in problem solving, if there is no instruction to use permutations and combinations in a question clearly, most likely students experience confusion. garfield & ahlgren (1988) stated that the combination material is difficult material for students and is a prerequisite material for studying probability and statistics. the difficulty of high school students in learning permutations and combinations is an important issue in probability lessons (ben-hur, 2006; busadee & laosinchai, 2013; fischbein, 1975). based on the results of preliminary research done by the authors showed that the error rate of students in solving the problem of combinatorics was quite high (meika & suryadi, 2018). combinartorics is considered one of the more difficult mathematical topics to teach and to learn (eizenberg & zaslavsky, 2004). one of the factors which causes difficulties in combinatoric learning is learning aid. learning aids that have been widely used by students in schools have not been able to assist them in rediscovering mathematical concepts, and according to fitria (2013) less optimal use of text book which supports teaching and learning process. in learning activities teacher usually explains the concept informatively, gives examples of problems, and provides exercises (herman, 2007). other problems which often occur is because most teachers are difficult to apply learning methods and teaching materials in accordance to lesson plan they have made (yulianti, zulkardi, & ilma, 2014). in order to make learning process runs optimally, teacher creativity is needed in the selection and use of appropriate learning resources with the development and needs of students, in this study we want to develop a special resource description on sub topic of the combination by using rme approach. the rme learning approach is one of the learning approaches that involves the active students who present student contributions both in constructing knowledge as well as in interaction with the learning environment. while local instruction theory (lit) is a special theory that can guide and help someone learn particular topic, in this case the topic studied is a combination. lit, gravemeijer (1999, 2004), developed in the context of design research, shows the means of teacher reference framework to design and involve students in a set of learning stages, learning activities are examples that support the development of student math focused on concepts. lit developed here contains characteristics of rme which uses real-life context, uses the model, there is students’ contribution, interaction and interrelationships among topics. 2. method this research uses design research method. the core of design research are cyclic process from the activities of designing or testing a series of learning activities and other aspects of designing. freudenthal (1991) mentioned that cyclic process in design research consists of idea/thought experiment and instruction experiment. gravemeijer & cobb (2006) stated that there are 3 stages in the implementation of design research, namely: (1) preparing for the experiment. this preliminary design serves to implement the initial ideas gained from the literature review before designing the learning activities; (2) teaching experiment. this stage aims to collect data to be used to answer the research questions. a series of instructional activities that have been designed, tested and revised, are applied in the classroom. research subjects in the teaching experiment were 19 students from the eleventh natural science-major graders at sman 1 pandeglang, banten; (3) retrospective analysis. the researcher analyzed the data obtained of the teaching experiment and used the results of the analysis to develop the next design. volume 8, no 2, september 2019, pp. 157-166 159 3. results and discussion the results presented include the stage of preparing for the experiment, teaching experiment, and retrospective analysis. 3.1. preparing for the experiment in preliminary stage, preparation and completion of teaching materials were done. activities undertaken at the preparatory stage: (1) analyzed the students by determining the research subjects of eleventh natural science-major graders at sman 4 pandeglang with cognitive level of heterogeneous students (2) analyzed the curriculum to find out that the combination material is in line with the 2013 curriculum and (3) analyzed the material to know that basic competence "describe and apply various combination rules through several real examples and present the flow of formulation of combinations through diagrams or other means" is appropriate to the research objective. then the researcher designed instructional materials in the form of material description or “uraian bahan ajar local instruction theory” (uba-lit) using realistic mathematics education approach and learning aids including syllabus, lesson plan or (rpp), and assessment instruments developed in accordance with rme characteristics and 2013 curriculum. furthermore, the learning aids that had been made were evaluated by the researcher. the result of selfevaluation is called first prototype. expert reviews were then performed where the first prototype was validated by four experts based on content, constructs and language. as the expert reviews stage was done, the first prototype was tested to five students who had received a combination lesson to check readability and work on teaching materials. researchers interacted with students to see the difficulties which might occur during the use of uba-lit to provide input or correction if anything needs to be fixed. after it was tested, the researcher asked the students to comment freely on commentary sheet that had been provided. the student comments including uba-lit was very helpful in finding the concept of formal combination without memorizing the formula. there were several less understandable words which gave various meaning. based on the validity test by the experts and the comments from the students it could be concluded that the first prototype teaching material product design developed was valid and had been revised into a second prototype based on the suggestions given. furthermore, the second prototype was tested (teaching experiment 1) to the students of eleventh natural science-major graders at sman cahaya madani banten boarding school consisting of 19 pupils. students were asked to solve problems on the teaching materials together in their group discussion to see the difficulties during the work and to simulate gradually the processing time. in the perspective of design research, the purpose of the initial design is to formulate lit that can be described and perfected during teaching experiment (gravemeijer & cobb, 2006). during this literature study, we also began designing learning activities. the sequence of this learning activity includes the conjectures of students’ thought and strategies developed and presented as the initial hlt. the hlt thought is dynamic and can be changed and adapted to the actual student learning process during the teaching experiment. hlt is a vehicle for planning student learning about the concept of combinatorics. the hlt component consists of learning objectives, description of activity, and conjectures of students’ thoughts will be explained in each learning activity. hlt during the teaching experiment serves as a guideline for teachers and researchers to determine the focus of teaching, interviewing, and observing. hlt is also used as a guideline and reference point in analyzing the entire data set collected during the teaching experiment. hlt in teaching experiments is described in table 1. meika, suryadi, & darhim, developing a local instruction theory … 160 table 1. overview of the hlt for learning the concept of combinations activity main goals description of activity conjectures of students’ thought demonstrating calculation in handshake (visual field activities) students are able to form a combination of problematic situations.  teacher asks five students to demonstrate handshake one another, other students notice and understand the handshake between a and b which is equal to handshake from b to a.  each group notices the demonstration carefully, and discusses how many handshakes occur from 2 people, 3 people, 4 people, 5 people and n people.  some students may quickly understand that the ab sequence is same as ba  some students may be able to conclude quickly the number of handshakes that occur.  some students may be able to visualize this activity and start building models from many handshakes from 5 people to n people. making color mixtures students are able to find understanding and formula of combinations. teacher asks students to make color mixtures using tree diagrams, tables and list of patterns, with guided reinvention:  2 of 3 colors provided  3 of 4 colors provided  2 of 4 colors provided  2 of 5 colors provided  r of n colors provided  some students may use tree diagrams to solve them.  some students may quickly find the number of color mixtures by using tables.  some students may quickly find the number of color mixtures by listing patterns.  some students may be able to visualize this activity and begin to build their models and do horizontal and vertical mathematization to find the multitude of color mixtures from r color of n colors provided.  some students may be able to relate the association of the color mixtures process with the combination concept. volume 8, no 2, september 2019, pp. 157-166 161 activity main goals description of activity conjectures of students’ thought resolving the problem in a story form students are able to calculate combinations of r element from different n element and solve problems in everyday life. do the calculations to find the combination formula of r elements from different n elements  some students may be able to see the connection between n! with (n-r)! or r! in calculations and able to simplify the calculations  some students may be able to associate prior activity for problem solving in this activity.  some students may accomplish it with tree diagram / table / pattern list.  some students may directly use the combination formula of r elements from different n elements 3.2. teaching experiment according to gravemeijer & cobb (2006), the purpose of teaching experiment is to test and improve the conjecture lit developed in the preliminary phase, and to develop an understanding of how it works. teaching experiment also aims to collect data to answer research questions. activity 1 in this activity the teacher initiated learning by asking four students (volunteers) to perform handshakes among them. let call these four students a, b, c, and d, shake hands each other once. the description of activities and parts of the lit groove developed in this activity are presented in figure 1. other students noticed their friends’ handshake demonstrations in front of class, and calculated the number of different handshakes of four model students. this handshake demonstration enables students to think that the handshake between a and b is equal to the handshake between b and a. the 1st round handshake was done by a to his friends b, c, and d. then 2nd round handshake from b to c and d, here students were very easy to understand that b did not need to do the handshake again with a because the handshake had been done from a to b in the previous round. the last, 3rd handshake was done by c and d. all students tried to answer the given problem and could determine the number of handshakes of 4 students easily. furthermore, to get more comprehensive understanding, the teacher provided additional activities in activities 2 and 3. meika, suryadi, & darhim, developing a local instruction theory … 162 figure 1. lit groove and handshake demonstration on combination learning activity 2 in this activity the teacher asked students to determine the number of color mixtures formed from some colors provided. the problem in this activity was to lead students to find the formal mathematical formula of the combination. this activity run well because students had got a concept from previous handshake activities. in addition, the problem of color mixtures could also be understood by students as a contextual problem and could be imagined by students easily. the lit design on this combination material was composed from a simple one and led to a general mathematical solution. the given problem was mixing 2 colors of 3 colors, mixing 3 colors of 4 colors, mixing 3 colors from 5 colors, and mixing 2 colors out of 5 colors. the result of the number of color mixtures obtained by the students were represented in the relationship table among the number of colors mixtures ( , the number of colors available ( ), and the number of formed colors mixtures. this table was made to facilitate students in vertical mathematization and find the formal mathematical formulas of combinations. figure 2 presents a table of results of student processes in finding formal mathematics from the concept of combination. this table is preceded by giving contextual problems so that students can find many ways manually. this is a vertical mathematization process where students are guided to find formal mathematics based on the pattern of problems given. student models demonstrate handshake in combination learning contextual problem combination observe presentation aktivity/trial/demonstration group discussion build combination calculations lit groove volume 8, no 2, september 2019, pp. 157-166 163 figure 2. vertical mathematization rocess of students in producing the formal mathematics of combinations activity 3 in this activity the teacher presented the material by giving daily life problem. this was to train students in solving problems related to everyday life, after this activity was completed, the teacher gave test. the one of questions of combination in the form of the story is given below: in a parent meeting at a high school attended by 25 people. the parents shook hands with each other. what is the number of greetings that occur if there are two pairs of husband and wife who attend the meeting and they do not shake hands with their partner?. figure 3. one of student’s answers in accomplishing story question translate in english: known : there are 25 people asked : if there were 2 pairs of husband and wife, how many handshakes happened? answer : 𝑛! 𝑟! 𝑛−𝑟 ! = 25! 2!23! = 300 300 − 2 = 298handshakes because there are 2 pairs of husband and wife who they do not shake hands with their partners meika, suryadi, & darhim, developing a local instruction theory … 164 figure 3 is an example of students' answers in accomplishing a story form question by using combination concept. students could solve problems using formal mathematical formula combinations of r elements of different n elements obtained from previous activities. 3.3. retrospective analysis the data analysis performed in this phase uses the initial hlt as a guideline and reference point in analyzing the entire data set collected during the teaching experiment. hlt is compared to the teaching and learning processes that occur in the classroom. the interpretive framework for understanding student learning process primarily is rme theory. the description of the analysis is not just about the examples that support the conjectures, but also the opposite. the conclusions of this analysis are used as the answers to the research questions. the main result is not primarily a successful design, but the reason of its works (cobb et al., 2003; gravemeijer & cobb, 2006). based on the results of activities in classroom learning activities, obtained lit flow diagrams before and after learning in combination material, presented in figure 4. in figure 4 can see the changes in the steps in learning, shown in the steps in the red box. this is the result of research findings that students better understand the concept of combination if it is built based on the right lit flow (after learning). figure 4. lit flow chart in combination material contextual problem combinatio n observe presentation aktivity/trial/demonstratio n group discussion build combination calculations tree diagram table register pattern picture (model) solution of combination formal combination formula informal way formal way solving problems with the formal formula after learning observe group discussion presentatio n contextual problem aktivity/ trial build combination calculations combinatio n tree diagram table register pattern picture (model) solution of combination formal combination formula informal way formal way solving problems with the formal formula before learning step change volume 8, no 2, september 2019, pp. 157-166 165 by using the right lit flow (figure 4), the results of hlt analysis and teaching experiment, students are able to construct a combination of problematic situations, find combinations and its formula, calculate combinations of r elements from different n elements, solve problems in everyday life associated with combinations. based on the students' work on the combination design of lit, students are able to work on the activities and problem solving which reaches 76%. students are familiar with the learning process, the process of determining the formal mathematical form of the pattern / model created previously. the use of relationship table between r!, n! and the number of ways obtained in this lit combination design can help achieve the vertical mathematical process. students are able to solve everyday problems by using the rules of multiplication or the sum of several combinations. the results of the test completion of combination which is given after learning process showed that students of low ability category were 64%, students of medium ability category were 70% and high ability category students were 80%. the results of this achievement indicate that learning using a combination of uba-lit teaching materials can develop students' understanding of combination. 4. conclusion the idea of combining comes when students directly observe the handshake demonstration process to determine that the handshake of a with b is equal to the handshake of b with a, this combination concept means that the ab sequence is equal to the ba sequence. the study also showed that students could remember what they had experienced but this did not mean that they could automatically transform their experience into abstract knowledge like the concept of combining calculations. although the students admitted that the calculation of combinations in determining the color mixtures in which the sequence of ab = ba, but they still felt a common misconception when they were given the problem in the form of a story. this raises the need to improve the design by adding more focused activities to allow students to validate this misunderstanding. in addition, group activities are intended to solve problems in story form as an application of combination concept. this activity allows students to communicate the subject of math with their friends during the learning process. this suggests that learning processes not only occur within individuals, but also involve social interaction among them. this is in line with the notion of social norms and socio-mathematics proposed by yackel & cobb (1996). references azhar, e. & kusumah, y. s. 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https://www.taylorfrancis.com/books/e/9780203088364/chapters/10.4324/9780203088364-12 https://www.taylorfrancis.com/books/e/9780203088364/chapters/10.4324/9780203088364-12 http://ejournal.sps.upi.edu/index.php/educationist/article/view/28 http://ejournal.sps.upi.edu/index.php/educationist/article/view/28 http://ejournal.sps.upi.edu/index.php/educationist/article/view/28 https://iopscience.iop.org/article/10.1088/1742-6596/948/1/012060/meta https://iopscience.iop.org/article/10.1088/1742-6596/948/1/012060/meta https://iopscience.iop.org/article/10.1088/1742-6596/948/1/012060/meta https://www.jstor.org/stable/749877 https://www.jstor.org/stable/749877 http://ejournal2.unsri.ac.id/index.php/jpm/article/view/819 http://ejournal2.unsri.ac.id/index.php/jpm/article/view/819 http://ejournal2.unsri.ac.id/index.php/jpm/article/view/819 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p207-224 207 evaluating graphing quadratic worksheet on visual thinking classification: a confirmatory analysis rina oktaviyanthi*, ria noviana agus universitas serang raya, indonesia article info abstract article history: received mar 4, 2023 revised apr 25, 2023 accepted may 25, 2023 published online jun 27, 2023 applying a graphing quadratic worksheet as a medium for learning the concept of a quadratic function clearer is an alternative instrument to accommodate the needs of developing students' mathematical visual thinking. in implementing graphing quadratic worksheet should show details of the dominant and recessive visual thinking classification aspects that develop in students. classification of dominant and recessive aspects of visual thinking needs to be completed to determine stages in improving the worksheet and learning instructions that are applied especially to recessive aspects. therefore, there is a need to evaluate the factors and trace the classification aspects of visual thinking that developed in students after practicing the graphing quadratic worksheet. the purpose of this research is to determine the categorization aspects of visual thinking in graphing quadratic worksheet items that develop and do not develop in students. confirmatory factor analysis was employed as a research method on 12 sub-variables from the three classifications of visual thinking. as research data, 93 student records were used. four main factors were formed as a result of the confirmatory factor analysis procedure, with the top two factors, namely factors 1 and 2, completely representing each aspect of the visual thinking classification and achieving the factor loading significance criteria. the implication is that the variables developed in the graphing quadratic worksheet for each aspect of the visual thinking classification have a strong relationship with the visual thinking ability overall. enhanced by a cumulative variance value for factors 1 and 2 specifically 56.88% of the total 81.78% for all factors. thereby it can be said that the categorization aspect of visual thinking that develops in students after implementing a graphing quadratic worksheet is achieved sensibly. keywords: confirmatory factor analysis, graphing quadratic, visual thinking, worksheet mathematics this is an open access article under the cc by-sa license. corresponding author: rina oktaviyanthi, department of mathematics education, universitas serang raya jl. raya cilegon km. 5, taktakan, serang city, banten 42162, indonesia. email: rinaokta@unsera.ac.id how to cite: oktaviyanthi, r., & agus, r. n. (2023). evaluating graphing quadratic worksheet on visual thinking classification: a confirmatory analysis. infinity, 12(2), 207-224. https://doi.org/10.22460/infinity.v12i2.p207-224 https://creativecommons.org/licenses/by-sa/4.0/ oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 208 1. introduction the quadratic function is one of the abstract mathematical concepts that students find challenging to learn. numerous research studies have revealed that mathematical visual thinking can address students' difficulties in learning quadratic functions (agus & oktaviyanthi, 2023; presmeg, 2020). students with mathematical visual thinking abilities are more potentially to be proficient in procedures for transforming information into other mathematical situations (frick, 2019). this mathematical situation promotes positive progress regardless of whether students are solving problems, but also when mathematical concepts are being constructed in their minds (hawes & ansari, 2020). according to heng and said (2020) and bjorklund (2022), learning through visual media can assist learners’ process and understand a concept skilfully as a consequence of visual stimulation influences their cognition area. conforming to elsayed and al-najrani (2021), mathematical visual thinking represented by a diagram or scheme is not merely a picture or illustration, but an accurate depiction of the quantity and relationship of certain mathematical problems. the goal is for motivating students to learn more, involving in the discovery of implicit mathematical ideas, supporting in reducing cognitive load, and improving higher-order thinking processes (anmarkrud et al., 2019; von thienen et al., 2021). the description's points confirm the importance of developing mathematical visual thinking in students. conceptually, visual thinking is expressed by cain (2019) and fernández-fontecha et al. (2019) as perception and discrimination, interpretation of the existence a thing (shape and object), and organizing mental images in various different modes through several processes such as deletion, addition, reflection, rotation and cutting, then working to find a relationship and translating it into positions and literal symbols to reach a conclusion. operationally, elsayed and al-najrani (2021) state visual thinking is the ability to transform information of all kinds into images, graphics, or other forms that can help communicate information. the classification of visual thinking is divided into three abilities: (1) the skill of visual discrimination (vd), the ability to detect differences and classify objects, symbols, or shapes (positions and patterns); (2) the skill of visual perception (vp), the ability to organize and interpret the information seen and to give meaning; and (3) the skill of visual analysis of shape (vs), the ability to connect abstract representations into concrete understanding or vice versa (elsayed & al-najrani, 2021; hermann & klein, 2015; lin, 2019). the following illustrates the graphing quadratic worksheet used in the study (see figure 1). (a) volume 12, no 2, september 2023, pp. 207-224 209 (b) (c) figure 1. illustrative worksheet: (a) vd aspect, (b) vp aspect, (c) vs aspect adopting a graphing quadratic worksheet as an illustrative medium for learning the concept of quadratic functions viewed appropriate based on the instrument's feasibility test to facilitate the future requirements of students' mathematical visual thinking (agus & oktaviyanthi, 2023). the results of incorporating a graphing quadratic worksheet on 93 firstyear students registered for calculus at universitas serang raya demonstrate that the instrument's validity and reliability values exceed the standard value (agus & oktaviyanthi, 2023). these conditions affirm that the questions or statements compiled in the graphing quadratic worksheet satisfy the visual thinking classification achievement indicator's appropriateness value (oktaviyanthi & agus, 2021; shanta & wells, 2022). elsayed and alnajrani (2021) elaborated the classification of visual thinking includes the skills of visual discrimination, visual perception, and visual analysis of shapes. however, the results of the graphing quadratic worksheet implementation do not provide information on which visual thinking classification aspects are dominant and recessive in students. furthermore, the implementation results do not indicate how consistent the classification aspects of visual thinking appear to students. in fact, for the purpose of optimizing students' mathematical visual thinking, teachers should first understand which aspects of visual thinking students have underdeveloped aiming to determine steps to improve the instruments and learning instructions employed. ghazali and nordin (2019) and vucaj (2022) both recommend conducting a comprehensive examination of an instrument or model with a focus on obtaining accurate information about the pattern of variables that appear or the interdependence of observed variables. factor analysis is commonly used in such techniques. brown (2015) defines factor analysis as a statistical technique for determining the interdependence of a structure (factor or dimension) of several variables observed simultaneously in time. the primary purpose of factor analysis is to simplify or reduce variables into smaller number of dimensions (crede & harms, 2019; hox, 2021)there are two types of factor analysis: confirmatory factor analysis (cfa), which is used to test the effects of methods or to construct validation from evaluation measurements, and exploratory factor analysis (efa), which is used to investigate general indicators into specific indicators (hox, 2021; jiang & kalyuga, 2020). oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 210 confirmatory factor analysis (cfa) was utilized to determine how established the visual thinking aspect of students was upon completion of a graphing quadratic worksheet. this is coherent with mustafa et al. (2020) who asserts that confirmatory analysis can be used to investigate the role of a variable on the indicators that constitute it. more studies have concentrated on the application of confirmatory factor analysis, particularly on the evaluation of an instrument's or variable's factors. on the strength of confirmatory factor analysis is appropriate for verifying the factor structure of a series of observed variables and allowing researchers to test the hypothesis of whether there is a relationship between the observed variables and the latent variables that emerge (brown, 2015; hox, 2021). ayebo et al. (2019) tested the psychometric characteristics and arrangement of the student attitude questionnaire towards statistics courses in several studies related to confirmatory factor analysis in 2019-2022. the research of zainudin et al. (2019) then measures the construct validity of students' mathematical creativity assessment instruments with a view to minimize assessment gaps. furthermore, asempapa and brooks (2022) research employs confirmatory analysis in the development of quantitative instruments to assess mathematics teachers' attitudes toward the practice of mathematical modeling. meanwhile, kaplon-schilis and lyublinskaya (2020) scrutinize predictors of technological pedagogical content knowledge in teacher preparation programs as an assessment of the effectiveness of technology-assisted learning. the design was then evaluated and validated by gonzález-ramírez and garcía-hernández (2021) for the level of student satisfaction with the mathematics learning system at universities. alwast and vorhölter (2022) also investigate the development of video-based instruments to ascertain teacher competence in the context of mathematical modeling. sari et al. (2022) then developed and determined construct validity for mathematical reasoning and proof instruments. several recent studies on confirmatory factor analysis confirm that this statistical technique is beneficial in assessing factors or testing instruments. derived from previous research, the use of confirmatory factor analysis was not found to confirm whether or not the visual thinking classification aspect was uniform in the graphing quadratic worksheet that students developed. the purpose of this research is to investigate aspects of visual thinking classification that develop in students after using a graphing quadratic worksheet, based on the exploration of research problems and in accordance with reviews of several studies related to confirmatory factor analysis in 20192022. it is expected that the research findings will not only fill a gap in the assessment of students' visual thinking skills, but will also contribute to improving the evaluation system for teaching and learning activities, both in terms of learning instruments and teaching instructions. 2. method to achieve the research objective of identifying aspects of mathematical visual abilities that develop with the implementation of a graphing quadratic worksheet, a quantitative research approach with a confirmatory factor analysis measurement model has been used. using a graphing quadratic worksheet, analysis activities were conducted to obtain student performance data in studying the concept of quadratic functions (agus & oktaviyanthi, 2023). student performance is obtained by completing worksheet with a rating for each visual thinking classification item. all items in the aspect of visual discrimination is 5, while every item in the aspect of visual perception and visual analysis of shapes is 10, then the total value in the worksheet is 100. this factor analysis study included 93 first-year students who had enrolled in calculus i courses at the university of serang raya. volume 12, no 2, september 2023, pp. 207-224 211 the research variable is a graphing quadratic worksheet indicator that corresponds to the visual thinking classification (see table 1). table 1. graphing quadratic worksheet indicators classification of visual thinking sub variable description of the graphing quadratic worksheet scheme the skill of visual discrimination. student's ability to identify differences or similarities in the form of a representation of a particular mathematical concept (vd) vd1 if a > 0, b > 0 and c > 0, create a quadratic function equation based on these conditions and then sketch the curve on the phet simulation coordinate plane. what can be connected from conditions a, b and c to the shape of the resulting curve? vd2 curve position of a quadratic function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is known. based on this information, determine the possible position of the curve if the value of a is positive (+), the value of b is negative (-) and the value of c is positive (+)? vd3 given a quadratic function = 6𝑥2 + 5𝑥 + 4 and 𝑦 = −6𝑥2 + 5𝑥 + 4. sketch the curve on the phet simulation coordinate plane. write down what is observed from the values a, b and c in each quadratic function with the shape of the curve. vd4 the curve position of the quadratic function with values a > 0 and a < 0 is shown. what facts can be found from the information on the shape of the curve and the value of a? the skill of visual perception. students' ability to investigate the implicit form of a certain mathematical concept (vp) vp1 given a quadratic function 𝑦 = 2𝑥2 + 2𝑥 − 2 and 𝑦 = 2𝑥2 − 4𝑥 − 6. sketch the graph and write down the components that make up the discriminant. determine the pattern that can be traced from the two quadratic functions based on the components of the discriminant. vp2 known the quadratic function and its graphical form. observe and write down the value of the discriminant constituent, the value of the discriminant and the value of a quadratic function. what is the confirmed regularity of the information? vp3 three equations of quadratic functions and their graphs are known. determine the discriminant value and square root of the three functions. what can be validated from the discriminant value relationship and the square root of the function obtained? vp4 various positions of the quadratic function graph are shown for values a > 0 and a < 0. determine the discriminant value of each graph. the skill of visual analysis of shapes. student's ability to determine part of the overall mathematical concept shown or vice versa (vs) vs1 create a quadratic function equation with the conditions a < 0, b < 0 and c > 0. sketch the graph and calculate the value of the function's discriminant. does the graph intersect the coordinate axes? how many points does the graph cross or touch on the x-axis? what can be related from the value of a and the value of the discriminant to the shape of the graph of the function? vs2 know the quadratic function and its graphical form. determine the value of the discriminant constructor, the discriminant calculation value, the intersection axis of the graph and the number of points through which the graph passes. what patterns can be identified from the information? vs3 given are six different quadratic functions along with their graphical representations. write down the criteria for a and ab values in the provided column. what can be identified from the value of a and ab to the position of the function graph? vs4 the various positions of the graph of the quadratic function for values a > 0 and a < 0 are shown. write down for each graph whether it intersects or touches the coordinate axes. oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 212 the procedure for the confirmatory factor analysis measurement model in this study refers to the explanation of hair et al. (2014) and the calculation process using spss assistance goes through the stages below: a. testing the assumptions of factor analysis using bartlett's test and kmo which are the main component tests. the bartlett test hypothesis namely: h0: the correlation matrix is an identity matrix. h1: the correlation matrix is not an identity matrix. rejecting criterion h0 is if the probability value (sig.) bartlett's test < 0.05 with the understanding that the correlation matrix is not an identity matrix so that principal component analysis can be confirmed. while in the kmo test, it is said to be middling if it has a kmo value range of 0.7 ≤ kmo < 0.8 (see table 2). table 2. classification of kmo values (arpaci, 2019) kmo value kmo level kmo value kmo level > 0.9 marvelous 0.6 – 0.7 mediocre 0.8 – 0.9 meritorious 0.5 – 0.6 miserable 0.7 – 0.8 middling < 0.5 unacceptable b. calculating the msa (measures of sampling adequacy) value for each variable with the condition that if the msa value is < 0.5 then the variable cannot be analyzed further (see table 3). the meaning of this value is the weak correlation between variables which has implications for reducing the variables and then re-analyzing the remaining variables. table 3. msa acceptance criteria (deutsch & beinker, 2019) msa value msa level = 1.0 a variable can be predicted without error by other variables. > 0.5 a variable can still be predicted and analyzed further. < 0.5 a variable is unpredictable and cannot be analyzed further, so it must be reduced or removed from the model. c. performing factor extraction using principal component analysis which can be seen from the value of communalities in the spss calculation output. communalities values < 0.5 are considered factors that are unable to explain indicators or variables. d. determining the number of main factors or components that are formed through eigenvalue, cumulative variance or scree plot at the spss calculation output. the main components that are worth choosing are those with an eigenvalue > 1 (chatfield, 2018) or those with a cumulative variance of more than 80% (matteson & james, 2014). e. examining the factors that are formed (loading factor) by rotating the factors using varimax rotation. the significance of the loading factor is fulfilled if the value of the loading factor at the output is > 0.5. this indicates that the loading factor being tested is significant or has an influence on the grouped variables. f. interpreting the results of the factor analysis. volume 12, no 2, september 2023, pp. 207-224 213 3. result and discussion 3.1. results 3.1.1. kmo and bartlett's test the first output of confirmatory factor analysis using ibm spss statistics 21 is the result of the assumption test of the appropriateness of the research variables to fulfill standard factor analysis procedures. according to table 4, the kmo value obtained is 0.750 which is in the range of values 0.7 ≤ kmo = 0.750 < 0.8. for the kaiser assessment category, the value of kmo = 0.750 is included in the middling data criteria for factor analysis. table 4. kmo and bartlett's test values for 12 variables kaiser-meyer-olkin measures of sampling adequacy. .750 bartlett's test of sphericity approx. chi-square 104.563 df 66 sig. 002 furthermore, the bartlett's test significance value = 0.002 < 0.05 which indicates a rejection of h0 where the correlation matrix between the test variables is not an identity matrix so that principal component analysis can be completed. in other words, the variables that are the focus of the test are not correlated with one another in the population. 3.1.2. msa the second procedure is the msa test (measures of sampling adequacy) which aims (1) to measure the sampling adequacy of each variable to be predictable and further analyzed and (2) to select variables that have a low correlation index between variables to be reduced and re-analyzed on the variables that remaining. the msa value in the spss output can be traced through the anti-image matrices table shown in table 5. table 5. msa values for 12 variables anti-image matrices vd1 vd2 vd3 vd4 vp1 vp2 vp3 vp4 vs1 vs2 vs3 vs4 anti-image correlation vd1 .710 a 0.282 -0.079 -0.039 -0.005 0.159 0.004 -0.099 -0.236 -0.002 0.077 -0.021 vd2 0.282 .537 a -0.163 0.133 0.27 -0.046 -0.004 -0.079 0.115 0.104 0.252 -0.563 vd3 -0.079 -0.163 .519 a -0.276 0.044 -0.252 -0.158 -0.413 -0.1 -0.102 -0.129 0.12 vd4 -0.039 0.133 -0.276 .489 a -0.171 0.272 0.059 0.174 0.098 0.331 -0.05 -0.264 vp1 -0.005 0.27 0.044 -0.171 .655 a 0.181 -0.005 -0.193 -0.132 0.2 0.235 -0.094 vp2 0.159 -0.046 -0.252 0.272 0.181 .662 a 0.137 -0.09 0.027 0.163 -0.051 -0.017 vp3 0.004 -0.004 -0.158 0.059 -0.005 0.137 .352 a -0.246 0.017 0.164 0.214 -0.082 vp4 -0.099 -0.079 -0.413 0.174 -0.193 -0.09 -0.246 .561 a 0.033 -0.12 -0.121 0.078 vs1 -0.236 0.115 -0.1 0.098 -0.132 0.027 0.017 0.033 .631 a 0.052 -0.055 -0.132 vs2 -0.002 0.104 -0.102 0.331 0.2 0.163 0.164 -0.12 0.052 . 510 a 0.067 -0.092 vs3 0.077 0.252 -0.129 -0.05 0.235 -0.051 0.214 -0.121 -0.055 0.067 . 536 a -0.247 vs4 -0.021 -0.563 0.12 -0.264 -0.094 -0.017 -0.082 0.078 -0.132 -0.092 -0.247 .419 a a. measures of sampling adequacy ( msa) table 5 shows that of the 12 variables that will be further tested, there are 3 of them that have msa <0.5, namely vd4, vp3 and vs4. based on the msa value criteria in table 3, the three variables with an msa value of <0.5 have a low correlation index between variables so they cannot be predicted and cannot be analyzed further. thus the three variables vd4, vp3 and vs4 must be reduced from the model and retested on the other 9 variables. oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 214 the results of retesting on 9 variables increase the value of kmo which is presented in table 6. reducing inappropriate variables makes the latest kmo values fall into the good (meritorious) data category so that it can be said that the 9 surviving variables are more feasible for factor analysis. table 6. kmo and bartlett's test values for 9 variables kaiser-meyer-olkin measures of sampling adequacy. .880 bartlett's test of sphericity approx. chi-square 64.332 df 36 sig. 003 meanwhile, the msa value for the 9 retested variables has a greater value of 0.5 (see table 7) which indicates that the 9 variables can be predicted and analyzed further. table 7. msa values for 9 variables anti-image matrices vd1 vd2 vd3 vp1 vp2 vp4 vs1 vs2 vs3 anti-image correlation vd1 .516 a 0.027 0.011 -0.08 0.12 -0.072 -0.126 -0.096 0.033 vd2 0.027 .617 a -0.034 0.194 -0.259 0.1 0.137 -0.134 -0.12 vd3 0.011 -0.034 .580 a 0.012 -0.106 -0.277 -0.056 0.038 -0.075 vp1 -0.08 0.194 0.012 .664 a 0.212 0.004 0.106 0.06 -0.073 vp2 0.12 -0.259 -0.106 0.212 .633 a -0.117 -0.056 -0.1 0.131 vp4 -0.072 0.1 -0.277 0.004 -0.117 .507 a 0.154 -0.102 -0.214 vs1 -0.126 0.137 -0.056 0.106 -0.056 0.154 .588 a 0.229 -0.027 vs2 -0.096 -0.134 0.038 0.06 -0.1 -0.102 0.229 .610 a 0.105 vs3 0.033 -0.12 -0.075 -0.073 0.131 -0.214 -0.027 0.105 .570 a a. measures of sampling adequacy ( msa) 3.1.3. factor extraction the purpose of factor extraction is to produce a number of factors according to the analysis criteria that are able to explain the correlation between the observed variables. in this study, factor extraction was achieved using principal component analysis which can be seen from the value of communalities in the spss calculation output. factor criteria that are able to explain the variable must have a value of communalities > 0.5. table 8 shows the communalities value of 9 variables > 0.5 which can be interpreted that all the variables used can be explained by the factors that are formed and have a strong relationship with these factors. the greater the value of communalities, the better the factor analysis, this is because the greater the characteristics of the original variables that can be represented by the factors formed. table 8. communality value communalities initial extraction initial extraction initial extraction vd1 1.000 .778 vp1 1.000 .556 vs1 1.000 .751 vd2 1.000 .578 vp2 1.000 .593 vs2 1.000 .656 vd3 1.000 .575 vp4 1.000 .646 vs3 1.000 .528 extraction method: principal component analysis. volume 12, no 2, september 2023, pp. 207-224 215 for example, the closeness of the relationship between the variable vd1 and the factors formed is 0.778. this value implies that the contribution of vd1 to the factor formed is 77.8 % or the variable vd1 can explain the factor formed is 77.8%. 3.1.4. number of factors the number of factors that can be formed can be seen from the eigenvalues. eigenvalue is a value that shows how much influence a variable has on the formation of a factor's characteristics. in the spss calculation output, the eigenvalues of the factors formed are known from the total variance explained (see table 9). the total variance explained table shows the percentage of total variance that can be explained by the variety of factors formed. the accepted eigenvalue significance criterion is > 1, while the eigenvalue < 1 is not used because it has the ability to explain lower variance than the ability of the initial variable. table 9. total variances explained components initial eigenvalues extraction sums of squared loadings rotation sums of squared loadings total % of variances cumulative % total % of variances cumulative % total % of variances cumulative % 1 1.89 36.996 36.996 1.89 36.996 36.996 1.723 35.145 35.145 2 1.429 19.882 56.878 1.429 19.882 56.878 1.446 32.071 67.216 3 1.206 13.402 70.280 1.206 13.402 70.280 1.338 30.87 70.303 4 1.035 11.502 81.782 1.035 11.502 81.782 1.053 11.479 81.782 5 0.892 6.906 88.688 6 0.719 5.983 94.671 7 0.658 3.309 97.980 8 0.649 1.211 99.191 9 0.523 .809 100 extraction method: principal component analysis. table 9 show that there are four factors that have an eigenvalue > 1, namely factor 1 with an eigenvalue of 1.89, factor 2 of 1.429, factor 3 of 1.206 and factor 4 of 1.035. the cumulative % column informs the cumulative percentage of variance that can be explained by factors. the amount of variance that can be explained by factor 1 is 36.996, the variance that can be explained by factors 1 and 2 is 56.878 and then the four factors are able to explain the total variance of 81.782. from these results it can be said that the four factors adequately represent the variance of the original variables. the visual representation of the scree plot shows the number of factors formed. from figure 2 it is known that the number of factors that must be maintained or stored in the main component is four. this is based on the extreme point of the curve line starting to slope shown in the fourth component. oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 216 figure 2. scree plot number of factor components 3.1.5. loading factor the factor that has been formed is called the loading factor whose value shows the correlation of each variable in the factor that is formed. table 10 displays the four factors that are formed and produces a loading factor matrix. the values in the matrix are the correlation coefficients between the variables and the four factors. if you pay attention to table 10, there are several variables that correlate with a factor but produce a correlation value with more than one coefficient interpretation. this makes it difficult to decide on the grouping of variables for each factor. for example, the vp2 variable correlates with factor 1 of 0.706 (strong correlation) and with factor 4 of 0.718 (strong correlation). in such conditions, it is difficult to decide whether the vp2 variable is included in the category of factor 1 or factor 4. the same situation occurs with the vs2 variable. because each of the factors formed cannot be clearly interpreted as the position of the variable representation, it is necessary to do factor rotation. table 10. matrix components component matrix a components 1 2 3 4 vd1 -.265 .160 -.198 .802 vd2 .667 -.148 .018 -.100 vd3 .302 .578 -.351 .162 vp1 -.568 .305 .351 -.130 vp2 .706 -.138 .248 .718 vp4 .309 .729 -.053 .128 vs1 -.328 -.158 .706 .347 vs2 .507 -.030 .573 .262 vs3 -.039 .615 .172 -.345 extraction method: principal component analysis. a. 4 components extracted. volume 12, no 2, september 2023, pp. 207-224 217 factor rotation uses varimax method aims to maximize the variance of loading factor on each factor so that the original variable only has a strong correlation with one particular factor and a weak correlation with other factors. table 11 show the result of factor rotation with varimax using spss which informs that each variable has a strong correlation with one factor. because each factor has been able to explain the variance of the original variables correctly, thus the loading factor of the rotation results is in table 11 used in the next analysis process. table 11 show that the loading factor value for each variable is at a value of > 0.5 which indicates the factors formed are significant to the grouped variables. table 11. varimax factor rotation rotated component matrix a components 1 2 3 4 vd1 .203 .061 .035 .856 vd2 .807 .033 .275 .180 vd3 .223 .707 .135 .083 vp1 .018 .053 .735 .110 vp2 .759 .114 .061 .017 vp4 .027 .753 .244 .137 vs1 .089 .052 .830 .224 vs2 .689 .026 .286 .314 vs3 .238 .696 .055 .337 extraction method: principal component analysis. rotation method: varimax with kaiser normalization. a. rotation converged in 5 iterations. 3.1.6. interpretation the interpretation of the factors formed from a series of previous confirmatory factor analysis processes is as follows: 1) factor 1 has a strong correlation with the variables vd2 (visual discrimination), vp2 (visual perception) and vs2 (visual analysis of shapes). the author named factor 1 as a strong factor in the visual thinking classification. the strong factor can explain the variance of data by 36.99 % with the largest loading factor value that appears in the vd2 indicator, namely 0.807. 2) factor 2 has a strong correlation with the variables vd3 (visual discrimination), vp4 (visual perception) and vs3 (visual analysis of shapes). the author named factor 2 as the medium factor in the classification of visual thinking. the medium factor can explain the variance of the data by 19.88 % with the largest loading factor value that appears on the vp4 indicator, which is 0.753. 3) factor 3 has a strong correlation with the variables vp1 (visual perception) and vs1 (visual analysis of shapes). the author named factor 3 as a sufficient factor in the visual thinking classification. the sufficient factor can explain the variance of the data by 13.40 % with the largest loading factor value that appears in the vs1 indicator, namely 0.830. 4) factor 4 has a strong correlation with the variable vd1 (visual discrimination). the author named factor 4 as a sufficient factor in the visual thinking classification. the low factor can explain the variance of the data by 11.50 % with the largest loading factor value appearing on the vd1 indicator, namely 0.856. furthermore, to ensure that the factors formed have no further correlation between one another, it is necessary to trace the values in the component transformation matrix in oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 218 table 12. correlation values on the main diagonal for each factor lie in the range of values 0.8 to 0.9 which belong to in the category of very strong correlation. the implication of the correlation value is that the three factors that are formed can be said to be precise and have a unique closeness relationship. table 12. component transformation matrix component transformation matrix components 1 2 3 4 1 .854 .255 .441 -.107 2 -.330 .937 .108 047 3 .358 .238 .880 -.199 4 .183 032 -.137 .973 extraction method: principal component analysis. rotation method: varimax with kaiser normalization. 3.2. discussion the confirmatory factor analysis procedure of the three aspects of the visual thinking classification which is detailed into 12 sub-variables produces the top four factors where the 9 variables in the four factors become predictors of the visual thinking classification aspects that develop in students after using a graphing quadratic worksheet. these four factors are hereinafter referred to as strong factors (factor 1), moderate (factor 2) and sufficient (factors 3 and 4). table 11 informed that factor 1 of the visual thinking classification is represented by one variable each, namely vd2, vp2 and vs2. this data implies that vd2, vp2 and vs2 are included in the strong factors that contribute the most to the optimization of students' visual thinking abilities. it can also be interpreted that these three variables are the variables that spread the most in the data group with a percentage of variance of 36.99 %. in other words, students can best understand the concept of quadratic functions and are able to explore their knowledge and demonstrate a more dominant visual thinking performance through the three items vd2, vp2 and vs2 on the graphing quadratic worksheet. the same condition is shown in factor 2 of the visual thinking classification which is also represented by one variable each, namely vd3, vp4 and vs3. these three variables explain the variance of data by 19.88 %. this value indicates that the questions in the graphing quadratic worksheet represented by vd3, vp4 and vs3 can be digested and achieved by students, especially to stimulate their visual thinking skills. furthermore, factor 3 of the visual thinking classification only correlates to the visual perception aspect represented by the vp1 variable and the visual analysis of shapes aspect represented by the vs1 variable with a percentage of variance of 13.40 %. this expresses that vp1 and vs1 still contribute to the classification of student visual thinking and can be used to explore this aspect. finally, factor 4 of the visual thinking classification has only one correlation in the visual discrimination aspect represented by the vd1 variable with a variance percentage of 11.50 %. based on the value of this variance, the item vd1 graphing quadratic worksheet is still in the fourth most spread variable category in the data group. recapitulation of the three aspects of visual thinking classification in the previous description which are represented by significant item variables in the graphing quadratic worksheet can be seen in table 13. the interpretation contained of the numbers displayed in table 13. is the visual thinking classification of the visual discrimination aspect can be improved through items vd2, vd3 and vd1 respectively based on the correlation values of volume 12, no 2, september 2023, pp. 207-224 219 the factors formed. furthermore, the visual perception aspect can be optimized through items vp2, vp4 and vp1. meanwhile, the visual analysis of shapes aspect can be supported by vs2, vs3 and vs1. the loading factor values of all the variables formed are significant and have a cumulative value of 81.78 % as shown in table 9. this percentage value indicates that students' visual thinking abilities in the quadratic function material are at a fairly high level because it can be explained by a variety of variables of more than 80%. table 13. visual thinking classification recapitulation classification of visual thinking sub variable factor factor loading information the skill of visual discrimination (vd) vd2 1 .807 significant vd3 2 .707 significant vd1 4 .856 significant the skill of visual perception (vp) vp2 1 .759 significant vp4 2 .753 significant vp1 3 .735 significant the skill of visual analysis of shapes (vs) vs2 1 .689 significant vs3 2 .696 significant vs1 3 .830 significant to explore which aspects of visual thinking classification are more developed in students after using a graphing quadratic worksheet can be seen in table 14. in factors 1 and 2 all aspects of the visual thinking classification meet the significance criteria for loading factor. this indicates that the variables per aspect of the visual thinking classification developed in the graphing quadratic worksheet have a strong relationship with the visual thinking ability as a whole. so that it can be said that the aspect of visual thinking classification that develops in students after using a graphing quadratic worksheet is achieved in a balanced way. this is also emphasized by the cumulative variance value of factors 1 and 2 to be exact 56.88% of the total 81.78% for all factors. as explained by deutsch and beinker (2019) the greater the eigenvalue, the greater the contribution of the cumulative variance, the implication being that it is increasingly able to explain the variance of the original variables. table 14. classification of visual thinking based on factor analysis factor sub variable factor loading information cumulative variance 1 vd2 .807 significant 36.996 vp2 .759 significant vs2 .689 significant 2 vp4 .753 significant 56.878 vd3 .707 significant vs3 .696 significant 3 vs1 .830 significant 70.280 vp1 .735 significant 4 vd1 .856 significant 81.782 a number of studies related to factor evaluation of an instrument either to review the validity of the instrument items or to investigate the extent to which the instrument can have an impact on users providing different types of knowledge and supporting the development of students' mathematical capabilities. the research results of tracing the most dominant oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 220 visual thinking classification aspect emerged from students after using the graphing quadratic worksheet. this is consistent with a number of previous studies. the research conducted by alsina et al. (2021) with a focus on analyzing learning assessment instruments to explore which mathematical processes among problem solving, reasoning and proof, communication, connection and representation are dominant and recessive in teaching practice. the results of the confirmatory analysis detected a change in scores on the mathematical connection process item. in addition, semeraro et al. (2020) used confirmatory factor analysis in examining the role of cognitive factors, non-cognitive factors and the quality of student and teacher interactions, which are the best predictors of student achievement in mathematics. the results of this study revealed that cognitive ability was the best predictor of students' mathematics achievement. furthermore, wan et al. (2022) analyzed the construct validation of the project-based stem learning experience scale from four dimensions namely scientific inquiry, technological application, engineering design and mathematical processing. the three studies have the same analysis pattern as this study, namely evaluating factors through testing the construct validity of the measuring instruments or instruments developed. 4. conclusion from the 12 visual thinking classification sub-variables, three variables were reduced that did not meet the msa score, namely vd4, vp3 and vs4, resulting in 9 variables that met the criteria for factor analysis. the results of the confirmatory factor analysis formed four main factors with eigenvalues > 1 and explained a total variance of 81.78 % with the understanding that the four factors adequately represented the variance of the original variables. factor 1 has a strong correlation with the variables vd2 (visual discrimination), vp2 (visual perception) and vs2 (visual analysis of shapes) which can explain the variance of the data by 36.99 % with the largest loading factor value appearing on the vd2 indicator, namely 0.807. this value denotes that the three items vd2, vp2 and vs2 on the graphing quadratic worksheet dominate the understanding of students in building an understanding of the concept of quadratic functions and stimulating their visual thinking abilities. factor 2 has a strong correlation with the variables vd3 (visual discrimination), vp4 (visual perception) and vs3 (visual analysis of shapes) which can explain the variance of the data by 19.88 % with the largest loading factor value appearing on the vp4 indicator, namely 0.753. this value indicates that the question items in the graphing quadratic worksheet represented by vd3, vp4 and vs3 can be digested and achieved by students. in factors 1 and 2 all aspects of the visual thinking classification meet the significance criteria for loading factor. this gives an understanding that between the variables per aspect of the visual thinking classification developed in the graphing quadratic worksheet has a strong relationship with the ability of visual thinking as a whole. so that it can be said that the aspect of visual thinking classification that develops in students after using a graphing quadratic worksheet is achieved in a balanced way. this claim is highlighted by the cumulative variance value of factors 1 and 2 is 56.88% of the total 81.78% for all factors. recommendations for further research are focused on tracing the relationship between latent variables that appear in this factor analysis using the structural equation modeling method so that the direction of the relationship between the three latent variables as a whole can be known. volume 12, no 2, september 2023, pp. 207-224 221 references agus, r. n., & oktaviyanthi, r. 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(2019). construct validity of mathematical creativity instrument: first-order and second-order confirmatory factor analysis. international journal of instruction, 12(3), 595-614. https://doi.org/10.29333/iji.2019.12336a https://doi.org/10.1080/01621459.2013.849605 https://doi.org/10.13189/ujer.2020.080115 https://doi.org/10.35445/alishlah.v13i1.483 https://doi.org/10.1007/978-3-030-15789-0_161 https://doi.org/10.20414/betajtm.v15i2.549 https://doi.org/10.1371/journal.pone.0231381 https://doi.org/10.1007/s10798-020-09608-8 https://doi.org/10.1007/978-3-030-62037-0_2 https://doi.org/10.1080/15391523.2020.1840461 https://doi.org/10.1007/s11165-020-09965-3 https://doi.org/10.29333/iji.2019.12336a oktaviyanthi & agus, evaluating graphing quadratic worksheet on visual thinking … 224 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p11-20 11 the students’ mathematical creative thinking ability of junior high school through problem-solving approach heris hendriana* 1 , fika muji fadhilah 2 1 institut keguruan dan ilmu pendidikan siliwangi 2 smp negeri 2 cilamaya kulon karawang article info abstract article history: received aug 30, 2018 revised sep15, 2018 accepted jan 12, 2019 this study aims to examine the achievement and improvement of mathematical creative thinking skills of students whose learning uses a problem-solving approach compared to those using ordinary learning and the implementation of learning steps using a problem-solving approach in the field. the method used in this study is a quasi-experimental method, because there is manipulation of treatment, where the experimental class uses learning with a problem-solving approach, and the control class uses ordinary learning. the population in this study were all seventh-grade students of smp negeri 1 cilamaya wetan. with the subject of the sample are two classes vii, while the sample was chosen 2 classes randomly where the experimental class obtained learning with a problem-solving approach, and the control class gained regular learning. class vii l as the experimental class and class vii i as the control class. the instrument in this study is a set of test questions in the form of a description consisting of eight questions. based on the results of research and analysis of pretest, posttest, and n-gain data, it was found that the achievement and improvement of students' creative mathematical thinking skills using problem-solving approaches compared to those using ordinary learning, and implementation of learning steps with problem-solving approaches in the field showed that the learning process is more effective and creative in mathematical problems solving. keywords: problem-solving approach creative thinking copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: heris hendriana, departement of mathematics education, institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, west java 40526, indonesia email: herishen@ikipsiliwangi.ac.id how to cite: hendriana, h., & fadhilah, f. m. (2019). the students’ mathematical creative thinking ability of junior high school through problem-solving approach. infinity, 8(1), 11-20. 1. introduction the ability to think creatively is an ability that is important for students to have so that students can solve problems faced in learning activities (istianah, 2013). in the learning process, especially in mathematics learning, the ability to think mathematically has a very important role to do and needs to be applied to students (choridah, 2013; darusman, 2014; istianah, 2013; nuriadin, 2015; rahman, 2012; sugilar, 2013; yenni & hendriana & fadhilah, the students’ mathematical creative thinking … 12 putri, 2017). in learning mathematics in schools students should be trained to have creative thinking skills in obtaining, choosing, and processing information in order to survive in an ever-changing and competitive situation (darusman, 2014). mathematical creative thinking ability of students at the junior high school level is still very low, this is because when students are given students do not want to work on the problem even students give up first before trying to solve the problem (rahman, 2012; sugilar, 2013). in addition, based on the research results of hidayat (2017), it was found that students' thinking abilities had not been achieved in the category of novelty indicators. this is difficult, because students still think the transition from concrete to abstract. so that when students are required to think creatively in a given problem, the solution is still following the usual algorithmic processes solved for routine problems (hidayat, wahyudin & sufyani, 2018). facts the show that students are less motivated to learn, students' attention to learning outcomes or values obtained by students seem to accept what they are and "resign" even when they get a score below the completeness criteria even if the student does not want to make improvements (sumarmo, hidayat, zukarnaen, hamidah, & sariningsih, 2012). so that the ability of mathematical creative thinking of junior high school students is still low. this can be seen from the value of the seventh grade daily math test of smp negeri 1 cilamaya wetan consisting of 32 students, 22 students of whom still have difficulties in solving problems in the form of high-level thinking questions that involve various concepts and only 9 other students who can solve the problem. thus, only 28% of students from 32 students were able to solve the problem. the main factor of the low ability of mathematical creative thinking of students is influenced by the learning approach used by the teacher. the teacher conveys the material and provides sample questions without involving student interaction on opinions and analysis, so that students think according to what is conveyed by the teacher, students' creative thinking skills are less developed. therefore it is necessary to develop a learning approach that involves students more actively and able to think creatively in the teaching and learning process and in accordance with mathematics learning (istianah, 2013; yenni & putri, 2017). one approach to learning mathematics that can be applied in anticipating problems that arise during the learning process of mathematics and can generate creative ideas is a problem solving approach. with the problem solving approach is expected to be able to attract the attention and interest of students in following the process of learning mathematics. 2. method the method in this study is a quasi-experimental method, because there is a treatment manipulation, where the experimental class uses learning with a problem solving approach, and the control class uses ordinary learning. before getting treatment first, both classes are given an initial test (pretest) and after receiving treatment, the two classes are given a final test (post test). the designs in the study are as follows: o x o ------------------ o o --: sampling randomly by class o : pretest = posttest mathematical creative thinking skills x : learning with a problem solving approach volume 8, no 1, february 2019, pp. 11-20 13 the population in this study were all students of smp negeri 1 cilamaya wetan. with the sample subjects are two class vii, while the sample is selected 2 classes randomly where the experimental class obtains learning with a problem solving approach, and the control class receives ordinary learning. class vii l as an experimental class and class vii i as a control class. examples of junior high school creative thinking ability test questions: a frame model of a beam is made of wire with a length (x + 5) cm, width x cm, and height (x 2) cm. if the length of the wire used is not more than 132 cm, determine the steps to determine the maximum size of the beam! the results of both pretest and posttest research were processed using spss 21.0 for windows software with the following steps: a. test the normality of the results data pretest with the aim to find out whether the sample comes from a population that is normally distributed. the normal assumption on the distribution of data to be analyzed is one of the requirements in quantitative analysis. the normality test used using the kolmogorov-smirnov normality test with a significance level of 0.05. b. if the two classes are normally distributed, then the homogeneity test is done to find out both the distribution of the experimental class and the control class whether the variances are the same or not. the homogeneity test used using the levene statistics test. c. if one or both classes are not normally distributed, followed by a two-mean similarity test using non-parametric test statistics. d. if the two classes are normally distributed and homogeneous, then the two-mean similarity test is continued using the t-test. e. if the two classes are normally distributed but not homogeneous, then the two similarity test uses the t-test. f. the two mean similarity test on pretest data used a two-party test. while the two mean similarity tests on posttest and n-gain data use the one-party test. the similarity test of the mean pretest score aims to find out that there is no significant difference in the initial ability of the experimental group and the control group. while the two mean similarity test in the posttest and n-gain to find out the achievement and improvement of mathematical creative thinking abilities of students who learn using problem solving approaches is better than those using ordinary learning. 3. results and discussion 3.1. results in the data analysis the results of mathematical creative thinking skills between students who use problem solving approaches with those using ordinary learning, the data obtained are data of mathematical creative thinking abilities obtained from the results of pretest and posttest. from the results of these two tests, another quantitative data is obtained, namely the data gain of students' mathematical creative thinking ability. data processing is done using spss 21.0 for windows software. before being analyzed, it will be presented first descriptive of the ability score of the pretest, posttest, and n gain in the following table. hendriana & fadhilah, the students’ mathematical creative thinking … 14 table 1. description of research results approach pretest postest n – gain ̅ sd smi ̅ sd smi ̅ sd problem solving 8.82 1.688 32 23.32 2.504 32 0.6273 0.0972 conventional learning 8.20 1.862 32 21.16 2.292 32 0.5457 0.8081 based on table 1 above, it can be concluded that the initial ability of mathematical creative thinking in both classes is not much different. after learning, the ability of experimental class mathematical creative thinking is better than control class means increasing the ability to think mathematically creative using a problem solving approach better than those using ordinary learning. in the analysis of the initial test data (pretest), after the normality test was obtained, the data was normally distributed, then continued with a variance homogeneity test. the results of homogeneity variance test obtained homogeneous data. furthermore, the two average difference test was carried out using spss 21.0 for windows software, so the following results were obtained: table 2. significant test results of the difference in two average pretest scores approach sd t-test interpretation tcount sig. problem solving 1.688 1.619 0.109 h0 accepted conventional learning 1.862 based on the results in table 2 it can be seen that the significant value of sig. (2tailed) with the t-test is 0.600. because the value of sig> 0.05, h0 is accepted with a significant level of 5%. thus it can be concluded that the ability of students to think mathematically creative in the initial test there is no significant difference between those who will use problem solving learning with those who will use ordinary learning. analysis of the final test data was conducted to determine the improvement of students' creative thinking skills after the approach process and to determine the equality of the sample. both data are assumed to be normal and homogeneous with a significant level of 5%. furthermore, a significant test of the two average differences was carried out using the one-t-test of the right party using the independent sample t-test assuming both homogeneous variance (equal variance assumed) with a significance level of 0.05. in this case the researcher uses the right-hand test with the aim of knowing which learning is better. table 3. significant test results for the difference in two average postes scores approach sd t-test interpretation tcount sig. problem solving 2.504 4.219 0.000 h0 rejected conventional learning 2.292 volume 8, no 1, february 2019, pp. 11-20 15 based on the results in table 3 it can be seen that the significant value of sig. (2tailed) with the t-test is 0.000. because the value of sig. (2-tailed) <0.05, h0 is rejected. so it can be concluded that the achievement of mathematical creative thinking abilities of students who use a problem solving approach is better than those who use ordinary learning. n-gain data analysis was conducted to determine the improvement of mathematical creative thinking skills between those who used problem solving approaches with those who used ordinary learning. in the n-gain data analysis, both data are assumed to be normal and homogeneous with a significant level of 5%. furthermore, a significant test of the two average differences was carried out using the one-t-test of the right party using the independent sample t-test assuming both homogeneous variance (equal variance assumed) with a significance level of 0.05. table 4. significant test results for the difference in the two average n-gain indexes approach sd t-test interpretation tcount sig. problem solving 0.972 4.284 0.000 h0 rejected conventional learning 0.808 based on the results in table 4 it can be seen that the significant value of sig. (2tailed) with the t-test <0.05 then h0 is rejected. thus it can be concluded that the increase in mathematical creative thinking skills of students who learn using a problem solving approach is better than those who use ordinary learning. 3.2. discussion learning activities carried out as many as 10 meetings with different learning (treatment), namely the experimental class using the problem solving approach and the control class using the usual learning approach. learning is done 8 times, 1 meeting for pretest, and 1 more meeting for postes. after the post-test, the data shows that the sample comes from a population that is normally distributed, the variance of the two sample groups is homogeneous, there are differences in the ability of creative mathematical thinking between the experimental class and the control class on the significance test of the two mean differences. it can be concluded that in the final test (posttest) the achievement of mathematical creative thinking abilities of students who use a problem solving approach is better than those who use ordinary learning. in the experimental class given learning using a problem solving approach. at the second meeting until the meeting of the seven students in the experimental class was grouped, students were given material and practice questions in the form of worksheets, students discussed with friends in a group doing the exercises on lks, then students had to present the results of the discussions they were working on. at the first meeting students in several groups were seen discussing with friends in the group, some students asked questions and gave opinions in solving problems, in addition students were more enthusiastic in issuing their opinions even though these opinions did not lead to problem solving, but some students still had feeling confused. this hendriana & fadhilah, the students’ mathematical creative thinking … 16 is because at the initial meeting, the majority of students were still in the form of adjustments to the learning provided. the difficulties faced by students are not solved in the form of comprehensive discussions. the results of the study by dilla, hidayat & rohaeti (2018) found that when students find it difficult to solve problems, they will guess the possible solutions. from this process, students actually have creative thinking skills. at the next meeting, some students begin to actively argue and ask questions, opinions and questions begin to lead to problem solving. learning that uses a problem solving approach shows a significant role in improving students' creative thinking skills. in learning that uses a problem solving approach, the focus of learning activities is entirely on students, namely thinking creatively in understanding a problem, planning the completion of a mathematical problem, solving problems and checking all the steps that have been done. because problem solving learning can challenge students' abilities and provide satisfaction to find new knowledge for students and can help students to develop new knowledge and be responsible for the learning they do. in addition, solving the problem can also encourage self-evaluation of both the results and the learning process. this is one of the advantages of problem solving that shows that the experimental class is better than the control class. during the research process there are several obstacles that arise, including constraints with time. in the learning process that uses a problem solving approach requires additional time as its application, because in the learning process students must understand the problems contained in the worksheet and students must plan the resolution of the problem. therefore, the learning process that uses a problem solving approach requires a little more time than ordinary learning (jonassen, & hung, 2015; karatas, & baki, 2017; loibl, roll, & rummel, 2017; özsoy, & ataman, 2017; riccomini, smith, hughes, & fries, 2015). in the control class, ordinary learning is given. during the second meeting until the seventh meeting students were given regular learning. the teacher only provides material and students only listen and get material in one direction. students are seen busy with each material provided by the teacher. in the ordinary learning process students become more passive, most students hesitate in solving problems in the form of problems, most students look saturated. students look familiar with activities that tend to be monotonous and boring. thus students will be difficult in capturing material, in the end students have no interest in following these subjects. so that the ordinary learning approach is not effective in developing the creativity of each student. it can be understood that ordinary learning is only oriented to explaining the subject matter, explaining the steps in solving the problem and providing examples of problem solving in a clear and detailed way and then students are asked to work on the questions that have been presented clearly and the problem solving is definitely uniform. in addition, students are not given the opportunity to express and conclude the material they have learned in their own language. this makes mathematics only seen as a set of formulas and rules that must be memorized and remembered by students not as activities or activities that must be done by students to find a concept (godino, batanero, & font, 2007; hendriana, 2017; kaiser, & sriraman, 2006; surya, sabandar, kusumah, & darhim, 2013; widyatiningtyas, kusumah, sumarmo, & sabandar, 2015; zan, brown, evans, & hannula, 2006). in the calculation of the gain of the experimental class and the gain of the control class that the sample comes from a population that is normally distributed, and the gain results have homogeneous variance and there is an increase in mathematical creative thinking skills students who use a problem solving approach better than those who use ordinary learning in the significance test difference of two averages. volume 8, no 1, february 2019, pp. 11-20 17 based on the data processing above, shows that the achievement and improvement of mathematical creative thinking abilities of students who are learning using a problem solving approach is better than those using ordinary learning, and the implementation of learning steps by using a problem solving approach in the field. this result is in line with opinions sumarmo, hidayat, zukarnaen, hamidah, & sariningsih (2012) stated that the implementation of mathematics learning using a problem solving approach can improve students' creative thinking ability in learning mathematics. based on observations of researchers in the field, it shows that: a. students can understand the problem, b. planning completion. c. resolve the problem. d. re-checking all steps that have been done based on the observations of researchers in the field, problem solving approaches are better than those who use ordinary learning approaches. the observations of researchers in the field are in line with sumarmo, hidayat, zukarnaen, hamidah, & sariningsih (2012) opinion, this is caused by: a. students play an active role in learning b. improving ability to work together. c. with problem solving learning, students better understand the contents of the lesson. d. problem solving learning can challenge students' abilities and provide satisfaction to find new knowledge for students. e. problem solving learning can increase student learning activities. f. problem solving learning can help students to develop their new knowledge and be responsible for the learning they do. in addition, solving the problem can also encourage self-evaluation of both the results and the learning process. g. problem solving learning is considered more enjoyable and liked by students. 4. conclusion based on the results of research and data analysis, it can be concluded that: (1) the achievement of mathematical creative thinking abilities of students who learn using a problem solving approach is better than those who use ordinary learning; (2) improvement of mathematical creative thinking skills of students whose learning uses a problem solving approach better than those who use ordinary learning; (3) the implementation of the learning steps using the problem solving approach in the field shows that the learning process is going well and students are more creative in solving mathematical problems. references choridah, d. t. 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(2017). adversity quotient dan penalaran kreatif matematis siswa sma dalam pembelajaran argument driven inquiry pada materi turunan fungsi. kalamatika jurnal pendidikan matematika, 2(1), 15-28. hidayat, w., wahyudin, w., & prabawanto, s.(2018). improving students’ creative mathematical reasoning ability students through adversity quotient and argument driven inquiry learning. in journal of physics: conference series, 948(1). istianah, e. (2013). meningkatkan kemampuan berpikir kritis dan kreatif matematik dengan pendekatan model eliciting activities (meas) pada siswa sma. infinity journal, 2(1), 43–54. https://doi.org/https://doi.org/10.22460/infinity.v2i1.23 jonassen, d. h., & hung, w. (2015). all problems are not equal: implications for problem-based learning. essential readings in problem-based learning, 7-41. kaiser, g., & sriraman, b. (2006). a global survey of international perspectives on modelling in mathematics education. zdm, 38(3), 302-310. karatas, i., & baki, a. (2017). the effect of learning environments based on problem solving on students’ achievements of problem solving. international electronic journal of elementary education, 5(3), 249-268. loibl, k., roll, i., & rummel, n. (2017). towards a theory of when and how problem solving followed by instruction supports learning. educational psychology review, 29(4), 693-715. nuriadin, i. (2015). pembelajaran kontekstual berbantuan program geometer’s sketchpad dalam meningkatkan kemampuan koneksi dan komunikasi matematis siswa smp. infinity journal, 4(2), 168–181. özsoy, g., & ataman, a. (2017). the effect of metacognitive strategy training on mathematical problem solving achievement. international electronic journal of elementary education, 1(2), 67-82. rahman, r. (2012). hubungan antara self-concept terhadap matematika dengan kemampuan berpikir kreatif matematik siswa. infinity journal, 1(1), 19–30. riccomini, p. j., smith, g. w., hughes, e. m., & fries, k. m. 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(2006). affect in mathematics education: an introduction. educational studies in mathematics, 63(2), 113-121. hendriana & fadhilah, the students’ mathematical creative thinking … 20 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p133-146 133 geometry exploration activities assisted with dynamic geometry software (dgs) in a teacher education classroom samsul maarif 1 , wahyudin 2 , muchamad subali noto 3 , wahyu hidayat 4 , herri mulyono 5 1,5 universitas muhammadiyah prof.dr. hamka, jl. tanah merdeka, jakarta, indonesia 2 universitas pendidikan indonesia, jl. setiabudi no. 229, bandung, indonesia 3 universitas swadaya gunung jati, jl. pemuda no.32, cirebon, indonesia 4 ikip siliwangi, jl terusan jenderal sudirman, cimahi, indonesia 1 samsul_maarif@uhamka.ac.id, 2 wahyudin_math@yahoo.com, 3 balimath61@gmail.com, 4 wahyu@ikipsiliwangi.ac.id, 5 hmuyono@uhamka.ac.id received: august 07, 2018 ; accepted: september 03, 2018 abstract the aims of this study were conducted to investigate the effectiveness of pre-service teachers’ geometric exploration activities assisted by dynamic geometry software (dgs) cabri ii plus computer application in constructing geometry proofs in a teacher education classroom. to these ends, mix-method design. a total of 72 pre-service teachers taking geometry course participated in the study. findings of the study show that students who participated in geometric exploration activities assisted by dgs cabri ii plus computer application had better achievement compared to their counterpart. the use of dgs cabri ii plus computer application was observed to enable the students to present diagrams of verification problems appropriately, determine the valid conjectures, and make justification regarding the statements in the written proof. more importantly, participating in geometric exploration activities assisted by dgs cabri ii plus computer application provide students with opportunities to explore alternative proofs related to geometry. keywords: dynamic geometry software, cabri ii plus, geometry. abstrak tujuan penelitian ini adalah mengetahui efektivitas ekplorasi geometri menggunakan dynamic geometry software (dgs) cabri ii plus dalam mengkonstruksi bukti geometri mahasiswa calon guru. metode penelitian ini adalah metode kombinasi dengan jumlah sampel sebanyak 72 mahasiswa calon guru yang mengambil mata kuliah geometri. temuan dalam penelitian ini menunjukkan bahwa siswa yang berpartisipasi dalam ekplorasi geometri menggunakan dgs cabri ii plus lebih baik dibandingkan dengan rekan mereka. penggunaan dgs cabri ii plus memungkinkan siswa menyajikan diagram masalah verifikasi secara tepat, menentukan dugaan yang valid, dan membuat jastifikasi pernyataan pada bukti tertulis. lebih penting lagi, siswa dapat berpartisipasi dalam kegiatan eksplorasi geometrik dengan menggunakan dgs cabri ii plus yang memberikan peluang untuk mengeksplorasi bukti alternatif yang terkait dengan geometri. kata kunci: dynamic geometry software, cabri ii plus, geometri. how to cite: maarif, s., wahyudin, w., noto, m. s., hidayat, w., & mulyono, h. (2018). geometry exploration activities assisted with dynamic geometry software (dgs) in a teacher education classroom. infinity, 7(2), 133-146. doi:10.22460/infinity.v7i2.p133-146. mailto:email-penulis-1@ymail.com mailto:email-penulis-2@ymail.com mailto:email-penulis-3@ymail.com mailto:wahyu@ikipsiliwangi.ac.id maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 134 introduction in the present study, the application of geometry exploration activities with dynamic geometry software (dgs) cabri geomtery ii plus computer technology to construct a geometry proof during a geometry course in a teacher education classroom was investigated. cabri geometry ii plus (henceforth cabri ii plus) is a computer application that helps students visualise abstract concepts on geometry, so students can easily understand and easier to arrange geometric proofs. (maarif, 2017). the computer application enables students to construct dots, lines, triangles, circles and other plane geometries complete with calculations (mariotti, 2002). cabrilog as cited in laborde (2002) suggests other benefits of cabri geometry ii plus, including simple and user friendly interface, understandable icons and fonts, support features for geometry learning, and export import facility. the term geometry proof in this paper is operasionalised as a valid argument that establishes the truth of the statement (jones & rodd, 2001). this argument is used to provide evidence or persuade other people to accept a belief (bell, 1978) through sereies of explanations, findings, relation between mathematical ideas and the geometry problems (marrades & gutiérrez, 2000) and systematic statements in an axiomatic system (knuth, 2002). bell (1978) suggests six alterantives that can help establish evidence or persuade people, inlcuding personal experience, acceptance of authority, observations of instances, lack of a counterexample, the usefulness of result, deductive argument. the later, the deductive argument is viewed as best alternative that can help to construct a geometry proof. it is because deductive reasoning employs the law of logic in relating the true statement to come to the right conclusion. in mathematics classroom context, several studies have shown that students experienced difficulty in compiling geometry proofs. students feel difficult to visualise the concept of geometry. consequently, they cannot analyse, define conjecture, justify the problem and moreover to compiling geometric proofs (knuth, 2002; mariotti, 2002; mariotti & balacheff, 2008). in preparing the proofs, students have trouble making intuition and only working on certain cases. as a result, the student can not determine the general form of proof, make a mistake in the proofs procedure and in selecting the technique, and the students also unable to apply counterexample in proofing. (moore, 1994; weber, 2005; selden & selden, 2008; arnawa, 2010; perbowo & pradipta, 2017). to address this difficulties, dgs, particularly cabri geometry ii plus has been used in classroom instruction to explore geometric materials. the objective of exploration through such an applicaiton is to construct conjectures, justify and formulate ideas of proof (jones, 2002, rodríguez & gutiérrez, 2006; mariotti, 2002; oldknow, 2009; baccaglini-frank & mariotti, 2010; kilic, 2013, maarif, 2017). according to sánchez & sacristán (2003), the use of tools such as dgs brings the possibility that students can understand the various geometry concepts that can help students in constructing a geometric proof. the activity of manipulating geometric shapes can help students to find and justify conjectures (maarif, 2017). this current study address two research questions as below: 1. does the preservice teachers’ application of geometry exploration activities with dgs cabri ii plus computer application affect their ability in constructing geometry proof? 2. how do the preservice teachers perceive the application of geometry exploration activities assisted by dgs cabri ii plus computer application? volume 7, no. 2, september 2018 pp 133-146 135 method a mixed-method design that combines two research strands i.e. quantitative and qualitative strand was adopted in this study (creswell, et al., 2007). the quantitative research was carried out to examine the effect of the application of geometry exploration activities assisted with dgs cabri ii plus in constructing a proof of geometry of pre-service teachers. in the quantitative stage, a quasi-experimental design was developed. a total of 72 preservice teachers attending geometry course participated in this study and were grouped into an experimental and control groups. the experiment group received an intervention where dgs cabri ii computer application was incorporated during preservice teachers’ geometric exploration activities in their attempts to construct geometry proofs. while in the control groups, the activity was conducted with a paper based-media. table 1. the difference of intervention procedure between two groups: intervention in the experimental group intervention in the control group 1. teacher explained the students about geometry theorem teacher explained the students about geometry theorem 2. teachers gave a geometry proof problem to the students teachers gave a geometry proof problem to the students 3. students constructed geometry theorem on dgs cabri ii plus worksheet students constructed geometry theorem manually on a piece of paper 4. students manipulated the figure they already constructed on the worksheet. the manipulation included labelling, determining the size of sides, angles and etc. using the software students manipulated the figure they already constructed on the worksheet. the manipulation included labelling, determining the size of sides, angles and etc. on the paper 5. students highlight conjectures by labeling the causal effect between them on the worksheet for the proof construction students highlight conjectures by labeling the causal effect between them on the worksheet for the proof construction 6. students reconstructed geometry shapes they made in dgs cabri ii plus, with information related to manipulation for constructing proof students reconstructed geometry shapes they made on the paper, with information related to manipulation for constructing proof 7. students developed the proof from conjectures they already determined students developed the proof from conjectures they already determined 8. students were encouraged to re-examine the geometry proof they already constructed students were encouraged to re-examine the geometry proof they already constructed 9. teacher guided students’ exploration activity to help them explore a proof teacher guided students’ exploration activity to help them explore a proof 10. students were given problems related to geometry proof students were given problems related to geometry proof 11. students operated dgs cabri ii plus to helped solve the problem they intended to prove students operated dgs cabri ii plus to helped solve the problem they intended to prove 12. students presented and explained the geometry proof students presented and explained the geometry proof 13. teacher and students together concluded the lesson teacher and students together concluded the lesson maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 136 in addition, in the qualitative stage, preservice teachers’ perception about of the exercise of geometric exploration activities assisted by dynamic geometry software (dgs) cabri ii plus computer application was explored. data collection instruments and analysis the collection of the data in the current study were gathered using two methods. the quantitative data were collected through two set of tests: pre-test and post-test. these two tests were purposefully developed to evaluate students’ ability in constructing a four item geometry proof, including conjecturing, formulating a statement, exploration, the selection and combination of coherent arguments, testing result, and writing a formal proof (reis & reinkl, 2002). the pre-test was distributed to all participants from both experimental and control class before the geometry course commenced and the post-test was given to the participants after eight weeks of application of geometry exploration activities assisted with dgs cabri ii plus ended. the collected quantitative data were analysed statistically using spss software. in addition to the quantitative data, the qualitative data were gathered through observation and interview. two of seventy two preservice teachers’ classroom activities from the experimental group were intensively observed and interviewed. both observation as well as interview were video-recorded. although the section of two out of seventy two preservice teachers rose concerns related to subjectivity as well as the validity of the finding from the qualitative data, the selection of small sample of two preservice teachers was purposefully to help the researchers focus and understand the detail of construction process of geometry proof with the use of dgs cabri ii plus computer application. the recorded observation and interview first were transcribed verbatim. the qualitative data were then coded and analysed using a thematic analysis. results and discussion research question one: does the preservice teachers’ application of geometric exploration activities with dgs cabri ii plus computer application affect their ability in constructing geometry proof? the first research question concerned whether the preservice teachers’ application of geometric exploration activities with dgs cabri ii plus computer application affect their ability in constructing geometry proof. to address the first research question, the research hypotheses were developed as below: h0: there is no mean difference on the preand post-tests between the experimental and control group, p-value > α with α is at 0.05 h1: there is a mean difference on the preand post-tests between the experimental and control group, p-value ≤ α with α is at 0.05 as mentioned earlier, the quantitative data were gathered using preand post-tests and the collected data were analysed statistically using t-test. table 2 and table 3 below present the statistical description of the data from the two groups and the t-test result respectively. volume 7, no. 2, september 2018 pp 133-146 137 table 2. the description of data table 3. t-test results of ability to construct geometry proof of experiment and control class table 3 as above shows that the calculation of t-test of preservice teachers’ post-test resulted the t value of 3.279 with the p-value of 0.002. in reference to the research hypothesis, the pvalue was observed to be lower than α = 0.05 (0.002 <α = 0.05), allowing the rejection of . the rejection of h0 hypothesis indicated that there was a difference in preservice teachers’ ability in geometry proof construction between the experimental and control groups. by observing the mean score as in the table 2, it was shown that the preservice teachers in experimental group achieved better than those in the control group (experimental group, mean score 15.278; the control group, mean score 13.917). in other words, the application of dgs cabri ii during geometric exploration activities helped teachers construct geometry proof better than those who did not use the application. to follow up the finding, the effect size test was calculated and the finding showed that the effect size was observed at .631 or at medium level. the quantitative finding of the current study is in line with earlier study by goldenberg (1995) and laborde (2002). goldenberg (1995) investigated the application of computer based mathematics learning at school suggested. the finding of his study suggested that the incorporation of computer technology in mathematics learning provided positive effect on students’ criticality as well as creativity. specifically, the finding showed that students’ who used computer application in their mathematics learning achieved better in mathematics test compared to those who did not. similarly, laborde (2002) study examined the effect of cabri geometry use in mathematics classroom on students’ mathematical representation ability. the result of her study showed that the cabri ii plus use in junior secondary mathematics classroom affected positively on students geometric learning compared to the conventional method. research question two: how do the preservice teachers perceive the application of geometric exploration activities assisted by dgs cabri ii plus computer application? the second research question explored teachers’ perception about the application of geometric exploration activities assisted by dgs cabri ii plus computer application. as n minimum maximum mean standard deviation eexperimental group 36 8.00 19.00 15.2778 3.36886 control group 36 6.00 19.00 13.9167 2.94109 valid n (listwise) 36 construct the geometry proof t df p-value (1-tailed) posttest equal variances assumed 3.279 70 0.002 equal variances not assumed 3.279 68.768 0.002 maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 138 discussed earlier in method section, two methods of data collection were employed to address the research questions namely observation and interview. thematic analysis was carried out to perform the qualitative data analysis with the predetermined themes as in table 4 below: table 4. predetermined themes participant themes illustrating a diagram of a proofing problem correctly (a1) determining a valid conjecture (a2) to justify the statement in the written proof (a3) find new ideas of proof, on students. (a4) dilla 1 0 0 1 lia 1 1 1 1 notes: a1 : help students draw a diagram of the problem of proof correctly a2 : help students determine valid conjectures a3 : assist students in the contribution of justifying the statement in the written proof a4 : helps students find new ideas of proofing on themselves. 0 : response to failure 1 : response to success the names of dilla and lia are pseudonymous the classroom narrative was employed to present the detail account of the application of geometry proof construction activity assisted by the dgs cabri ii plus. the classroom instruction was below: prove that: if on , the sides ac and bc each constructed equilateral triangle i.e. and outside , then . the following subsections detail dilla and lia’s exploration in their attempts to construct geometry proof. detailed account of dilla’s exploration dilla constructed an equilateral triangle abc. on the ab and bc sides, she create an equilateral triangle abd and cbe. then, dilla attempted to determine the length of the side of pe, ap, cq and dq by using existing services on cabri ii plus. figure 1 below draws dilla’s exploration using cabri ii plus. figure 1. results of dilla exploration using cabri ii plus volume 7, no. 2, september 2018 pp 133-146 139 as shown in the figure, the exploration shows that ap = pe = cq = dq = 3 cm. this was reflected as dilla thought cq + qd = pe + ap and this led her into a conclusion of ae = dc. although dilla misinterpreted the problem that the triangle abc was known to be an arbitrary triangle not just an equilateral triangle, dilla presented its conjecture by making an equilateral triangle. throughout the exploration activities using cabri ii plus, dilla invented an idea of new proofing on herself (see table 4, a4 / 1), where the idea has never been thought of before. figure 2 below details dilla’s construction process. figure 2. dilla's construction process translation of the figure 2: interpretation of the problem: is an equilateral triangle is an equilateral triangle is an equilateral triangle will be proven: ae = cd maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 140 proof: =) see and bc = bc (coincide) ecbacb  (an equilateral triangle) ebacba  (an equilateral triangle) ____________________________________________ , thus ap = pe =) see and ab = ab (coincide) abdcba  (an equilateral triangle) cbadab  (an equilateral triangle) ____________________________________________ , thus cq = dq from the above data, we have, cq=dq=ap=pe, thus dc = cq+dq ae= ap+pe it means ae = cd as in the figure 2 above, dilla was shown to present that , suggesting bc=bc (coincide), (an equilateral triangle) and (an equilateral triangle), and thus ap = pe. further, dilla assumed that because ab=ab (coincide), (an equilateral triangle) and (an equilateral triangle), and thus cq=dq. with these two arguments, she concluded that cq=dq=ap=pe, cd=cq+dq and ae=ap+pe, consequently cq = dq. from dilla's answer, it was observed some inaccuracies in her writing of proof argument. the interview script below highlights dilla’s argument for doing so. interviewer : the answer you have given is very impressive to me, so it is interesting to dig deeper. in the answer, you write that bceabc  causes ap = pe and abdabc  results in cq = qd. what is your explanation? dilla : here is my explanation, ap is a high line of abc triangle and pe is a high line of bce triangle. because bceabc  then ap=pe (coded a3/0). interviewer : so do you think if the two triangles are mutually congruent, then the height of the two triangles is the same? dilla : yes sir, always the same. interviewer : could you explain it in more detail? dilla : while exploring with cabri ii plus, i measured the triangle's high line. then i noticed that all three triangles have the same height. thus, since all three are equilateral triangles, they are the same height (coded a2 / 1). from the interview above, it was shown that dilla apparently employed a different perspective about the concept of the two triangular congruences. typically, when it comes to volume 7, no. 2, september 2018 pp 133-146 141 the concept of congruence of two triangles, dilla should have connected to the corresponding sides length, and the corresponding angles which allowed her to obtain the fact that the sides were actually at the same length, and the angle was the same. however, dilla argued that if two triangles were congruent, then the height of the one corresponding side would also be equals. her argument indicated that exploration with cabri ii helped dilla constructed a novelty answer. geometry exploration activities with cabri ii plus thus may shape the novelty of individual thinking which thus promote the development of a geometric proof ability. with regards to determination of the conjecture, dilla misconstrued the proof from which she was observed to pay a little attention to the notation on the construction of the image. as suggested in the dgs cabri ii plus manual, users are required to focus on the correct geometric image notations. in the cabri ii plus application, users could use angle and label to determine geometric image notations. despite of misconstruction dilla had made, dilla exploration activities in cabri ii plus had helped dilla in creating a diagram of the verification problem correctly (see table 4, a1/1 and figure 2). interviewer : i am impressed with the idea you had described. however, i observe a less complete use of the notation in your answer. for example, you assume pe is a high line on the abe triangle, but the notation abpe  does not exist. what is your opinion? dilla : oh, .. yes sir! i forgot the notation should be written to help me. probably because there is no sign in the software, but there is a "mark angle" pack, i forgot to make it (coded a1/1). interviewer : in your opinion, to what extent does the notation function in geometry drawings help you construct geometric proofs? dilla : very helpful sir. when we describe the geometric shape of the problem of proof, the drawing must be precise complete with the notation. usually, it makes me confused when the notation false. interviewer : what kind of confusion? dilla : for example like i cannot make conjectures, and usually, later the theorem used is wrong. in addition, during her exploration activity, dilla was observed to take one case of an equilateral triangle. while, the problem of proof is fundamentally an arbitrary triangle which also included types other than an equilateral triangle. the positive aspect from dilla’s exploration however, dilla was able to use her knowledge to determine a new conjecture in relation with the concept of two triangular congruences. detailed account of lia’s exploration during an observation on lia’s exploration activity, it was observed that lia had made several attempts to explore the problem by creating any arbitrary triangle abc. lia then adjusted the triangle into an equilateral triangle, each was on ab and bc side. lia put forward a conjecture put to indicate two congruent triangles. different colours were applied to highlight a mutually congruent. in the next step, lia was observed to determine the length of the ab=bd= 6 cm, bc =be=7 cm and . such a length determination enabled lia to make a conclusion that that led ae=be. to this stage, lia was shown to have an ability to illustrate the problem in reference to the image that she had constructed using cabri ii plus. throughout the exploration activities, lia was enabled to explore ideas herself in maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 142 order to help her solve the problem of proof. lia’s result of exploration is drawn in figure 3 below. figure 3. results of lia’s exploration using cabri ii plus to better understand about lia's exploration, we conducted an interview. here are the results of the interview. interviewer : could you please explain your answer? lia : yes sir, when i did the exploration with cabri ii plus, i suspected that cbdabe  that's why i put the colour, the first was yellow, and the other was blue (coded a1 /1). interviewer : what is your purpose of giving a different colour to the two triangles? lia : to make it easier for me to highlight both triangles (coded a1/1). interviewer : what does it mean to see both triangles? what do you want to see? lia : yes sir, i mean to make it easier to determine which side is the same and which angle is the same. so i specified the side size that i need as well as the angle with the "distance or length" and "angle" buttons. interviewer : i am very impressed with your exploration, does your exploration help you in determining the conjecture for proof? lia : i found that there were two equal sides and one same angle, ab = bd = 6 cm, bc = be = 7 cm and 0 103 cbdmabem . so that the two triangles were congruent because it met the requirements of side, angle, side. (code a2 /1) from the above interview, students’ exploration with cabri ii plus was shown to enable them to obtain conjectural solutions for a proof. applying colours to the two triangles that they considered congruent, was a creative step during the exploration stage. moreover, the exploration tools in cabri ii plus seemed to have helped the students in determining the length of the sides or the degree of the angles in order to justify the validity of the argument (coded volume 7, no. 2, september 2018 pp 133-146 143 a2/1) and the statement in the proof (coded a3/1). after the exploration, lia wrote down the proof to explain her exploration activities and it is presented in the following figure 4. figure 4. lia’s construction process translation of figure 4. interpretation of the problem: is an arbitrary triangle is an equilateral triangle is an equilateral triangle will be proven: ae = cd proof: see and ab = bd ( is an equilateral triangle) be = bc ( is an equilateral triangle) ____________________________________________ , so that ae = cd as in the figure 4 above, lia's proof was observed as simpler but easy to accept compared to dilla’s. from her work of cabri ii plus, lia seemed to have already understands comprehended the concept of conjecture of two congruent triangles which helped her determine the triangles. lia was observed to construct the proof by conjecturing ab = bd shwoing that triangle abd was an equilateral triangle. it was therefore, as observed in lia’s work, cbdabe  and be = bc, suggesting that bce was an equilateral triangle. accordingly, cbdabe  (s.a.s) which shows that ae=cd. the concept of lia’s proof is shown in an interview with her as below. maarif, wahyudin, noto, hidayat, & mulyono, geometry exploration activities assisted … 144 interviewer : i am very interested in the answers you have given, can you explain again in detail? lia : if we already know which triangle is congruent, actually it is simple sir. i've been exploring with cabri ii plus, so just need to be drawn and just write the proof. interviewer : can you explain a bit clearer? lia : we see triangles abe and cbd. since ab = bd then triangle abd is an equilateral triangle, cbdabe  and be = bc because the bce triangle is an equilateral triangle, so it can be inferred cbdabe  (s.a.s) so ae=be (coded a3/1). interviewer : it’s interesting. but how can you explain that cbdabe  lia : okay, here's my reason sir [while pointing to the picture on the answer sheet] abd and cbd is an equilateral triangle, right?, so automatically, the size of each corner of both triangles 60 0 which mean cbeabd  . now, let’s see abe is a sum of abd and abc . also, angles cbd are the sum of cbe and abc . so it can be concluded that cbdabe  . interviewer : does it mean cbdmabcmcbemabcmabdmabem  so that cbdabe  ? lia : yes, sir. that’s right the qualitative finding as revealed in the current study suggest benefits from incorporating dgs cabri ii plus in students’ geometry learning. the use of such a computer application helped students in creating and presenting the geometry proof construction. this finding corresponds mariotti & balacheff (2008) thought the existence of visual tools may help students to understand and construct the logic of proofing. it is because, as marrades & gutiérrez (2000) views, geometry proof functions to explain, discover, as well as communicate mathematical alternatives to address issues in geometry. to this view, visual tool helps students to develop a theorem that they will use to construct a proof. the qualitative finding of the current study has shown that with the use dgs cabri ii plus, students were given opportunities to create and manipulate figures which allowed them to point out valid conjectures. more importantly, from students’ exploration activity it was shown that cabri ii plus enabled them to construct a valid argument by dragging the figure construction they had already created, adding guidelines and setting up the lengths and angles (maarif, 2017). the finding is also in line with what several authors (e.g. jones, 2002, rodríguez & gutiérrez, 2006; mariotti, 2002; baccaglini-frank & mariotti, 2010; kilic, 2013; maarif, 2017) have suggested that the use of dgs in geometry learning facilitate geometry exploration activity, particularly in reference to determining conjecture as well as developing alternative to construct a geometry proof. conclusion this current research addressed two research questions. first the research question concerns with whether the preservice teachers’ application of geometric exploration activities with dgs cabri ii plus computer application affect their ability in constructing geometry proof. findings from the experimental research in the study showed that there were mean difference volume 7, no. 2, september 2018 pp 133-146 145 in geometry tests between students who used dgs cabri ii plus in constructing geometry proof and those who did not. in other words, students’ use of dgs cabri ii plus in constructing geometry proof helped improve their attainment in geometry test. the effect size of such use of dgs cabri ii plus was at a medium level. second research question asks how the preservice teachers perceive the application of geometric exploration activities assisted by dgs cabri ii plus computer application. findings from observation and interview showed that students’ exploration activities in dgs cabri ii plus benefited students in constructing geometry proof. in particular, it helped students to create correct diagram of the problem, determine a valid conjecture, justify the statement in a written proof, and facilitate the students in exploring alternative ways to construct geometry proof. references arnawa, i. 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(2005). problem-solving, proving, and learning: the relationship between problem-solving processes and learning opportunities in the activity of proof construction. the journal of mathematical behavior, 24(3-4), 351-360. sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 12, no. 2, september 2023 e–issn 2460-9285 https://doi.org/10.22460/infinity.v12i2.p225-242 225 towards numeracy literacy development: a single-case study on the use of the living book homeschooling model andi harpeni dewantara1*, farida agus setiawati1, sari saraswati2 1universitas negeri yogyakarta, indonesia 2universitas hasyim asy'ari, indonesia article info abstract article history: received mar 8, 2023 revised may 7, 2023 accepted may 25, 2023 published online jul 5, 2023 public schools are not always believed to be able to support the development of an individual's potential comprehensively. homeschooling, an educational program where students learn from home, is currently an alternative education. this study aims to reveal why parents choose to homeschool their children and describe how a homeschooler parent as a single tutor develops her child’s numeracy literacy skills in living book homeschooling. this research is a holistic single-case study with two subjects: a homeschooler (j) and his mother (upl) as the tutor. data were collected through in-depth interviews and document analysis of j’s learning activities. thematic analysis with atlas.ti software was employed. findings reveal that the parents’ main reasons for homeschooling are dissatisfaction with public school instruction and flexibility to comprehensively develop homeschoolers’ skills. in addition, the integration of rme (realistic mathematics education) in the living book homeschooling model is a very powerful support to students’ literacy numeracy development. practically, there are three main strategies implemented; the use of real contexts and concrete teaching aids, as well as an emphasis on conceptual understanding and high-order thinking skills. keywords: homeschooling, living book, math literacy, numeracy this is an open access article under the cc by-sa license. corresponding author: andi harpeni dewantara, doctoral program of educational evaluation and research, universitas negeri yogyakarta jl. colombo yogyakarta no.1, sleman, daerah istimewa yogyakarta 55281, indonesia. email: andiharpeni.2022@student.uny.ac.id how to cite: dewantara, a. h., setiawati, f. a., & saraswati, s. (2023). towards numeracy literacy development: a single-case study on the use of the living book homeschooling model. infinity, 12(2), 225-242. 1. introduction formal schools are not always trusted to be able to comprehensively support the development of children's potential. for various reasons, many parents feel dissatisfied with the academic system in formal schools and then decide to send their children to school at home (aram et al., 2016; boulter, 2017; pozas et al., 2021). home-schooling, an educational program where children learn from home, is one of many viable alternatives for education. although the number is still limited when compared to formal schools, homeschooling has become an alternative educational program offering various advantages. https://doi.org/10.22460/infinity.v12i2.p225-242 https://creativecommons.org/licenses/by-sa/4.0/ dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 226 the homeschooling program provides a wider opportunity for parents to improve the educational quality, develop their children's moral and religious values (razi, 2016), meet the needs of children with medical problems or special needs (jamaludin et al., 2015), and maintain childrens’ mental health and keep them away from various potential negative environmental influences (ray, 2015). in addition, the main uniqueness and strength offered by homeschooling is ‘customized education’, where learning design is adapted to children's potential and the surrounding environment (purwaningsih & fauziah, 2020). parents have the flexibility to adopt the best teaching methods and learning activities based on their children's needs, which is why homeschooling is believed to be more capable of maximizing children's potential. some studies even claim that children who are homeschooled have better academic performance than students who are formally schooled (ray, 2015; wichers, 2001). it further emphasizes that customized learning in homeschooling has great potential to support students’ achievementnot only for academic aspect but also for various other life skills. the implementation of homeschooling programs should ideally be supported by adequate resources, especially in terms of parents' readiness to optimize children's learning experiences (pozas et al., 2021). parents play a major role as teachers, managers, dynamists, facilitators, and motivators for homeschoolers. but in fact, not all homeschooler parents are ready to optimize their children's learning process. several previous study results reveal not all students who attend homeschooling have good academic achievements, because not all of their parents have adequate experience and good pedagogical competence to run their role as tutors/teachers. some homeschoolers’ parents tend to apply the conventional way to teach their children at home by replicating the learning activities and classroom environment in formal schools. in other cases, many of them simply hire private tutors without considering the needs and conditions of their children. while most homeschooler parents entrust private tutors to teach their children, mrs. ‘upl’ -a housewifededicates most of her time to independently create the best learning environment to facilitate her child ‘j’ to attend homeschooling program. the initial interview with ms. upl revealed that trust issue toward formal, public schools is one of the main factors to elect homeschooling program. she claimed that customized home-based learning could optimize j’s learning experience based on his interests and talents. the homeschooling instruction designed by ms. upl focuses on literacy development, a fundamental competency that should be honed from an early age but is sometimes even neglected in the formal school. ms. upl employed a variety of learning methods based on various instructional theories. these innovative efforts explicitly have positive impacts, hence the literacy development-based learning applied by ms. upl in homeschooling class is fairly interested to investigate further. this article specifically focuses on the development numeracy literacy undertaken by mrs. upl in the homeschooling program for j. this issue is important to be thoroughly explored due to the fact that numerical or mathematical literacy has become a global concern in the field of mathematics education, as it is considered an essential skill that must be acquired by every mathematics learner. numeracy literacy emphasizes the significance of learning mathematics within a real-world context (amaral & hollebrands, 2017; widjaja, 2013). despite its importance, it is a common challenge that many mathematics learners struggle to apply mathematical concepts in their everyday lives (benson-o'connor et al., 2019; bolstad, 2023; dewantara et al., 2015; wijaya et al., 2014). the poor performance of students in numeracy literacy has become a global issue. therefore, any efforts or strategies aimed at enhancing students' numeracy literacy skills are worth to discuss. numeracy or mathematical literacy development have been addressed by numerous studies (chen & chiu, 2016; decoito & richardson, 2018; solano et al., 2018). several volume 12, no 2, september 2023, pp. 225-242 227 previous studies also have examined homeschooling programs which highlight the homeschooler parents’ perspective. those previous researches could generally categorized to four aspects: diversity of parental profession as main tutors in home-schooling (purwaningsih & fauziah, 2020), parents’ perspective on homeschooling (jolly et al., 2013; neuman & guterman, 2017a), the impact of homeschooling program (letzel et al., 2020), parents’ motivation for home education (collom, 2005), as well as parents’ experiences of homeschooling amid covid-19 pandemic (fontenelle-tereshchuk, 2021; parczewska, 2021; reaburn, 2021; thorell et al., 2022). however, there are no studies that specifically investigate how to develop numeracy literacy of a young learner in a homeschooling program. therefore, this study aims to reveal why parents choose to homeschool their children and to examine how a homeschooler parent as a single tutor develop her child’s numeracy literacy skills in the living book homeschooling model. 2. method 2.1. research method this is a qualitative research with holistic single-case study approach. holistic single-case study is a type of case study approach that investigate a unit as single global phenomenon (using single unit of analysis) (depoy & gitlin, 2016; yin, 2014). this study is categorized as a single case study due to it focuses on single-case which is considered unique and has special distinction compared to other similar cases. 2.2. participants the researcher tried to explore information about the literacy development process through the living book homeschooling program which was carried out by two subjects involved in this study: mrs. upl and her son j. mrs. upl is a 35 year old housewife domiciled in yogyakarta, indonesia who chose to be a tutor for her son j (7 years and 3 months) who is currently undergoing a homeschooling program. 2.3. instruments and procedures this study used two instruments: interview guidelines and documents (syllabus, teaching materials and homeschooler j’s portfolio documents). in-depth interviews were conducted with two research subjects using semi-structured interview technique to reveal how the homeschooling learning process was undertaken by j and to deeply analyze how the numeracy literacy development model was implemented by mrs upl in the homeschooling program. the interviews were conducted three times online via zoom meeting (twice with mrs. upl and once with student j, accompanied by mrs. upl), on october 21, november 15, and november 22, 2022. the duration of each interview ranged from 50 to 70 minutes. to ensure the informants felt free and comfortable during the interview process, the researcher aimed to create a conducive atmosphere (creswell & poth, 2016). to support the results of these interviews, the researcher analyzed homeschooler j's portfolio documents, particularly in the mathematics course. the portfolio documents refer to j's completed mathematics problems provided by mrs. upl on various topics/materials. in addition, there were several short videos that demonstrated j's problem-solving process for some mathematics questions. the selection of learning documents was purposive, aligned with the specific topic addressed in this study, namely integer operations (addition and subtraction). data validation was carried out using the triangulation method: source and method/technique triangulation. the interview data from mrs. upl and j were cross dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 228 referenced with the learning documents and videos of j's learning process to gain the credible data. the obtained data were subsequently analyzed qualitatively. 2.4. data analysis data analysis employed thematic analysis using atlas.ti (version 22.2.5) software. analysis is inductive, where the results of qualitative research emphasize meaning rather than generalization. data analysis consisted of three stages: data reduction, data display, and drawing conclusion (miles et al., 2019). the initial step of data analysis in this research involved transcribing the recorded interview and j’s learning video into a written transcript, enabling easy analysis of the content. the researchers thoroughly reviewed all the transcripts multiple times. in the data reduction stage, the researchers reduced the collected information from the first stage (including interview results, learning documents, and videos) by sorting and selecting relevant and significant data, identifying themes and patterns, and excluding irrelevant data. the identified important information was then grouped into central themes. the reduced data was displayed in the form of descriptions and diagrams (see figure 1) that illustrated the relationships between themes or categories. the core of the description entails capturing the essence of the informants’ (mrs. upl and j) experience and examining how they deal with numeracy development in the book living homeschooling. subsequently, interpretation was conducted on the reduced and displayed data. in the final stage, the researchers performed verification to ensure the accuracy of the analysis results, drew conclusions, developed findings, and assigned meaning based on the data analysis. 3. result and discussion 3.1. results 3.1.1. parents’ reason for homeschooling mrs. upl, the subject of this research, decided to homeschool her son for some reasons. being the only parent in her village to decide to independently school her children with a homeschooling system is based on several considerations, one of which is the degradation of trust in the meaningfulness of the learning process from regular formal schools. in an interview, mrs. upl stated that "my husband and i are interested in homeschooling because we think that the lessons in schools are not relevant and not lasting. even if most of us are asked what lessons we remember, we mostly forget them, even though school is for years." furthermore, mrs. upl emphasized that most or almost all of the formal school programs are focused on academic development. however, in her opinion, to optimize a child's potential, it is not enough to simply develop their academic abilities, but it is also necessary to develop their talents, interests, and life skills. therefore, according to her, homeschooling is the right choice because the academic learning sessions can be balanced with the child's other needs. mrs. upl's decision to school her child through homeschooling was welcomed by her son, j. initially, mrs. upl also gave j the freedom to choose whether he wanted to try formal schooling like the other children in the village, or to go to school with a homeschooling system. without thinking too long, j chose to attend school independently at home because he is a child who likes to explore the environment. with the flexibility in time that he has when learning through homeschooling, before starting academic hours (at 09:00), he is free to play and explore the environment such as going to the fields, small rivers, gardens, and around. such activities are certainly difficult for him to do freely when restricted to school hours if he were to attend formal school. volume 12, no 2, september 2023, pp. 225-242 229 3.1.2. development model of homeschooler’s numeracy literacy one of the main focuses of the learning applied by mrs. upl in homeschooling is the development of literacy and numeracy, basic skills that should be cultivated early on but are often overlooked, including in the formal learning system in schools. this article specifically examines the development model of numeracy literacy applied by mrs. upl in the living books homeschooling program. based on the results of data analysis using software atlas.ti, the mapping of numeracy literacy development model in the living books homeschooling conducted by mrs. upl and j can be visualized in the following network diagram (see figure 1). figure 1. development model of j’s numeracy literacy in homeschooling based on interviews with mrs. upl, it was found that the process of developing numeracy literacy in the living books homeschooling program is carried out using a realistic approach, namely rme (realistic mathematics education). figure 1 also provides information that in practice, the strategies for developing numeracy literacy applied by mrs. upl generally consist of three things, as follows: a. using a real context, which is a familiar context that can be imagined by the students, or is close to the students' real experiences and consistently connects various mathematical topics with real student situations. b. using concrete manipulatives to bridge students' understanding of concepts from informal concrete forms to formal mathematical abstract forms. c. emphasizing concept understanding and high order thinking skills, not just focusing on computational or memorizing skills only. the first strategy for developing numeracy literacy applied is the use of realistic contexts. according to mrs. upl, context is the starting point for introducing a material to be taught. therefore, the selection of context must be familiar so that it is easily understandable and imaginable by j. the context selection strategy carried out by mrs. upl is to use contexts from things that j likes or are close to his real experiences. for example, j is a child who enjoys fishing in a pond, looking for worms in the fields, and flying kites. the dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 230 activities from these hobbies then become context ideas in the numeracy learning model designed by mrs. upl. according to mrs. upl's admission, the context of activities such as fishing, looking for worms, and playing kites, which are used as the initial situation to introduce a mathematical concept, has proven to be effective in attracting j's attention. j was enthusiastic to ask further about the material to be learned if mrs. upl gave narratives or stories about things he likes at the beginning of the learning. the use of contexts that are close to the students' experiences make the students interested and motivated to learn mathematics. students view the mathematics being learned as applied and useful for solving problems related to real world conditions and their daily experiences. therefore, the use of the right context can motivate students to participate in mathematics learning activities. with good context, mathematics learning become more meaningful and enjoyable. in addition to being used as a starting point for learning, realistic situations are also commonly used as the context for practice problems. here is an example of a problem using the context of j's daily activities developed by mrs. upl (see figure 2). figure 2. context problems figure 2 indicates that the realistic context is not only found as the starting point of learning designed by ms. upl, but it is also used as the context for assignment or task problems. the context used to construct the situation in the problem is referred to as the problem context. the use of realistic context by mrs. upl is closely related to the characteristics of the living books model, the homeschooling approach being followed. the focus of this model is the involvement of students in various activities to gain real-life experiences. the concept of living books can be realized in various mathematical activities that are close to the students' everyday experiences. some examples of daily activities used as learning contexts by mrs. upl include fishing in the river, shopping at a store, playing with toy cars, making kites, and accompanying mrs. upl shopping at the market. mrs. upl uses these daily activities as realistic contexts to introduce various mathematical concepts. in some cases, these activities are also used as 'learning environments' in developing numeracy literacy. mrs. upl calls it 'playing while learning' and 'engaging in daily activities while learning mathematics.' in many cases, j was involved in various mathematical activities on a daily basis. for example, mrs. upl introduced j to currency topic. as a follow-up activity, mrs. upl gave j some money and asked him to go shopping at a store, recorded the purchases and told about his activities after returning from shopping. from this simple activity, many applied mathematical content could be taught to j in an enjoyable way. another example is the activity of playing gasing (a traditional game in indonesia). mrs. upl and j competed in playing gasing and recorded the time of their play in each translation 3. jembar collected his fishing bait in his father's pond. according to the fishing journal he made, the number of fish in the pond should be 26. however, when the pond was cleaned, there were only 17 fish. how many fish died? 4. jembar made 12 fishing tools from wood. he had already installed 3 of them on his fishing gear. he gave 4 fishing tools to his friends. how many fishing tools are left? volume 12, no 2, september 2023, pp. 225-242 231 session (see figure 3). through this activity, mrs. upl introduced the concept of time measurement through an enjoyable game. the context of shopping and playing gasing are just two out of many learning contexts used by mrs. upl to design mathematical learning in the living books homeschooling program. figure 3. game activity (time measurement) record the second strategy applied is the use of various concrete manipulatives as learning aids. mrs. upl uses manipulatives as a support to bridge j's understanding towards the formal mathematics stage. mrs. upl believes that abstract mathematical material will be much easier to understand with the help of manipulatives. the manipulatives used are tailored to the teaching material and easily found or made using simple materials. figure 4. manipulatives (1. beads, 2. abacus, 3. mini legos) some examples of concrete manipulatives (see figure 4) used by mrs. upl and j are: a. beads. this manipulative is used to recognize numbers as a sequence and as a tool to perform addition and subtraction operations 1 100. b. a 10 x 10 abacus, used for counting numbers up to 100. c. mini legos as a tool to learn place value and perform operations counting 20 1,000. the mini legos are a simple and inexpensive modification of the relatively expensive dienes blocks. d. stopwatches and and broken clocks are used to help learn about time measurement. 1 2 3 dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 232 e. rulers, popsicle sticks, rulers, and sewing meters are used as manipulatives to learn about length measurement. the third strategy applied by mrs. upl is emphasizing understanding of concepts and high-level thinking skills such as reasoning and problem solving, not just computational skills. in order for math learning to be more meaningful, the learning goal should not just focus on memorizing formulas or proficiency in applying mathematical algorithms. instead, students should have meaningful experiences in the learning process, one of which is discovering mathematical ideas/concepts with the help of the teacher as a facilitator. in the context of homeschooling being undertaken by j, mrs. upl plays an important role in facilitating and stimulating the development of independent learning so that j is expected to be able to construct his own mathematical concepts being learned. as the sole tutor of math in the living books homeschooling program, mrs. upl plays a central role in designing learning activities for j. using realistic contexts and manipulatives, mrs. upl designs learning with activities that stimulate j to construct his cognitive ideas in understanding mathematical concepts from abstract to formal mathematics. in an interview, mrs. upl explained that the activities designed for mathematics education used a realistic mathematics education (rme) approach, which focused on the development of numeracy literacy. rme was chosen as the learning approach to create meaningful learning. in the rme view, mathematics is a human activity. in the rme approach, the focus of the learning activities is based on students' real-life experiences and the use of real-life situations. therefore, the learning activities began with a realistic or reallife context, aided by manipulatives, and then developed into a model using the manipulatives, culminating in the discovery of abstract, formal mathematical concepts. this series of learning activities is known as a learning trajectory. one example of a learning trajectory with rme approach oriented towards the development of numeracy literacy developed by mrs. upl is on the addition and subtraction of numbers 1-999 topic. the learning is started with various activities using realistic contexts, that is, contexts that are close to real situations or everyday experiences. for example, context situations used for the subject of addition and subtraction are various activities that j likes. for example, adding up the fish or worms caught by j, adding up the passengers in a train car, subtracting the papers used to make a kite, and so on. next, the counting operations are carried out using concrete manipulative materials or replicas of real objects, such as mini legos. after j is proficient in using the concrete manipulatives, the next step is to create a model/illustration of the manipulatives (a representation of legos) and provide a sign as a representation of the counting operation being carried out. a picture model of the legos he made refers to a model-of. creating an illustration/model of manipulatives was shown by j in solving substraction problems in his learning document (see figure 5). figure 5. creating model-of a substraction problem 5a 5b volume 12, no 2, september 2023, pp. 225-242 233 the way of his thinking beyond the picture and answer j made were then confirmed through interview as follows. transcript 1 [1] r : why did you use picture/ illustration to solve 25-13=12 problem? [2] j : because i did not use legos anymore. [3] r : ok, could you explain what picture is this? (pointing illustrations 5a and 5b). [4] j : lego. the bigger one (pointing illustration 5a) is lego “tens”, the big lego. the smaller picture (pointing illustration 5b) is lego ‘ones’, the small lego. [5] r : then, how to solve the problem using these pictures? [6] j : mmm…the problem is 25. i drew 2 big legos and 5 small legos. then minus 13. mmm..that was 1 lego and 3 small legos. [7] r : what did you do with 13? [8] j : we crossed it out. 1 big lego and 3 small legos. [9] r : if it is a substraction, do we need to cross it out? [10] j : yes, it is, mrs. [11] r : then, how to get the final answer? [12] j : see how many are left, which are not crossed out. 1 big lego, 2 small legos. [13] r : so, what is the answer? [14] j : 12. 1 big lego is 10, 2 small legos are 2. in the next step, j is assisted by mrs. upl in creating a model that is directed towards finding solutions to add or subtract. in the final step, j works with procedures and mathematical symbols as a form of formal abstract mathematics. at this formal level, j is able to write mathematical sentences for addition or subtraction operations, such as 25+13=38 and 25-13=12. the learning trajectory can be depicted in the form of an iceberg in the following figure 6. figure 6. iceberg of learning trajectory (counting 1-999) formal mathematics model for model of concrete dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 234 3.2. discussion 3.2.1. parents’ reason for homeschooling the decision of parents to school their children with a homeschooling system is motivated by many factors. mrs. upl, the subject of this research, is also such a case. one of primary reasons is the willingness to optimize her son’s potential, not only for academic aspect but also talent, interest, and life soft-skills such as problem solving, creativity, selfregulated, independent, and so forth. homeschooling then is considered as the most fit program to develop all those potential in a flexible way. this reason is in line with the results of previous studies that lack of trust in formal schools is one of the major reasons that parents decide to school their children with a homeschooling system (nuhla et al., 2020). in addition, the trend of homeschooling has become popular because it is considered able to support a child's competence and potential more optimally. the flexibility in time and material makes homeschoolers have more opportunities to explore and develop their potential based on their unique learning style, talent, and interests. homeschooling is considered able to develop a child's cognitive, affective, and psychomotor domain aspects in a balanced way, so homeschooling education provides a real experience that is beneficial for a child's life skills. the homeschooling education system can be categorized from structured homeschooling (like school at home) to unstructured homeschooling (unschooling) (neuman & guterman, 2017b). in practice, the approaches used are varied. mrs. upl applies a homeschooling system called the living books model. this model developed by charlotte mason focuses on the use of real experiences. the approach used is teaching good habits, basic skills (reading, writing, math), and activities to gain real experiences such as going for a walk in a park, visiting a museum, shopping at a market, and so on (muhtadi, 2012; na'imah, 2019). 3.2.2. development model of homeschooler’s numeracy literacy one of strategies to develop numeracy skill is through home numeracy activities. this strategy is suitable to apply in homeschooling program. the finding of this study reveals that mrs. upl tried to develop j’s numeracy skill through home numeracy activity with three main strategies: 1) using a real context which is familiar and can be imagined by the students, 2) using concrete manipulatives to bridge students' understanding of concepts from informal concrete forms to formal mathematical abstract forms, and 3) emphasizing the problem solving and other high order thinking skills such as reasoning, critical thinking, and creativity. the first strategy is the use of realistic contexts. context is the starting point for introducing a material to be taught, so that is why the context should be interesting, contextual, and easily understandable by the student. the proper selection of context is the main thing to consider in mathematics learning. cheng (2013) proposed that real-life context problems refer to problems embedded in real life situations that have no ready-made algorithm. the context selection strategy implemented by mrs. upl using realistic situations or everyday problems, hobbies, or real experiences of her child j is in accordance with the theory of previous experts. mrs. upl's strategy of using realistic context aligns with the characteristics of realistic mathematics education which uses the real world as a source or starting point for the development of mathematical concepts (van den heuvel-panhuizen & drijvers, 2014). the use of context is one of the five characteristics (tenets) of rme. the context used in rme is real life situations or phenomena that children are familiar with. volume 12, no 2, september 2023, pp. 225-242 235 in rme, real context is used as a starting point to construct students' mathematical concepts and ideas. it has been confirmed by trung et al. (2019) that teachers need to use context or create context that can help students construct their mathematical knowledge. however, in rme it is necessary not only to motivate students with everyday life contexts, but also to associate with experimentally real contexts and use them as the starting point for progressive mathematization (gravemeijer, 2001). a good context serves as a bridge for students to be involved in mathematical activities. as previous research has shown, the use of real-life context such as congklak traditional game (dewantara & mahmud, 2020), traditional market (wibawa et al., 2022), origami activity (afriansyah & arwadi, 2021), baratayudha war stories, and uno stacko game (risdiyanti & prahmana, 2020) has been proven to help students better understand mathematical concepts through a learning trajectory that stimulates students to construct mathematical ideas from the abstract to the formal stage. in conclusion, the use of appropriate context plays a crucial role in constructing mathematical knowledge. the appropriate use of context can support the construction of ideas, knowledge, and understanding of the material being studied by students. students will easily be involved in learning activities when the material they are studying is related to their daily experiences. laurens et al. (2017) claimed that the ease of learning can be experienced if learning content and context are related to students' daily activities. as a starting point for learning, mamolo (2018) also proposed that the use of realistic contexts would help students to get involved in meaningful mathematical activities, including exploring, modeling, visualizing data, abstracting, and concluding. in addition, the realistic context is not only found as the starting point of learning. ms. upl also utilized the context for tasks. the context used to construct the situation in the problem is referred to as the problem context. the problem context is problem in which its situation is real to the student (gravemeijer & doorman, 1999). in the rme approach, contextual problems are mathematical problems presented in real-life situations that children are familiar with. these problems can be a word problem, a game, a drawing, a newspaper clipping, a graph, or a combination of such elements (yilmaz, 2020). however, the term 'real' does not have to be literally interpreted as a situation that has actually been experienced by students. the context refers to a problem situation or event in the task, which often comes from real life or might be from imaginary situations (e.g., fairy tales) (van den heuvelpanhuizen, 2020; vos, 2020). to sum up, a realistic context is a conceivable or familiar situation which might be from students' real experiences or simply an imaginary situation. the contexts of the tasks should refer to situations that the students can imagine and that are truly meaningful to them. the context used in the questions or problems posed to the students is supposed to be locally based. the students' culture and their practices, which are closer to them, support the development of a more meaningful learning situation for the students (bonotto, 2010). contextual problem solving is known to have a positive influence on students' ability to understand mathematics. contexts of tasks that are close to students' daily life situations motivate them and lead them to engage in solving the tasks (da ponte & brocardo, 2020). in addition, context plays a crucial role in leading students to engage in the mathematical world. the use of realistic context can increase the meaningfulness of mathematical tasks. contexts drawn from a wide spectrum of areas in real life reflect that mathematics can be applied and used anywhere in society. vos (2020) proposed that the task context can also demonstrate the use of mathematics in students' future lives outside of school. as a result, students will feel that what they are learning can be useful for solving problems around them. for example, when j learns about money and its operations, he feels that his mother will trust him with a certain amount of money and he can go shopping on his own at a store to buy his favorite snacks. dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 236 the use of realistic context is closely related to the characteristics of the living books model. this model focus on the involvement of students in various activities to gain real-life experiences. this finding shows that mrs. upl stimulates homeschooler j to view mathematics as a social activity. mathematics is not just about memorizing formulas or performing calculations algorithms without meaning, but learning mathematics should be seen as an activity that requires interaction and communication in the context of various reallife activities. it then implies that in choosing tasks, students' experiences and reference frameworks have to be carefully considered and taken into account to ensure that these situations indeed make sense to them and have the potential to engage them in a purposeful mathematical activity (ainley et al., 2006). thus, the use of appropriate context can motivate students, spark students' de interest in learning mathematics, connect students in constructing mathematical ideas, and develope a more meaningful learning situation for the students. the second strategy is the use of various concrete manipulatives. the use of various manipulatives such as mino legos, abacus, and beads are able to foster the student’s interest to learn. besides, the use of manipulatives also support to bridge homeschooler j's understanding towards the formal mathematics stage. the finding also reveals that homeschooler j are more easyly understand the abstract mathematical material with the help of manipulatives. jones and tiller (2017) claimed that students understand mathematics better when they can see and touch it, not just memorize formulas. mathematics essentially contains abstract concepts. manipulatives can support building such abstract concepts from concrete activities (simon, 2022). in addition, by using various shaped and colored objects as manipulatives, learning becomes more enjoyable and able to increase j's learning activity, such as observing, manipulating form or demonstrating a mathematical concept. in other words, manipulatives are used as a mathematics exploration medium for children. larbi and mavis (2016) asserted that the use of manipulatives can enhance learning by providing opportunities for students to explore and actively engage in observation and discovery. the role of manipulatives is so essential that almost every topic/mathematics material studied by j is used manipulatives as a learning aid. mrs. upl employs manipulatives to engage homeschooler j in hands-on learning of mathematics. it is confirmed by jones and tiller (2017) that the use of manipulatives is able to foster learners' interest and encourage active engagement in lesson. this finding indicates that the use of concrete manipulatives has a significant contribution in making learning more effective. students are more easily able to understand abstract mathematical concepts because manipulatives help bridge the students' thinking construct from concrete to abstract models. based on the findings of the researcher's interview with j (refer to transcript 1), it is noticeable that lego acts as a cognitive bridge, facilitating the transition from abstract to formal concepts. initially, j employed manipulative lego objects, offering a concrete representation. as j progressed, a shift occurred towards a semi-concrete approach, utilizing lego illustration/model as visual aids. with the assistance of lego and its illustration, j was able to effectively convey comprehension of the subtraction concept, symbolically represented through the act of 'crossing out' subtracted numbers. in addition, the use of manipulatives can help attract students' attention and increase their motivation to learn, increase their activity and involvement in learning by providing opportunities for exploration, holding, manipulating. engage students not only physically but also emotionally. the third strategy is emphasizing understanding of concepts and high-level thinking skills such as reasoning and problem solving, not just computational skills. in this strategy, mrs. upl use rme approach to develop the homeschooler j’s numeracy skill. numerical literacy, as called as mathematical literacy, relates to an individual's ability to formulate, employ, interpret (and evaluate) mathematics in a variety of real-world situations (oecd, volume 12, no 2, september 2023, pp. 225-242 237 2019). formulate relates to what extent individual is able to recognize and determine opportunities to use mathematics in problem situations and then make structured mathematical model to formulate the contextualized problem. employ indicates how well students solve a mathematics problem by computing, manipulating, and applying the concepts and facts. interpreting refers to how evaluate mathematical solutions or conclusions and reflect them whether reasonable and/or useful in the context of the real-world problem (oecd, 2018). doing home activity numeracy using rme approach can directly facilitate the development of students' mathematical literacy, which essentially means being able to know, understand, and use basic mathematical concepts in daily life. the characteristics of pmri applied by mrs. upl in developing literacy are through the use of realistic contexts that are close to students' real lives. the strategy used by mrs. upl is to always connect the use of mathematical material in a personal, social, occupational, or scientific context that is appropriate for j's cognitive development. for example, mrs. upl uses context problems to stimulate j's numeracy literacy after learning about addition and subtraction. mrs. upl will introduce how addition and subtraction are used in real life, such as counting the toys they own, counting the fish in the aquarium, counting the number of books they have, and so on. in addition, mathematical literacy can also be developed in the form of intertwinning, one of the characteristics of pmri (gravemeijer & van eerde, 2009; sembiring et al., 2008). mrs. upl trains j to be able to connect the mathematical ideas/concepts they are learning with other ideas/concepts, connecting or integrating a mathematical topic with a topic, material, or even other fields. for example, when j plays a game about addition and subtraction using number cards, mrs. upl deliberately creates cards with various shapes with the aim of introducing simple geometric shapes. another example is when j is learning about currency and how to apply it through shopping at the store, mrs. upl connects it with simple financial literacy concepts. in this way, j will realize the wide use of mathematics in real life. using rme approach, mrs. upl could designing attractive learning trajectories to create meaningful mathematics lesson. when rme approach is integrated with living book model homeschooling, it is very powerful to support students’ numeracy literacy development. a wide opportunity given to homeschooler doing home numeracy activity. one major challenge in implementing homeschooling programs is the lack of professional and pedagogical competency of parents as tutors (basham et al., 2020; pozas et al., 2021). a common issue is that parents are unable to effectively fulfill their role as teachers and facilitators of learning (basham et al., 2020). however, this issue was not experienced by mrs. upl. as both a parent and the sole tutor for mathematics education, she was able to demonstrate her role as a planner of learning activities, facilitator, and teacher for homeschooler j. 4. conclusion the findings of this study reveal that the parents’ main reason for homeschooling are a dissatisfaction with public school instruction model and flexibility to provide a better personalized learning program to develop the homeschooler's cognitive, affective, and psychomotor domain aspects in a balanced way. this study result also confirm that the implementation of living books homeschooling and rme approaches strongly supports the successful development of numeracy literacy in homeschooling education. practically, there are three main strategies for developing numeracy literacy employed by the homeschooler in this study: use of context real and concrete teaching aids, as well as an emphasis on conceptual understanding, reasoning, and problem solving. the learning trajectory with the rme approach is clearly able to support homeschooler j to construct his own ideas from the dewantara, setiawati, & saraswati, towards numeracy literacy development: a single-case … 238 concrete situational stage to the abstract stage of formal mathematics. the integration of rme approach and living book model homeschooling is very powerful to support students’ literacy numeracy development. acknowledgements the authors would like to thank the indonesian endowment fund for education (lembaga pengelola dana pendidikan lpdp) indonesian ministry of finance for the financial support is gratefully acknowledged. references 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(2014). case study research: design and methods (applied social research methods). sage publications thousand oaks, ca. https://doi.org/10.1007/978-3-030-33824-4_3 https://doi.org/10.22342/jme.4.2.413.151-159 https://doi.org/10.54870/1551-3440.1317 https://doi.org/10.22342/jme.11.1.8690.17-44 infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p69-82 69 analysis of abstract reasoning from grade 8 students in mathematical problem solving with solo taxonomy guide imam kusmaryono 1 , hardi suyitno 2 , dwijanto 3 , nurkaromah dwidayati 4 1 sultan agung islamic university, jl. kaligawe raya km.4, semarang 50112, indonesia 2,3,4 semarang state university, jl. sekaran gunungpati, semarang, indonesia 1 kusmaryono@unissula.ac.id, 2 hhardisunnes@yahoo.com, 3 dwijanto5@gmail.com , 4 noengkd_unnes@yahoo.co.id received: july 03, 2018; accepted: august 11, 2018 abstract this research was a descriptive research. description of research result was presented quantitatively and qualitatively. subjects of the research were 30 (thirty) 8th graders of smpn 10 (state junior high school) in semarang, indonesia. data were collected through tests, documentation, observations, and interview. student answers documents were observed and analyzed with solo taxonomy guidance. the objective of the study was to analyze and provide an interpretation of students abstract reasoning level in cognitive development based on intended learning outcomes. the result of findings from students’ answers basically showed that students' abstract reasoning on the lower, middle and upper level, was alike to stages of structure complexity improvement. there were two main changes from concrete thinking to abstract thinking: quantitative stage (uni-structural and multi-structural) occurred first, as the number of details in student responses increased and then changed qualitatively (relational and extended abstract) because the detail was integrated into a structural pattern. keywords: abstract thinking, abstract reasoning, problem solving, solo taxonomy. abstrak penelitian ini adalah penelitian deskriptif. deskripsi hasil penelitian disajikan secara kuantitatif dan kualitatif. subjek penelitian sebanyak 30 siswa kelas 8 pada smp negeri 10 di kota semarang, indonesia. pengambilan data melalui metode tes, dokumentasi, observasi dan wawancara. respon jawaban siswa dalam tes penalaran abstrak diamati dan dianalisis dengan berpandu taskonomi solo. tujuan penelitian untuk menganalisa dan memberi interpretasi tingkat penalaran abstrak siswa dalam perkembangan kognitif berdasar capaian pembelajaran yang diinginkan (intended learning outcomes). hasil temuan dari tampilan respon jawaban siswa secara mendasar menunjukkan bahwa penalaran abstrak siswa pada kelompok bawah, tengah dan atas, serupa tahapan peningkatan kompleksitas struktur. ada dua perubahan utama dari penalaran konkret menuju penalaran abstrak yaitu tahap kuantitatif (uni-structural dan multi-structural) terjadi pertama, seperti jumlah detail pada respon siswa meningkat dan kemudian mengalami perubahan kualitatif (relational dan extended abstract), karena detail tersebut terintegrasi menjadi pola struktural. kata kunci: berpikir abstrak, penalaran abstrak, pemecahan masalah, taksonomi solo. how to cite: kusmaryono, i., suyitno, h., dwijanto, d., & dwidayati, n. (2018). analysis of abstract reasoning from grade 8 students in mathematical problem solving with solo taxonomy guide. infinity, 7(2), 69-82. doi:10.22460/infinity.v7i2.p69-82. kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 70 introduction research on the issues of developing abstract thinking has been widely carried out, but they have different ways to stimulate the most appropriate solution for development of this thought (gilead, liberman, & maril, 2014; lermer, streicher, sachs, raue, & frey, 2016). abstract thinking level is closely related to students’ academic achievement. how students understand and learn depends on cognitive processing ability and abstract thinking level (darwish, 2014). there had been many studies confirming that the level of abstract thinking predicted students' academic achievement in mathematics and science as well as other fields of science (gilead et al., 2014; lermer et al., 2016). in some cases, students' abstract thinking abilities are confronted with cognitive obstacles, didactic, psychological and epistemological obstacles (komala, 2018). therefore, students must be conditioned through practice, giving scaffolding and raising awareness of learning through the investigation process. discussing about abstract thinking is certainly related to cognitive development. the experts defined cognitive development as a process of acquiring advanced thinking skill and intellectual thinking skill with an ability to use problem-solving approaches in life situations from early to adult age (gilead et al., 2014; susac, bubic, vrbanc, & planinic, 2014). based on the aforementioned discussion, it is deemed necessary to evaluate the abstract thinking that focuses on abstract reasoning process of students at every level of school education. abstract reasoning development of students' reasoning abilities should not be considered normal. however, the school must encourage teachers to organize learning that provides opportunities for students to exercise their reasoning skills (adegoke, 2013). reasoning is a thinking process arises from emphirical observation resulting in some concepts and understandings. based on similar observations, proportions will be formed, according to the determined proportions (regarded as correct), a new proportion could be drawn. this process is called as reasoning (markovits, thompson, & brisson, 2015). abstraction is a construction process in one's mind, which involves reasoning in determining the relationship between mathematical objects and changing this relationship into a specific expression that is independent of mathematical objects (yilmaz, argun, & role, 2018). abstract reasoning refers to the ability of information analysis, detecting pattern and relation, and solving problems on complex level (datta & roy, 2015). abilities included in abstract reasoning are: (1) being able to formulate theories about natures of object and idea, (2) being able to understand meanings underlying a happening, statement or object, (3) being able to identify the correlation of verbal and nonverbal ideas, and (4) being able to detect pattern and relation underlying among happening, ideas and objects (simanjuntak, abdullah, & maulana, 2018). abstract thinking ability is a result of brain maturation (piaget, 1964). while abstract reasoning is a part of (individual) abstract thinking ability that showing certain abstract thinking level in a certain domain, that will be similarly potential on reasoning ability in other domain (datta & roy, 2015). piaget introduced four stages of cognitive development that determined reasoning and mental development skills of a person from his childhood to adulthood (joubish & khurram, 2011; piaget, 1964; simatwa, 2010). in particular, ages 11-14 years old (8th grade students) as an age in which a transition occurs in their cognitive development from stage of concrete operational to formal operational (susac et al., 2014). meanwhile, relating to cognitive development, many junior high school students have not yet acquired the ability to understand volume 7, no. 2, september 2018 pp 69-82 71 abstract concepts without a real basis (darwish, 2014; simanjuntak, abdullah, & maulana, 2018). according to neo-piagetians, a person in charge of abstraction-based reasoning task could be possibly on concrete operational stage while temporarily becomes an expert of formal operational in solving problems based on other kind of abstraction (joubish & khurram, 2011). profile of cognitive development reveals level of individual abstract thinking skill. this cognitive development level can be measured using a number of intellectual tasks (ojose, 2008). in this study, the quality of student responses toward a number of intellectual tasks were evaluated with structure of the observed learning outcomes (solo) taxonomy guidance. the solo taxonomy the solo taxonomy is an important tool to assess students' knowledge and skills, by examining their answers in depth (biggs & tang, 2011; chalmers, 2011). the assessment in solo taxonomy is based on the quality and structure of the answers given by students toward the questions (korkmaz & unsal, 2017). by solo taxonomy, teachers can identify their answer responses (learning outcomes) so that the students' understanding level toward a given problem can be determined (özdemir & yıldırz, 2015). the solo taxonomy classifies the ability of students' responses to problems into 5 different hierarchical levels: level 0 : pre-structural. at this level, students use knowledge without understanding, just repeat the given questions (goff, potter, & pierre, 2014; potter & kustra, 2012), and even wrong answers or those of not answering the question (korkmaz & unsal, 2017). level 1 : uni-structural. at this level, students have a limited understanding. students focus only on the use of data related to the question (biggs & tang, 2011). level 2 : multi-structural. at this level, students can focus on more than one aspect to the question but these are not related to each other (biggs & tang, 2011; chalmers, 2011). level 3 : relational. at this level, students understand how to build the whole and the relationships between the structures that make up the whole (biggs & tang, 2011). level 4 : extended abstract. students can reason by considering abstract characteristics and can make generalizations. students can view topics from many perspectives, hypothesize, and make generalizations (biggs & tang, 2011; brabrand & dahl, 2009). in content standard of indonesia curriculum 2013, learning outcomes are grouped into 5 (five) categories based on solo taxonomy: pre-structural (level 0 is kindergarten class), unistructural (level 1 is grade i and ii), multi-structural (level 2 is grade iii and iv), relational (level 3 is grade v and vi), and extended abstract (levels 4 and 5 are grade vii, viii, and ix) (bsnp, 2013). considering this, it is clearly emphasized that, learning mathematics in grade 8, teachers should encourage students to acquire learning achievement on high level thinking pattern or extended abstract. the description of the learning cycle in the learning achievement of the solo taxonomy begins before the cycle, which is pre-structural, in the cycle of the quantitativephase (unistructural and multi-structural), the qualitative phase (relational), and out of the cycle of kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 72 qualitative (abstract expanded). the following is presented in figure 1 of the learning cycle along with the explanation of each level in the achievement of the solo taxonomy learning. quantitative phase qualitative phase figure 1. bigg’s solo taxonomy: learning outcomes assessment (goff, potter, & pierre, 2014) bases on figure 1, level of uni-structural and multi-structural response was included to quantitative (concrete) thinking phase. level of relational and extended abstract response was categorized as qualitative (abstract) thinking phase. (goff, potter, & pierre, 2014). in this study, student answer responses on abstract reasoning ability test were evaluated and analyzed, then the answers would be categorized into one of solo taxonomy levels. the objective of the study was to analyze and provide an interpretation of students abstract reasoning level in cognitive development based on intended learning outcomes (ilo) guided by solo taxonomy.the results can be used by teachers as a reference to manage mathematics learning to the level of student cognitive development based on grade level. method this research uses quantitative and qualitative descriptive approach (creswell, 2014). based on purposive sampling technique, 30 students of 8-e class in smpn 10 (state junior high school) semarang, indonesia. the subjects were all at the age range of 12 to 14 years old. in this study, mathematics learning was conducted by applying group investigation learning model which was done for five weeks discussing about polyhedra topic. at the end of the lessons, students were assigned to do intellectual tasks in forms of mathematical reasoning tests. data were collected by tests, documentation, observation and interview. the test instrument was an intellectual task to assess students' ability in using formal (abstract) reasoning strategy. the test consisted of 2 (two) points of mathematical reasoning problem about polyhedral geometry. the test instrument had been strictly evaluated by a team of experts (instrument validators) in terms of contents, construction, concurrent and predictive validity. based on test results, the subjects were divided into three groups: the lower group was filled by students with low cognitive ability, middle group was those with moderate cognitive ability, and upper group was those with high cognitive ability.students' response or answers as the document of pre-structural fail, lack of knowledge uni-structural one relevant aspect multi-structural several relevant and independent aspects relational integrate into a structure extended abstract generalize to a new domain level learning outcome assessment volume 7, no. 2, september 2018 pp 69-82 73 intellectual tasks were observed and analyzed by guidance of learning achievements according to solo taxonomy. data analysis was described qualitatively as a result of learning achievement on students' abstract reasoning. in order to collect deeper information about abstract reasoning, each group selected 2 (two) students to be interviewed.the following is the test instruments used to measure students' abstract reasoning abilities, in table 1. table 1. test instruments to measure abstract reasoning problem-solving task intended learning outcomes problem 1: abdullah wants to make a tent of fabric with the model and size as shown in the picture below. how many square of fabric is required by abdullah to make one tent with its base? relational problem 2. take a look at the abc.def prism image below. the prism contains water as high as ch. comparison of length ch: hf = 3: 1 the base plane abc with the elbow at point c. length ac = 8 dm, length ab = 10 dm, and height ad = 16 dm. if the volume of water in triangular upright prisms is transferred into a length beam of 16 dm base plane, base width of 6 dm and height of 8 dm, then how high is the water in the block? extended abstract kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 74 results and discussion results after having a test in form of intellectual tasks, result of student responses or answers were observed and analyzed based on solo taxonomy. the observation and analysis on student responses showed a result that is distributed in table 2. table 2. scores of student response on abstract reasoning test cognitive level students problem number response rate answers means score ps us ms r ea lower 6 1 0 1 5 0 0 63 2 0 2 4 0 0 53 middle 18 1 0 0 4 14 0 81 2 0 2 13 2 1 64 upper 6 1 0 0 0 4 2 90 2 0 0 1 1 4 92 total 30 means 0% 8% 45% 35% 12% 73 notes: ps : pre-structural r : relational us : uni-structural ea : extended abstract ms : multi-structural based on the result of the test on mathematical reasoning ability (see table 2), it was obtained an average score of 30 students is equal to 73 with high category. furthermore, students' abstract reasoning on the lower, middle, and upper group was evaluated more deeply. discussion lower group students considering table 2, students in the lower group with gained average score by 63 on problem solving task number 1 and on problem number 2 by 53. thus, it could be said that students' mathematical ability in reasoning component was in low category. based on solo taxonomy point of view, students' responses on the problem number 1 and 2 were just at uni-structural (25%) and multi-structural (75%) levels. the following is an example of the results of the response of the students' answers (selected subject) from the lower group, in figure 2. subject (s.12) response answer on problem 1 figure 2. response of subject answers (s.12) in the lower group volume 7, no. 2, september 2018 pp 69-82 75 referring to figure 2, it was explained that subject (s.12) could handle various aspects of the topic but could not establish relationships. subject (s.12) could focus on more than one aspect to the question but could not be related to each other (biggs & tang, 2011). guided by solo taxonomy, subject (s.12) answers were at multi-structural level. at this level, subject (s.12) were only able to use ideas from concrete instructions to solve problems. subject (s.12) still thought based on concrete facts and had not been able to establish relationships between aspects of one another. they still operated at the level of quantitative (concrete) thinking and had not yet operated at an advanced stage of cognitive development (qualitative or abstract). below is an excerpt interview between researcher (r) and subject (s.12). r : did you use correct way in second step (calculating area ii)? (s.12) : i don’t know and i don’t understand what i wrote. r : you wrote (s.a. i + s.a. ii = 224). but your final answer was 200 m 2 . could you please explain your reason? (s.12) : it needed a long time. i knew, and i drew a conclusion that it was right 200 m 2 . based on analysis on interview result, in this case, subject (s.12) had an obstacle, that they were not able to understand that every concept could have many interpretations. a previous research (susac et al., 2014) pointed out that many students used a very concrete strategy such as inputing data in form of numbers already in the question. subject (s.12) could not solve problems in creative ways, and failed in logical conclusion. according to subject (s.12), to apply abstract thinking should take a lot of time. subject (s.12) answer response in this case is right and correct. however, the subject (s.12) could not provide justification (clarification) or reasons on his answer. this situation led to a tought that students were difficult to use reasoning based on deduction (darwish, 2014). students in the bottom group, have difficulty in abstract reasoning and subject (s.12) are still weak in developing abstraction. in other words, subject (s.12) had not succeeded in developing or improving abstract thinking and logical reasoning. based on the result of subject (s.12) assignment in this lower group, teachers need to pay serious attention to the lower group students in mathematics learning. assistance in learning could be provided by scaffolding to improve cognitive development to an advanced level (abstract) (chang, wang, & chao, 2009). the hope is not to keep students at a concrete level of thought throughout their study year, which will distract their efforts to solve more complex problem. middle group students regarding to table 2, students in the middle group obtained an average score on problem solving task number 1 of 81 and problem number 2 of 64. the average score ‘moderate’ category. thus, it could be said that the student's mathematical ability in reasoning was in ‘fairly good’ category. reviewed by solo taxonomy, students' answers to problems 1 and 2 were at levels of uni-structural (6%), multi-structural (47%), relational (44%), and extended abstract (3%). the following is an example of the results of the response of the students' answers (selected subject) from the middle group, in figure 3. kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 76 subject (s.08) response answer on problem 2 figure 3. response of student answers (s.08) in the middle group considering figure 3, subject (s.08) had used steps of problem solving well.however, the subject (s.08) unfortunately failed in decision taking, then the final answer was wrong. consider the following excerpt of interview between the researcher (r) and the subject (s.08). r : did you check your answers carefully? (s.08) : i did not do a re-check on my answers. r : what do you know about water volume if it is moved? (s.08) : i did understand. the water volume will be the same although the container is different. r : your answer is correct, perfect. please check your answer. (s.08) : it means that the water volume after being moved was 16 x 6 x 6 = 576. then, the result was not equal to 288 (water volume before movement). so my answer was illogical. subject (s.08) could understand the problem in context to triangular prism and moved in context to a cuboid volume. however, subject (s.08) was too soon to draw conclusion without re-checking. so, the answer was wrong and invalid. this shows the subject (s.08) is still weak in identifying the relationship between verbal and nonverbal ideas. guided by the solo taxonomy, table 2 shows the responses of students in the middle group being in the quantitative phase of 57% and 43% of them were already operating at the stage of cognitive development qualitative (abstract). but dominantly, students had not been able to see topics from many perspectives, hypothesize, and had not made generalizations. so, cognitive ability had not reached the maximum extended abstract. this result might indicate that students were delayed in achieving the expected level of cognitive development of abstract thinking and would develop cognitive abilities as age increased. according to piaget, at the age of 14, most individuals should be at a formal operational level (abstract level) (joubish & khurram, 2011; mascolo & f., 2015; piaget, 1964; simatwa, 2010). considering this situasion, teachers should still believe that students' cognitive development could still be improved through a learning process that focused on improving reasoning ability. volume 7, no. 2, september 2018 pp 69-82 77 upper group students table 2 presented an average score gained by upper group students on problem solving task number 1 was 90 and on problem number 2 was 92. average score indicated‘very high’ category. thus, it could be said that students' mathematical ability in the reasoning component was in ‘high’ category. overviewed by solo taxonomy level, student responses towards problems 1 and 2 were at level of multi-structural (8%), relational (42%), and extended abstract (50%). a total of 92% students in the upper group had operated at the stage of cognitive development qualitative (abstract). the following is an example of the results of the response of the students' answers (selected subject) from the upper group, in figure 4. subject (s.024) response answer on problem 2 figure 4. response of student answers (s.24) in the higher group on figure 4, the subject (s.24) had done the steps of problem solving well, systematically, and easily to understand. the subject (s.24) understood how to construct aggregate and correlation among structures that construct the aggregate (biggs & tang, 2011). this ability is a high level of abstract reasoning ability. according to ylvisaker and hibbard (refered to in darwish, 2014) abstract reasoning ability relate to the ability of moving what have been learnt from one context to another. the answer by the subject (s.24) question number 2 was said to reach extended abstract level. to support this argument, consider the following excerpt of interview between the researcher (r) and the subject (s.24). r : how did you come up with this answer? (s.24) : i just imagined that the water volume did not change as 288 litres. r : what strategy did you use to solve the problem? (s.24) : i thought that there was a ratio between water volumes (v1) and (v2) r : is your answer logical? (s.24) : yes, it is logical. the water volume should not change, it was just the height of water that changed because of different containers. kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 78 r : how did you prove that your answer was correct? (s.24) : i re-checked the steps of problem solving. then, i formulated volume (v1) and (v2) equation model. reviewing figure 4, it could be described that subject (s.24) could focus on more than one aspect for interrelated questions. subject (s.24) understood how to build the whole and the relationship between the structures that make up the whole (biggs & tang, 2011). subject (s.24) could reason with considering abstract characteristics and could make generalizations. subject (s.24) understood that each concept could have many wider conceptual interpretations or conceptual understandings. subject (s.24) solved problems in a more creative way. subject (s.24), taking a more complex problem. subject (s.24) have been able to use abstract things that are not written in direct facts. subject (s.24) can do the abstraction contained in the problem very well. so it can be said that the subject (s.24) has developed advanced mathematical thinking (smith, wigboldus, & dijksterhuis, 2008). based on the excerpt of the interview, it can be said that the subject (s.24) has good metacognitive abilities including self regulation and controls the thinking process through repeated checking and reflection (lukum, laliyo, & sukamto, 2015; qohar & sumarmo, 2013). by solo taxonomy guidance, in the upper group there is an answer response reaching the 50% abstract extended level, but there are still few students in the quantitative phase is the multi-structural level (8%). similarly, in middle group students, this result might indicate that some students were delayed in achieving the expected level of cognitive development of abstract thinking and would develop optimally in teen age (joubish & khurram, 2011; mascolo & f., 2015; piaget, 1964; simatwa, 2010). meanwhile, according to darwish, teenagers gradually developed the ability to use hypothetical-deductive reasoning, and extended their logical thinking to abstract concepts. but this did not mean that there would be no further change in their cognitive (darwish, 2014). they could seek any excuse, real or imaginary, and had the ability to use scientific reasoning to solve relatively complex problems. this finding shows that abstract reasoning ability plays an important role in the achievement of their mathematics learning outcomes (widodo, 2017; yumiati & noviyanti, 2017). overviewing on the results of evaluation on students' abstract reasoning thoughts on the bottom, middle and upper cognitive levels, it could be explained that results of their answers were similar with stages of structure complexity improvement. there were two main changes from concrete thinking to abstract thinking: (1) the quantitative (uni-structural and multistructural) stage occured first, as amount of detail in student responses increased and then changed (2) qualitatively (relational and extended abstract) because the detail was integrated into a structural pattern. conclusion regarding on reviews of previous research result by experts and the results and discussion in this study, it could be said that abstract reasoning of 8th graders was not reaching 100% as expected in content standard of indonesia curriculum 2013 suggesting that, 8th graders should reach extended abstract level. in the lower group, the abstract reasoning of students was still in phase of quantitative thinking (concrete) with the achievement of solo taxonomy was at uni-structural level (25%) and multi-structural level (75%) and no students reached qualitative stage of abstract thinking. thus, it could be said that lower group students had low abstract reasoning levels. in the middle group, the abstract reasoning of students based on the volume 7, no. 2, september 2018 pp 69-82 79 achievement of the solo taxonomy was at an abstract level that extended abstract by only 3%, but by 44% (relational level) they were already operating at the stage of qualitative cognitive development, so the level of abstract thought was quite good. in the upper group, 92% of students have reached the qualitative phase (abstract thinking) which includes 42% at the relational level and 50% of the extended abstract level. students in the upper group always work with high reasoning and rich abstractions. basically, result of evaluation on students' abstract reasoning in the lower, middle and upper cognitive level could be concluded that, as student answers’ result, it was similar to stages of structure complexity improvement. there were two main changes from concrete thinking to abstract thinking: (1) the quantitative (unistructural and multi-structural) stage occured first, as amount of detail in student responses increased and then changed (2) qualitatively (relational and extended abstract) because the detail was integrated into a structural pattern. acknowledgments thanks to many who have supported this research are promoters, validators, dissertation reviewers and leaders of the sultan agung islamic university of semarang indonesia which has provided funds for the implementation of this research. references adegoke, b. a. 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(2017). abilities of reasoning and mathematics representation on guided inquiry learning. journal of education and learning, 11(3), 283–290. kusmaryono, suyitno, dwijanto, & dwidayati, analysis of abstract reasoning … 82 sebuah kajian pustaka: journal of mathematics education p-issn 2089-6867 volume 8, no. 1, february 2019 e–issn 2460-9285 https://doi.org/10.22460/infinity.v8i1.p21-30 21 the effectiveness of realistic mathematics education to improve students’ multirepresentation ability muhtarom* 1 , nizaruddin 2 , farida nursyahidah 3 , nurina happy 4 1,2,3,4 universitas pgri semarang article info abstract article history: received sept 30, 2018 revised jan 28, 2019 accepted jan 30, 2019 this research aimed to evaluate the effectiveness of realistic mathematics education (rme) to improve students' multi-representation ability. a quasiexperimental design was used in this research. sixty-four samples from the seventh-grade students of junior school were randomly selected and divided into two classes: experimental class was treated using rme and control class was treated using conventional learning, with each class consisting of thirtytwo students. the essay test was used to measure the multi-representation ability of students and the questionnaire was used to measure students' responses in rme learning. the data from the essay test were analyzed by n-gain test and t-test in which normality and homogenity test were conducted previously, while the students' learning completeness and student responses were presented descriptive quantitative. the result of the research concluded that the multi-representation ability of students who get rme learning is better than the multi-representation ability in students who get conventional learning. 87.25% of students who get rme learning with the developed device have completed the kkm, and many students are very enthusiastic and interested in rme based learning, thus increasing their learning spirit in a learning process. keywords: multi-representation rme copyright © 2019 ikip siliwangi. all rights reserved. corresponding author: muhtarom, departement of mathematics education, universitas pgri semarang, jl. sidodadi timur no. 24, semarang, indonesia. email: muhtarom@upgris.ac.id how to cite: muhtarom, m., nizaruddin, n., nursyahidah, f., & happy, n. (2019). the effectiveness of realistic mathematics education to improve students’ multi-representation ability. infinity, 8(1), 21-30. 1. introduction guided discovery, didactic phenomenology, and the principle of mediation model are the three basic principles of realistic mathematics education (rme). lessons are developed through mathematical concepts that originate from the real world in accordance with the indonesian cultural context. the selected context is easily recognized by the students and can be imagined by students, languages, and diagrams presented very clearly to provide support in the development of mathematical concepts (sembiring, hadi & dolk, 2008). rme is more effective in improving student learning outcomes than conventional learning (laurens et al., 2018; ginting et al., 2018; zakaria & syamaun, 2017). laurens et mailto:muhtarom@upgris.ac.id muhtarom, nizaruddin, nursyahidah, & happy, the effectiveness of realistic mathematics … 22 al. (2018) stated that realistic mathematics education (rme) improve students’ mathematics cognitive achievement; improving the reasoning ability of elementary school student (ginting et al., 2018) and improving students’ achievement and attitudes towards mathematics (zakaria & syamaun, 2017). however, this study has not focused on multirepresentation skills of students. this is possible because learning tools that contain multiple representations trained to students are rarely found. whereas rme should be developed in accordance with the needs of students there is no exception for the development of multi-representation skills of students (sembiring et al., 2008). the development of multi-representation skills of students also reinforced by neria & amit (2004) study which states that only 153 students (44%) answered correctly with verbal representation, 131 students (37%) correctly answered with symbol representation. in addition, nizaruddin, muhtarom & murtianto (2017) study states that the majority of students tend to use symbol representations to solve math problems, rather than using other representations. even when using verbal representation, students find difficulties composing sentences while students have not been able to solve problems when using visual representations. students are not able to accommodate to reconstruct their cognitive structure (muhtarom, murtianto & sutrisno, 2017), including in the process of translation between representations. it shows that basically students still do not have multirepresentation skills, students are still focused on one of the representations they think are suitable. table 1. focus of multi-representation ability representation description visual  the ability to represent data or information in the form of diagram, graphics or table.  able to use visual representation to solve the problem.  able to draw to clarify and facilitate its solution verbal  able to identify the problem based on data or given representation  able to write the representation of given representation  able to write steps of math problem solving symbol  able to make math equation or model from other given representation  able to solve a problem by involving math expression (milrad, 2002) on the other hand, keller & hirsch (1998); bransford & schwartz (1999) and ainsworth (2006) strongly recommend a teacher to use more than one representation in the learning process of mathematics. the use of representations of more than one type at the same time is said to be multi-representation (brenner et al., 1997). it is further emphasized that the ability of multiple mathematical representations is very important for students because they can develop mathematical concepts, relationships between concepts, using varied representations and help in communicating their way of thinking (nctm, 2000). hwang et al., (2007) divide the representations used in mathematics education into five types, i.e representations of real-world objects, concrete representations, symbol representations, verbal representations and visual representations. among the five representations, the last three are more abstract and have higher difficulty levels. the representation of symbols is the skill of presenting the mathematical problems in the formula. the verbal representation is the skills of translating the nature and relationships in mathematical problems into the language or vowel, the visual representation is the skill of presenting math problems in pictures or graphs (kaput & romberg, 1999; milrad, 2002). volume 8, no 1, february 2019, pp. 21-30 23 furthermore, these three representations will be the focus of this research, in which the description of each representation has been described in table 1. thus, rme-based devices containing visual representations, verbal representations and symbol representations were developed to facilitate different student learning styles. this representation begins with a realistic situation close to the student so that they can develop other representations. students build their confidence in problem-solving (muhtarom, juniati & siswono, 2017) through their chosen form of representation, fearlessness, and beliefs in explaining the answers (supandi et al., 2018). rme-based devices are expected to contribute positively to students in gaining understanding of mathematics, improving learning interactions, and developing multi-representation capabilities. based on the above description, the problem in this research are: a. is there any difference in the multi-representation skills of students with rme and conventional learning? b. how is the mastery of students with rme and conventional learning? c. how is the improvement of multi representation skills of students with rme and conventional learning? d. how is the student's response to rme-based tools developed? 2. method 2.1. general background of research the first stage of this research is the development of rme based learning tools that include lesson plans, modules based multi-representation, media, test description and student response questionnaire. the overall stages of this study use the concept of analysis, design, development, implementation, and evaluation (almomen et al., 2016), in which the tools are developed based on local indonesian wisdom and contain several mathematical representations. figure 1 clearly outlines the stages of device development up to the evaluation of the effectiveness of rme learning tools developed. 2.2. sample of research the sample of this research consisted of sixty four students grade vii junior school in pati regency of central java province, indonesia. the sample is divided into two classes: experimental class was treated using the rme learning and control class was given treatment using the existing learning strategy (conventional learning), with each class consists of thirty-two students. the research sample was selected using cluster random sampling technique to ensure the objectivity of the research, avoiding bias in the research and giving equal opportunity to a group of students who were collected in the class to be a research sample. prior to treatment, the researchers tested the normality by the lilliefors method to ensure that the sample came from a normally distributed population, tested homogeneity with bartlett's test to ensure that both homogeneous samples, and t-test to show that both samples had the same initial ability. muhtarom, nizaruddin, nursyahidah, & happy, the effectiveness of realistic mathematics … 24 figure 1. research step 2.3. instrument and procedures the learning device developed include lesson plans, media, and rme-based modules. prior to use, the device has been validated by three validators. they conclude that the device is eligible to use, in condition provided the text size should be enlarged. furthermore, comments and suggestions from experts should be considered to improve the device so that the quality of media and rme module get better. the essay test is structured referring to the syllabus of mathematics subjects in the 2013 curriculum in which the solution uses visual, verbal and symbol representations. researcher uses rme-based long questions to measure students' multi-representation skills in experimental and control classrooms. prior to use, the test question has been validated by three experts, already said three lines above then tested to determine the reliability, level of difficulty and the differentiation of the item. the analysis of essay test instrument result is presented in table 2 which clearly indicates that there are five items used as pre-test and post-test in this study. table 2. analysis of essay test instrument question reliability difficulty level differentiation of item remark r criteria score criteria score criteria 1 0.65 reliable 0.93 easy 0.25 enough used 2 0.70 medium 0.59 good used 3 0.71 medium 0.43 good used 4 0.42 medium 0.45 good unused 5 0.29 difficult 0.54 good used students only master one skill representation the need of learning material that triggers multi-representation producing learning devices that improve students’ multirepresentation ability (verbal, picture and symbol) analysis design learning devices based on rme to improve students’ multi-representation ability learning device validation based on rme, reliability instrument test; differentiate item test and difficulty instrument test rme modules validation rme media validation questionnaire student’s response validation applying learning devices based on rme at school knowing the effectively of learning device based on developed rme development implementation evaluation volume 8, no 1, february 2019, pp. 21-30 25 question reliability difficulty level differentiation of item remark r criteria score criteria score criteria 6 0.24 difficult 0.50 good unused 7 0.36 medium 0.48 good unused 8 0.88 easy 0.39 enough used student responses are needed to analyze the readability of rme devices that have been made and to know how the students respond to the rme-based devices. quantitative data scoring obtained from the results of the questionnaires using likert scale. before the questionnaire was used it was validated by three validators who concluded that the questionnaire was worth using. the prerequisite test includes the normality test and homogeneity test, which aims to find out the statistical tests to be used in the data analysis process. parametric statistical tests are used if samples from classes with conventional learning and rme classes come from normally distributed populations, and the variance of both homogeneous groups. if the normality test requirement is not met, it will be used non-parametric statistical test. the t-test is used to find out whether there is a difference of mean of multirepresentation ability between rme class and conventional class. the data tested is the post-test result, in the following way: h0 : the mean of multi-representation ability of rme class is less than the average of conventional class. ha : the mean of multi-representation ability of rme class is better than the average of conventional class students are said to master learning if they get multi-representation ability at the value of 75, and mastery learning is classically met if at least 85% of all students complete the study. 75 is the minimum criteria of mastery learning (mcml) established by the school (hernawan, 2008). to calculate the improvement of students’ multi-representation skills before and after learning, it is calculated by the normalized gain formula (meltzer, 2002), namely: scoretestprescoreidealmaksimum scoretestprescoretestpost )g(gainn    the result of n-gain calculation then interpreted on table 3. table 3. n-gain representation (g) amount of n-gain (g) interpretation g ≥ 0,7 high 0,3 ≤ g < 0,7 medium g < 0,3 low (meltzer, 2002) after the questionnaire is completed by the students, then it is analyzed and counted in percent. to be able to provide meaning and decision making, the researcher uses the provision as an indicator of student responses presented in table 4. muhtarom, nizaruddin, nursyahidah, & happy, the effectiveness of realistic mathematics … 26 table 4. percentage range and student response criteria interval criteria 81% 100% very enthusiastic and interested 61% 80% enthusiastic and interested 41% 60% quite enthusiastic and interested 21% 40% less enthusiastic and interested < 21% not enthusiastic and interested (arikunto, 2010) 3. results and discussion 3.1. results table 5. normality test result learning strategy n lobs ltable hypothesis remark rme 32 0.149 0.157 h0 accept conventional 32 0.148 0.157 h0 accept table 5 presents that lobs < ltable, with α = 0.05 and n = 32. this means that sample from a class that uses conventional learning and a class that uses rme come from a normally distributed population. table 6. homogenity test result learning strategy n varians fobs ftable hypothesis remark rme 32 80.32 0.883 1.822 h0 accept conventional 32 71.35 table 6 shows that fobs = 0.883, and ftable = 1.822, therefore h0 is accepted. it can be concluded that both goup varians are homogen. table 7. the results from the t-test of the post-test scores learning strategy n mean tobs ttable hypothesis remark rme 32 80.78 3.296 1.99 9 h0 reject conventional 32 73.59 table 7 presents the result of t-test with the dependent variable is the students’ multi-representation ability. it is clear that there is a significant difference between the multi-representation ability of the students, where sp = 8.723, tobs = 3.296, with the value of v = 32 + 32 2 = 62 and α = 0.05, obtained t(0.05, 62) = 1.999; thus h0 is rejected. it means that the multi-representation ability of students who get rme learning is better than the multi-representation ability of students who get conventional learning. student learning mastery is seen from the pre-test score taken before the students are given rme study, while post-test value is taken after students are given rme learning. table 8 clearly shows the value of pre-test and post-test taken from both research classes. it is clear that the average post-test score is higher than the average pre-test. related to the achievement of student learning, in the class with rme the percentage of students who achieve mastery of 87.25% which means that almost all students complete the kkm. while in the conventional learning class percentage of students who achieve completeness only amounted to 53.125%. volume 8, no 1, february 2019, pp. 21-30 27 table 8. mean of pre-test, post-test, and mastery learning percentage learning strategy mean mastery learning percentage pre-test post-test rme 45.47 80.78 87.25% conventional 19.69 63.59 53.125% after getting the value of pre-test and post-test, then in each class is tested with ngain test which aims to see improvement of multi-representation ability of students. table 9 provides an overview of the multi-representation skills of students on rme learning. consider that the image representation increases by 0.74, the increase in verbal representation by 0.79 and the increase in symbol representation by 0.95 or the increase in the high category. table 9. improved students’ multi representation skill in rme class representation n-gain interpretation visual 0.74 high verbal 0.79 high symbol 0.95 high table 10. improved students’ multi-representation in the conventional class representation n-gain interpretation visual 0.50 medium verbal 0.25 low symbol 0.89 high table 10 gives the depiction of students’ multi-representation skill improvement in conventional learning. it is seen that the picture representation improvement amounted 0.50 or categorized as medium improvement, verbal representation improvement amounted 0.25 and symbol representation improvement as much as 0.89. while the comparison of multi-representation capability improvement of each class is presented in table 11. it shows that almost all students who get rme learning have improved their multirepresentation ability. table 11. improved students’ multi-representation learning strategy n-gain interpretation rme 0.80 high conventional 0.40 medium the rme learning implemented in the experimental class is equipped with rmebased tools. as already described that this device has been validated and feasible to use. the results of the assessment of 32 students who received learning with rme-based tools showed that 87.5% or 28 students stated very enthusiastic and interested, and 4 students stated quite enthusiastic and interested in learning with tools based on rme, thus increasing the learning spirit in the learning process. muhtarom, nizaruddin, nursyahidah, & happy, the effectiveness of realistic mathematics … 28 3.2. discussion our preliminary analysis indicates that many students only mastered the representation of symbols in a math problem. thus, teaching materials are needed that triggers the multi-representation abilities of students. this teaching material contains several representations (verbal, pictures and symbols) so that it is expected to be able to improve students’ multi-representation ability. the development of rme based learning tools that include lesson plans, modules based multi-representation, media, test description and student response questionnaires. the rme tools developed have been validated and declared feasible to be implemented in the learning process. the results showed that multi-representation ability in students who got rme better than the ability of multi-representation in students who received conventional learning. this indicates that the rme tools developed have been able to foster students' beliefs and confidence in the use of multiple representations to solve mathematical problems (nizaruddin et al., 2017). supporting the description is shown that 87.25% or almost all students who get rme learning with the developed device have completed the kkm as determined by the school that is the value of 75; this is inversely proportional mastery of students with conventional learning is only equal to 53.125%. further data show that many students are very enthusiastic and interested in rme based learning, thus increasing their learning spirit in learning process. furthermore, the implementation of rme has been able to improve the multi-representation ability of students, which is obtained by an increase of 0.8 with high category in the application of rme and only an increase of 0.4 in the moderate category on conventional learning. the number of representations students use to solve math problems is a reflection of their understanding of mathematical concepts and procedures (brenner et al., 1997). the learning process begins using symbol representation, then using verbal and visual representations that help students in translating their representations. this is in line with the opinion of keller & hirsch (1998); bransford & schwartz (1999); ainsworth (2006); and hwang et al., (2007) saying that the use of more than one representation can avoid the limitations of one type of representation so as to build student understanding. during the student learning process in groups, students actively solve math problems, feel enthusiastic and more challenged to do using some kind of representation. students actively develop an understanding of the concepts and their relationships so that they have multiple representational skills, and the multi-representation capabilities themselves can anticipate mistakes in understanding mathematical concepts (hwang et al., 2007). this is shown when one group presents the results of the discussion, the others actively respond to what has been described; so that the mutual process of cooperative knowledge formation can be realized. while the conventional learning process shows students passively receive the knowledge described by teachers and students do not do the construction of knowledge (muhtarom, juniati & siswono, 2017). thus, students have been able to make different representations because they possess good mathematical knowledge and have knowledge of the kinds of representations as well as the nature of the relationships between their chosen representations (janvier, 1987; nizaruddin et al., 2017; supandi et al., 2018), as was done during the process learning rme. 4. conclusion multi-representation-based rme is one of the main factors in improving students' ability in learning mathematics. this research shows the fundamental differences in the multi-representation skills of students who get rme lessons and students who get volume 8, no 1, february 2019, pp. 21-30 29 conventional learning. furthermore, rme learning by incorporating multi-representation is expected to be applied continuously by the teacher as an alternative in improving the quality of mathematics learning at school. teachers need to be encouraged to always trill the ability of the representation, so that students are challenged to elicit multirepresentation skills, especially in solving math problems. acknowledgements we would like to thank the rector of universitas pgri semarang (upgris) and the dean of the faculty of mathematics education, natural science, upgris. references ainsworth, s. 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(2017). the effect of realistic mathematics education approach on students’ achievement and attitudes towards mathematics. mathematics education trends and research, 1(1), 32-40. infinity journal of mathematics education p–issn 2089-6867 volume 7, no. 2, september 2018 e–issn 2460-9285 doi 10.22460/infinity.v7i2.p83-96 83 students’ performance skills in creative mathematical reasoning heris hendriana 1 , rully charitas indra prahmana 2 , wahyu hidayat 3 1,3 institut keguruan dan ilmu pendidikan siliwangi, jl. terusan jenderal sudirman, cimahi, indonesia 2 universitas ahmad dahlan, jl. pramuka kampus 2 unit b kav 5 pandeyan, yogyakarta, indonesia 1 herishen@ikipsiliwangi.ac.id, 2 rully.indra@mpmat.uad.ac.id, 3 wahyu@ikipsiliwangi.ac.id received: april 10, 2018; accepted: june 10, 2018 abstract this study aims to examine mathematics teacher-candidate students’ mathematical creative reasoning ability based on the level of adversity quotient (aq). this study uses a mixed method of sequential type by combining quantitative and qualitative methods in order. population in this study is all students attending the course of calculus in mathematics education of study program at stkip siliwangi that consist of 270 students divided into six classes. the results are aq gives effect to the achievement of students’ mathematical creative reasoning abilities based on the whole and the type of aq climber, champer, and quitter. the achievement of students’ mathematical creative reasoning abilities and based on aq, the champer and climber fall into the medium category, while on the quitter type, it falls into the category of low. on the other hands, the achievement of students’ mathematical creative reasoning abilities is yet to be achieved well at the indicator of novelty. keywords: adversity quotient, creative reasoning. abstrak penelitian ini bertujuan untuk menelaah kemampuan penalaran kreatif matematis mahasiswa calon guru matematika berdasarkan tingkat adversity quotient (aq). penelitian ini menggunakan metode kombinasi tipe sequential explanatory dengan menggabungkan metode kuantitatif dan kualitatif secara berurutan. populasi dalam penelitian ini adalah semua mahasiswa yang mengikuti program kalkulus di program studi pendidikan matematika stkip siliwangi yang terdiri dari 270 mahasiswa dibagi menjadi enam kelas. hasil penelitian menunjukkan bahwa aq memberikan efek pada pencapaian kemampuan penalaran kreatif matematis mahasiswa berdasarkan keseluruhan dan tipe aq climber, champer, dan quitter. pencapaian kemampuan penalaran kreatif matematis mahasiswa berdasarkan aq climber dan champer termasuk ke dalam kategori sedang, sementara pada jenis aq quitter termasuk ke dalam kategori rendah. di sisi lain, pencapaian kemampuan penalaran kreatif matematika mahasiswa belum tercapai dengan baik pada indikator kebaruan. kata kunci: adversity quotient, penalaran kreatif. how to cite: hendriana, h., prahmana, r. c. i., & hidayat, w. (2018). students’ performance skills in creative mathematical reasoning. infinity, 7(2), 83-96. doi:10.22460/infinity.v7i2.p83-96. hendriana, prahmana, & hidayat, students’ performance skills in creative … 84 introduction an environment conducive created to learning is essential to learners' academic achievement (visser, juan, & feza, 2015). furthermore, the purpose of learning mathematics in indonesia among others are to train one's understanding in thinking and reasoning and concluding, develop learners’ creativity through imagination, intuition, and the inquiry, and develop problem-solving and communication abilities (prahmana, kusumah, & darhim, 2017; soedjadi, 2000). on the other hands, numerous researchers have documented stated that the teaching profession has been recognized as key to improving the quality of education worldwide (acuña ruz, 2015; hendriana, hidayat, & ristiana, 2018; prahmana & kusumah, 2016; prahmana, kusumah, & darhim, 2017; weybright, caldwell, xie, wegner, & smith, 2017; young, 2017). school dropout is a crisis whereby grade 12, only 52% of the appropriate age population remain enrolled in south africa (weybright et al., 2017). the capabilities a person need in solving a problem be understanding of the concept (conceptual understanding), procedural fluency, strategic competence, adaptive reasoning, and productive disposition (hendriana, rohaeti & hidayat, 2017; kilpatrick, swafford & findell, 2001; runisah, herman & dahlan, 2017). it is why one of the capabilities that are considered important in the formation of one's mindset is mathematical reasoning ability. the reasoning is a pattern of thinking activity in drawing a conclusion or making a new statement based on some previously known statements that are considered correct. on the other hands, educational inclusion as an educational device that breaks with exclusionary practices and spaces in the educational system and the challenges that offer to pre-service teacher's formation and also 'controversial issues' are topics under which different groups have built irreconcilable arguments on (infante, 2010; toledo jofré, magendzo kolstrein, gutiérrez gianella, & iglesias segura, 2015). therefore, the reasoning abilities including educational inclusion are also needed in everyday life so that everyone in everywhere can respond and analyze any problems that arise in a comprehensive, critical, objective and logical way. the creative reasoning is a reasoning which emphasizes on a problem-solving process that includes novelty, plausible and based on mathematical foundation (lithner, 2008; fathurrohman, porter & worthy, 2017). in south africa, mathematics mastery is a growing concern (kotzé, 2007). bergqvist (2007) suggested a framework for mathematical reasoning as follows in figure 1. what is meant by creative reasoning is a type of reasoning that finding a solution to a mathematical problem is not only conducted by imitating solution such as samples of exercise and item tests contained in the textbooks as well as considering the algorithm or the steps of a solution? volume 7, no. 2, september 2018 pp 83-96 85 figure 1 framework of mathematical reasoning the consistent attitude of a person in teaching-learning process is one determinant of success so that it can harmonize his or her attitudes and behavior to reach the expected goals (robbins, 2010). furthermore, syah (2010) also stated that there are several factors determining the success of one's learning in mathematics, namely, internal & external factors, and learning approaches. one part of students’ internal factors is adversity quotient (aq). stoltz (2004) argued that adversity is one of the difficulties faced by someone so that there are some who have broken the spirit to face and solve the challenge. in the meantime, aq is one’s persistence in facing all obstacles to achieve success. also, stoltz (2004) also suggested that aq has four key dimensions which form the basis of aq’s measuring tool. first, control is that a person's response to adversity, either slow or spontaneity. second, origin and ownership are that the extent to which a person feels can improve the situation. third, coverage (reach) is that the extent of the difficulties encountered in life effects. last, endurance is that reflects how a person perceives his predicament and can persist through those difficulties. aq is the predictor of success of a person in the face of adversity, how he behaves in a tough situation, how he controls the situation, is he able to find the correct origin of the problem, whether he takes his due ownership in that situation does he try to limit the effects of adversity and how optimistic he is that the adversity will eventually end (phoolka & kaur, 2012). aq related to the level of a person. there are three types or levels, namely the climber (high), the camper (medium) and a quitter (low). students who have high levels of climber would be able to overcome the difficulties faced but must still address in a way given the additional task of enrichment. in addition to the person who has a level of climber, aq can also tap as a peer tutor to his friends who have a camper and quitter aq level. method the purpose of this study is to determine and analyze in depth about the mathematics teachercandidate students’ mathematical creative reasoning abilities based on the level of adversity quotient (aq). this study uses a mixed method of the following type by combining quantitative and qualitative methods in order. the first stage of this research, a quantitative method, is conducted to obtain measurable data. the second phase, a qualitative method, is mathematical reasoning creative reasoning imitative reasoning global creative reasoning local creative reasoning algorithmic reasoning memorized reasoning hendriana, prahmana, & hidayat, students’ performance skills in creative … 86 carried out to explore the findings obtained from the first stage (prahmana et al., 2017). the materials used are linear inequality and absolute value. from the data analysis, it will result in the creative mathematical reasoning skill of the students related to adversity quotient level namely the climber (high), the camper (medium) and a quitter (low). the research population in this study was all students attending the course of calculus in mathematics education of graduate study program stkip siliwangi in the academic year of 2016/2017. the total members of the population consist of 270 students divided into six classes. clusters random sampling uses to determine the experimental class as a sample. of the six available classes, one class, a2, is selected to be the experimental one. the results of measurements of creative mathematical reasoning skill and the categorization of adversity quotient, both are combined to determine subject that meets the predefined categories. the data analysis was done based on topics which fit the category. besides, we further conducted interview towards students selected as a representative of each level in the process of creative mathematical reasoning skill. it is done to delve into some of the constraints associated with delve and their adversity quotient categories respectively. results and discussion the findings regarding creative mathematical reasoning skills of students reviewed by the adversity quotient (aq) presented in table 1. table 1. students’ mathematical creative reasoning ability based on aq type of adversity quotient mean sd n climber 6,204 (62.04 %) 1,111 9 champer 6,111 (61.11 %) 0,894 12 quitter 4,815 (48.15 %) 0,694 9 total 5,750 (57.50 %) 1,080 30 notes: ideal score: 10 based on the description in table 1, the interpretation obtained is that the development of students’ mathematical creative reasoning skills, both overall or by type of adversity quotient (aq) at the type of climber and champer, fall into the category of the medium. however, for the type of adversity quotient (aq) at the quitter types, it falls into the category of low. regarding supporting the description of students’ mathematical creative reasoning skills that describe in table 1, it is necessary to analyze data regarding students’ mathematical creative reasoning skills through the mean test. after testing the normality of the data distribution of students’ mathematical creative reasoning skills, it finds that the data normally distribute. based on these findings, one-way anova uses to calculate the mean test of the ability above (table 2). volume 7, no. 2, september 2018 pp 83-96 87 table 2. the summary of one-way anova test of students’ mathematical creative reasoning ability based on the type of aq source jk dk rjk f value sig adversity quotient (aq) 11.271 2 5.635 6.754 0.004 inter 22.527 27 0.834 it appears that the significance level of 5%. the adversity quotient (aq) has a significant influence towards the achievement of students’ mathematical creative reasoning skills (table 2). the skill is evident from the value of obtained sig which is 0.004. so, to see which type of aq is significantly different, then the post hoc test is conducted through scheffe test, presented in table 3. table 3. scheffe test of the achievement of students’ mathematical creative reasoning ability based on the type of aq type of aq (i) type of aq (j) mean difference (i – j) sig interpretation climber champer 0.0925 0.974 not different champer quitter 1.2953* 0.013 different climber quitter 1.3878* 0.012 different based on table 3, it concluded that at a significance level of 5%. there are significant differences between the students’ mathematical creative reasoning ability at the type of aq climber & quitter with champer with quitter compared to aq type of climber and champer. the implication is that the students’ mathematical creative reasoning skills at aq type of quitter have developed more than aq at the type of climber and champer. the achievement of the students’ mathematical creative reasoning skills based on the indicators of novelty, plausible, and mathematical foundation presented in table 4. table 4. the indicators achievement of the students’ mathematical creative reasoning ability based on aq item number indicators of mathematical creative reasoning mathematical foundation plausible novelty climber 30.56 % 52.78 % 16.67 % champer 27.08 % 62.50 % 10.42 % quitter 58.33 % 38.89 % 2.78 % total 37.50 % 52.50 % 10.00 % overall, the achievement of students’ mathematical creative reasoning ability that includes novelty (the ability of plausible and mathematical foundation) is still low. it can be seen from the percentage of achieved novelty indicator that reached 10%. while the plausible indicators (including mathematical foundation capabilities) reached 52.50% and ability that only hendriana, prahmana, & hidayat, students’ performance skills in creative … 88 includes the mathematical foundation reached 37.50%. it shows that there is still some students (37%) who solved problems based on mathematical foundation but cannot provide a reasonable excuse which has novelty. similarly, based on the overall, the students’ mathematical creative reasoning skills students based on the type of adversity quotient (aq) climber and champer are still likely high at the ability of plausible (which also includes the ability of mathematical foundation) which for the climber is 52.78% and for champer is 62.50. as for the ability of novelty (which includes the ability of plausible and mathematical foundation), it is still significantly less compared to those who only have the ability mathematical foundation. in contrast to the type of quitter at the type of adversity quotient (aq), which indicates that the ability of only mathematical foundation (of 58.33%) is higher than plausible capability (which also includes the ability of mathematical foundations) amounting to 38.89% and novelty (which also includes the ability of mathematical foundation and plausible) amounting to 2.78%. it shows that students who have ag of quitter type are still having trouble in solving problems by providing a reasonable excuse which has a novelty. it is clear that the students still have difficulties to solve the problems based on reasons that make sense (plausible) and which have a novelty. it is also evident from the problem-solving process is done by them presented in figures 2, 3, 4, 5, 6, and 7. based on the description of the students’ mathematical creative reasoning skills, it seems that students have a tendency to solve problems that are given only through the routine procedures (usually done by the lecturers), but it fell into the category of reasonable (plausible) and based on mathematical foundation. however, there are also some students who master mathematical creative reasoning capabilities that include novelty. it is evident from the students’ answers in figure 2. figure 2. the results of the students’ work which belong to the novelty and plausible of mathematical creative reasoning ability volume 7, no. 2, september 2018 pp 83-96 89 it is different from the majority of the other students’ work who answered through the solving problems process in figure 3, 4, and 5. after the interview to students regarding the results of their work conducted, the researchers conduct an interview process to gain a deeper understanding of the work that they already did. the results of the interview transcript, based on figure 2, talk about student who has mathematical creative reasoning abilities with novelty indicator presented as follows: lecturer : why did you do with a move like that? student 1 : i did this work with based on the definition of absolute values. lecturer : but if it is linked to the work of other friends who are equally based on absolute values, there is still a difference in the final settlement. what underlies your thoughts to use the concept of an incision in the inequality (1), (2) and (3) and described it on the number of the line. (shown in figure 2). student 1 : it is based on intuition, and i assume that just because the problems are about the inequality, then i think to describe it on the number of lines and an incision in the inequality (1), (2) and (3). however, although i do the work intuitively, i feel that the answers i give are reasonable and correct. based on interviews conducted by the student 1, it can be concluded that the creative reasoning which falls into the category of novelty is the process of problem-solving activity done by a person which he or she thought to be a new thing. figure 3. results of student work which are classified as plausible on creative reasoning ability the results of the interview transcript, based on figure 3, talk about student who has mathematical creative reasoning abilities with plausible indicator presented as follows: hendriana, prahmana, & hidayat, students’ performance skills in creative … 90 lecturer : do you believe that you are doing it right? student 2 : i believe that what i am doing is correct. lecturer : what underlies that the work you are doing is correct? student 2 : i based on the process and steps were taken by the steps on the definition of absolute value, but when connecting its inequality, i associate the limits stated in absolute value with its inequality. for example, i associate -5x + 2 to limit x <0 the inequality "<7" so -5x + 2 <7. then, it produced x> -1 and after that, i looked at the cut between x <0 and x> -1. so, it was obtained -1 <x <0. i did that on the boundary 0 ≤ x <1 and x ≥ 1. moreover, in the end, i made a combination of all limitations that met the value of x and produced the answers -1 <x <9/5. lecturer : why did you base on such a step? student 2 : i did that by discussions in groups, and the form of such items has once ever been described by my former high school teacher who taught inequalities in absolute values. lecturer : is the solving process of such problems a new thing to you? student 2 : it is not a new thing, but i have to make modifications in the process and procedure completion. based on interviews conducted by the student 2, it can be concluded that the creative reasoning which falls into the category of plausible is the process of problem-solving activity done by a person that he or she thought is reasonable in the process of solving it but it is not routine work. also, a problem-solving process conducted can be done procedurally through the concepts that he or she has in mind using any possible modification. figure 4. result of student work classified as novelty but it has errors on the answers given the results of the interview transcript, based on figure 4, talk about student who has mathematical creative reasoning abilities with novelty indicators but there are some errors in the final settlement presented as follows: lecturer : do you believe that you are doing it right? volume 7, no. 2, september 2018 pp 83-96 91 student 3 : i believe that i am doing is correct. lecturer : what makes you so sure that you are doing it correctly? student 3 : on the problems you gave to me, it was previously assumed that my answers were considered correct. the solving process of these problems is as follows in figure 5. figure 5. results of student work on previous problems lecturer : why did you do such steps, so that the final settlement is like that? student 3 : on the final settlement i assume that the inequality (1) and (2) meets the same x value, namely, the first x value is x <6 and the second value of x> 2. thus the definition is based on the absolute value: | x | = x, x ≥ 0 (for which meets the positive x value), and | x | = x, x <0 (for which meets the negative value of x). so i conclude that: 2 <x = x <6 2 <x <6. lecturer : can the solving process on the problem of 3 | x 4 | <6 that you did be applied to the problem of 3 | x | + 2 | x 1 | <7? student 3 : i think it could be applied, sir. lecturer : have you rechecked the answers that you produced? student 3 : i did not check it, sir. lecturer : try to check the results of the answer to the problem of 3 | x | + 2 | x 1 | <7? then, the student 3 examines the results of the answer, and the obtained results are in the following figure 6. hendriana, prahmana, & hidayat, students’ performance skills in creative … 92 figure 6. results of the recheck on students’ answers lecturer : how is the result of the answers recheck that you produced on the problem 3 |x| + 2 |x – 1| <7? student 3 : it turns out that for x = 2 does not meet the inequality of 3 | x | + 2 | x 1 | <7. so, i admit that my answer was incorrect. based on interviews conducted by the student 3, it can be concluded that the correctness of creative reasoning on the category of novelty and plausible needs reviewing. so, the foundation of creative reasoning ability is the truth of the conclusions obtained, although the conclusion is obtained from the mathematical foundation, plausible and novelty. figure 7. results of student work categorized only on mathematical foundation in creative reasoning ability the results of the interview transcript, based on figure 7, talk about student who has reasoning abilities but do not belong to the creative mathematical reasoning presented as follows. volume 7, no. 2, september 2018 pp 83-96 93 lecturer : are you convinced of the correctness of completion that you produced? student 4 : i believe that the answer i give is correct. lecturer : what is the basis for your belief that the answer you gave is correct? student 4 : what underlies me is that the work i did is by the settlement algorithms contained in textbooks. based on interviews conducted by students 4, it can be concluded that he does not have a creative reasoning ability yet; he only has imitative reasoning ability. it is in line with lithner (2008) who found that imitative reasoning is a reasoning process where someone goes through the stages of memorizing or follows an algorithmic process which is based on textbooks. the research results show that the students can make the process of reasoning quite well. it is evident from the completion process by the students containing logical statements which can defend the truth of the results of the settlement. ponte, pereira and henriques (2012) support the result that state who argued that mathematical reasoning is a process to arrive at conclusions derived from solving problem process. also, it is in line also with velez & ponte (2013) who described that mathematical reasoning is a logical statement derived from the propositions of problems given and followed by formulating and testing assumptions from certain cases so that it can produce general conclusions. if we analyze in depth, the results of research explain that the students’ reasoning process still categorize as heterogeneous thinking. it seen that they resolve the problems through algorithm procedures that can be memorized routinely. the activity shows that the students’ mastery of creative reasoning is still relatively weak. it is in line with the opinions of bergqvist (2007) who also suggested that creative reasoning which includes routine procedures in the process of problem-solving is a low creative reasoning. if the students’ completion procedures in solving problems are always conducted in the same way, it is feared that the completion process will be classified into imitative reasoning abilities. the results supported by lithner (2008) that state who found that imitative reasoning consists of memorized and algorithms reasoning. memorized and algorithm reasoning look no different because the procedures done by the students in performing problem-solving strategies looks the same (figueroa & aillon, 2015; hidayat & prabawanto, 2018; lithner, 2008). the results of the study on students’ mathematical creative reasoning skills based on grouping of aq show that students with aq type quitter often encounter difficulties and easily give up in solving the problems given; in contrast with students with aq camper, and climber. it supported by phoolka & kaur (2012) who argued that the aq be a supporting factor for someone to reach success of a person in dealing with difficulties; how he or she behaves in difficult situations, how he or she can control the situation, how he or she is able to find quite well the basic appearance of a problem, whether or not he or she can take a role in resolving the problems on that conditions and situation, and whether or not he or she can deal with the difficulties and how he or she can remain optimistic and confident that the difficulties will end. also, parvathy & praseeda (2014) also suggested that a person who has good aq will be able to survive in facing difficulties in learning mathematics. conclusion the adversity quotient (aq) gives effect to the achievement of the students’ mathematical creative reasoning abilities based on the overall type of adversity quotient (aq), namely hendriana, prahmana, & hidayat, students’ performance skills in creative … 94 climber, champer and quitter. the achievement of the students’ mathematical creative reasoning skills based on the overall and type of adversity quotient (aq) of champer climber type falls into the category of the medium, while the quitter type falls into the category of low. the achievement of the students’ mathematical creative reasoning skills, on indicators of novelty (which also includes the ability of plausible and mathematical foundation), has not been achieved quite well. it makes the urgency of the problems that can be solved through some efforts in the form of innovative learning which can result in a meaningful learning activity. finally, it is highly expected that the innovative learning can improve students’ adversity quotient so that it directly gives a good effect on students’ mathematical creative reasoning skills. references acuña ruz f. 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