International Journal of Interactive Mobile Technologies (iJIM) – eISSN: 1865-7923 – Vol. 15, No. 05, 2021 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial and Problem-Solving Skills https://doi.org/10.3991/ijim.v15i05.19037 Muhamad Ikhsan Sahal Guntur (), Wahyu Setyaningrum Yogyakarta State University, Yogyakarta, Indonesia ikhsan.guntur@gmail.com Abstract—The researchers implemented the quasi-experimental method of research. This research was conducted at SMAN 1 Ngemplak, Indonesia. The research was conducted from March to April 2020. In this study, the samples consisted of 70 students divided into two classes, namely Class A and Class B. The students in Class A received treatment by using the Augmented Reality Mod- ule (ARM). Meanwhile, Class B students were taught conventionally using a con- ventional module from the government and GeoGebra application. The instru- ments used in this study were written tests, the first test was in the form of mul- tiple choices and consisted of 25 questions to measure students’ spatial skills, and the second test was in the form of essays, consisted of five questions in order to measure students’ problem-solving skills. The reliability of the tests had been tested before collecting the data. The data were analyzed using descriptive statis- tics and inferential statistics (normality test, homogeneity test, paired t-test, and independent t-test). Based on the analyses' results, it can be concluded that Aug- ment-ed Reality Module has a significant effect on students' problem-solving and spatial skills since there is an improvement in students' mean scores in the pre- test and post-test results. Besides, it can be concluded that Augmented Reality learning is more effective than conventional learning since the students are usu- ally taught using conventional modules from the government and GeoGebra ap- plication. Keywords—Augmented Reality, Spatial Skills, Problem-Solving Skills 1 Introduction Mathematics plays a fundamental role in a child's development and communication skills in future life. Necessary numeracy skills play a role in developing children's learn- ing outcomes in order to be competent adults. A broader acknowledgment can be de- veloped with other works of literature. However, building a robust Mathematical foun- dation in early childhood is critical for children's future educational success and their economy, health, and career [1]. Children who enter school with strong Mathematics skills tend to have a higher chance of success, starting from kindergarten to the next levels. iJIM ‒ Vol. 15, No. 05, 2021 159 https://doi.org/10.3991/ijim.v15i05.19037 mailto:ikhsan.guntur@gmail.com Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… One of the strategies in improving students’ Mathematics skills is by utilizing tech- nology in the learning process [2][3]. Technology utilization in the classroom can be applied using smartphones, tablets, and laptops [4] [5]. Previous research revealed that it could improve students' Mathematics skills rather than laptops by utilizing smartphones and tablets [6]. Guntur stated that one of the smartphone features that can improve students’ Mathematics skills is Augmented Reality [7]. Augmented Reality (AR) is a technology designed to combine virtual objects into real objects. Thus, the virtual objects blend into the real things in the display [8]. AR has great potential to be developed since it does not involve any sophisticated tool or expensive hardware. AR can be accessed on a tablet or smartphone, and it can be used in each level of education [9][10]. According to Wu, AR's positive effects in improving students' Mathematics skills had been identified in previous research, such as developing students' skills and knowledge, enhancing students' experiences, and enhancing collaboration skills [11]. Using AR in Mathematics learning can improve students' understanding of abstract concepts [12][13]. Besides, Lee stated that Augmented Reality plays a role in enhancing students' understanding, presenting the teacher's material, making the while teaching efficient, and creating a new and interactive atmosphere in the Mathematics learning process [14]. Moreover, several cognitive skills should be possessed by students in learning Math- ematics. According to NCTM, one of them is a problem-solving skill [15]. Problem- based learning should be implemented in Mathematics learning at school since it is the fundamental core in learning Mathematics [16]. Besides problem-solving skills, the students should possess spatial skills and be ac- commodated in the learning process. This is in line with Kaufman's previous research, revealing that AR plays a role in improving students' spatial skills [17]. Also, AR plays a role in stimulating students' interest in learning and improving students' understanding of Geometry [18]. According to previous researches, the studies on exploring Augmented Reality in the Mathematics teaching and learning process, especially in improving students' problem- solving skills and spatial skills, are still limited. Therefore, this research was conducted to identify the Augmented Reality Module's effectiveness in improving students' spatial skills and problem-solving skills. This study's results are expected to be a reference in determining alternatives and innovations in learning mathematics to improve students' spatial skills and problem-solving skills. 2 Literature Review This section discusses several important and related topics in previous literature, such as Problem-Solving Skills, Spatial Skills, and Augmented Reality. 160 http://www.i-jim.org Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… 2.1 Problem-solving skills The essential core of learning mathematics is to solve a problem (problem-solving) [19, p. 400]. Problem-solving skills should be taught in school to improve students' understanding of other subjects [20]. NCTM categorizes problem-solving skills as the main focus in Mathematics learning in school [21]. According to [22, p. 112], when problem-solving is categorized as the focus in teaching and learning skills, it can be applied using several strategies. These strategies include teaching for solving a prob- lem; delivering the materials related to problem-solving basic concepts, such as knowledge, understanding, and skills that can be applied in solving problems; focusing on students' process in solving a problem, and applying problem-solving skill as the strategy that can help students in solving any problems [23][24]. As stated by [15], problem-solving skills are categorized as the essential core com- petencies in the mathematics learning process. Problem-solving skills can also be con- sidered as a process in seeking the combination of several rules that can be applied to solve any situation [25]. Hence, in order to comprehend Mathematics, problem-solving skills should be taught. [19] The curriculum, tasks, or activities related to problem- solving should be developed in the teaching and learning process, and the learning out- come will have resulted from the problem-solving process. Problem-solving skills are the main components in the Mathematics curriculum and Mathematics necessary skills [26]. Hence, based on previous literature, it can be concluded that problem-solving is the main objective in learning Mathematics, and it should be taught in school to improve students' mathematics skills. 2.2 Spatial skills Linn & Petersen [27] stated that spatial skills refer to someone’s skills in simplifying, modifying, producing, and memorizing non-linguistics symbols. Meanwhile, Van de Walle [19] stated that spatial skill could be defined as an intuition about shapes and the relationships among figures. Besides, Mulligan said that spatial skills refer to some- one’s skill to recognize and manipulate (in thought) the spatial characteristics of an object and the spatial correlation between objects [28]. Spatial skills are the skills of solving spatial problems by using the perception of two and three-dimensional shapes and the understanding of the perceived information and relations [29]. Sarama & Clem- ents stated that there are five components in spatial skills, such as Spatial Perception, Spatial Visualization, Mental Rotation, Spatial Relation, and Spatial Orientation [30] 3 Method The researchers implemented the quasi-experimental method of research using a nonequivalent pretest-posttest design. This study's independent variable was the Aug- mented Reality Module, and the dependent variables were students' spatial skills and problem-solving skills. iJIM ‒ Vol. 15, No. 05, 2021 161 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… 3.1 Samples of the study This research was conducted in SMAN 1 Ngemplak, Indonesia. This research was conducted from March to April 2020. 2 (two) classes were selected by applying random sampling. After that, two classes were selected for receiving the treatments. Students in Class A received treatment by using the Augmented Reality Module. Meanwhile, Class B students were taught conventionally using a conventional mod- ule from the government and using Geogebra as the learning medium. The teacher used the 3D GeoGebra application since it could be compared with the Augmented Reality Module. The samples consisted of 70 students, while each class consisted of 35 stu- dents. 3.2 The steps of the research The steps conducted in this research were: compiling the research instruments (Les- son Plan, Augmented Reality Module, Pretest, and Posttest for each variable, and scor- ing rubrics for each variable); conducting validity test for the instruments by involving two expert judgments, conducting reliability test for the instruments, conducting pre- liminary studies and asking for stakeholders’ permission in the school, conducting pre- test, conducting research, conducting post-test, analyzing data, and drawing conclu- sions. 3.3 Research instrument Several tests were conducted to collect data on students’ problem-solving and spatial skills before receiving the treatment. The test on measuring students’ problem-solving skills consisted of 5 questions in the form of essays, and the test on measuring students' spatial skills consisted of 25 questions in the form of multiple choices. The reliability of the test has been tested by using Cronbach Alpha. The result of the reliability test is shown in Table 1. Table 1. Reliability test Spacial Skill Test Problem Solving Skill Test Cronbach’s Alpha N of Items Cronbach’s Alpha N of Items 0.921 25 0.628 5 Based on Table 1, Cronbach's Alpha value was higher than 0.6. Hence, the test is categorized as reliable [31]. The test's validity had also been tested by expert judgments, such as two Mathematics lecturers. The result of the validity test indicated that the in- strument was categorized as strongly valid. The three experts also validated the module used in the experimental class, and the AR module was feasible to be used. Meanwhile, students in the control class were taught using the government's conventional module, and the validity test was not needed. Since the technology used in the control class is the 3D GeoGebra application, it has been suitable to be compared with the other class to identify each medium's effectiveness. 162 http://www.i-jim.org Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… 3.4 The technique of analyzing data The data were analyzed using descriptive statistics and inferential statistics. The data descriptions are arranged by describing the real data, such as calculating the mean score, the highest score, the lowest score, and standard deviation of the data obtained before and after receiving the treatment. This study's inferential statistical analysis was a t-test, which consisted of 3 tests that aimed to test 3 different hypotheses. The first is a paired t-test to determine the Aug- mented Reality module's effectiveness in improving students' spatial skills and prob- lem-solving skills, evaluated from pre-test and post-test results. Then, for the second time, a paired t-test was used to find the effectiveness of conventional learning methods in improving students' spatial skills and problem-solving skills, evaluated from the pre- test and post-test results in the control class. Third, an independent t-test was used to identify any difference in students' mean scores based on the post-test results to measure students' spatial skills and problem-solving skills in each class. The decision-making and conclusion-drawing were at the significance level of ( 0.05 2 = 0.025), while the decision rule is if the significance value is more significant than 0.025, it meets the normality assumption. Meanwhile, if there is any difference in the mean score, it should be identified whether it is negative or positive to identify which class shows a better result. Univariate normality analysis was conducted by using the Shapiro-Wilk test. The decision-making and conclusion-drawing were at the signif- icance level of 0.05, while the decision rule is if the significance value is more signifi- cant than 0,05, then it is met the normality assumption. In order to identify the homo- geneity of the variants, Levene-Test was used. The decision-making and conclusion- drawing on the hypothesis test were at the significance level of 0,5. The decision rule is if the significance value is more significant than 0.05., then it is met the matrix vari- ance of covariance assumption and the homogeneity of the variance. 4 Result and Discussion In learning Vector using Augmented Reality Module, the activities were conducted based on the lesson plan. The data were collected by conducting pre-test and post-test to identify students' problem-solving skills and spatial skills. The results of the tests were analyzed and presented as descriptions before conducting an inferential statistics test. 4.1 Result The research results are processed and presented in descriptions to give an initial view for the researchers. The descriptions of the data of students’ problem-solving skills in the three classes are shown in table 2. iJIM ‒ Vol. 15, No. 05, 2021 163 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… Table 2. Problem Solving Skill Data Description N Min Max Mean Std. Dev PRE_ PSS _ARM 35 40 85 63.9714 12.084 POST_PSS_ARM 35 66 100 85.9714 9.0861 PRE_ PSS _MC 35 50 85 66.0857 10.268 POST_ PSS _MC 35 68 90 79.0571 6.3660 Valid N (listwise) 35 In the description, PRE_PSS_ARM is students’ score of problem-solving skills pre- test in the class taught using Augmented Reality Module. POST_PSS_ARM is stu- dents’ score of problem-solving skills post-test in the class taught using the Augmented Reality module. PRE_PSS_MC is students’ score of problem-solving skills pre-test in the control class, and PRE_PSS_MC is students’ score of problem-solving skills in the control class. Based on Table 2, it can be identified that students’ mean score on problem-solving skills in ARM and Control class before receiving treatment was below the average value (75), since students’ mean score of pre-test in ARM class was 63.97, and in the Control class was 66.08. However, after receiving treatment, students’ mean score of post-test in ARM class is 85.97, or greater than 75. Meanwhile, students’ mean score of post- test in Control class is 79.05. As a result, it indicates an improvement based on the pre- test and post-test results on students’ problem-solving skills variable. The descriptions of students' spatial skills in the Mathematics learning process in three classes are shown in Table 3. Table 3. Description of spatial skill data N Minimum Maximum Mean Std. Dev PRE_SS_ARM 35 54 84 71.82 8.7194 POST_SS_ARM 35 66 100 84.62 10.020 PRE_SS_MC 35 50 88 67.31 11.190 POST_SS_MC 35 55 100 75.6 13.737 Valid N 35 Based on Table 3, the results of the Spatial skills test indicate that students' mean score in ARM class and Control class before receiving the treatment was below the average score (75), while students' mean score in ARM class was 71.82, and students' mean score in Control class was 67.31. Meanwhile, after receiving treatment, their mean score is increased above 75, while students' mean score in ARM class was 84.62, and students' mean score in the Control class was 75.6. As a result, the result of descrip- tive statistics indicated an improvement in each variable. However, to ensure the con- clusion of the hypothesis, the inferential statistical test was conducted. The first step was conducting a normality test and continued testing the data's homogeneity using an independent t-test. Furthermore, the normality tests were conducted for four classes using the Shapiro- Wilk test. The normality tests were conducted for two classes and each variable, both in pre-test and post-test. The results of the normality tests are shown in table 4. 164 http://www.i-jim.org Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… Table 4. Normality test with Shapiro-Wilk test Statistic df significance. POST_SS_ARM .943 35 .068 POST_SS_MC .939 35 .053 PRE_SS_ARM .939 35 .052 PRE_PSS_ARM .963 35 .275 POST_PSS_ARM .939 35 .052 PRE_SS_MC .947 35 .089 PRE_PSS_MC .945 35 .078 POST_PSS_MC .948 35 .099 Table 4 indicated that the whole data's significance value (in all classes) is greater than 0.05. Hence, it is met with the normality assumption. Furthermore, the homogene- ity test for problem-solving skills post-test in ARM class and control class and homo- geneity test for spatial skills post-test in ARM class and control class were conducted using independent t-test. The results of the homogeneity test in each class are shown in Table 5. Table 5. Test of homogeneity of variances Levene Statistic Df1 Df2 Sig. PSS 0.074 1 68 0.786 SS 3.199 1 68 0.078 In the description, SS is Spatial Skills, and PSS is Problem-solving Skills. Table 5 indicates that from the result of the Levene-test, the significance value is more signifi- cant than 0.05. Hence, it is met the homogeneity assumption of the variance, both in ARM class and Control class. The results of ARM's effectiveness in improving students’ problem-solving skills and spatial skills are shown in Table 6. Table 6. Paired samples t-test in ARM class Mean Std. Dev t df sig PRE_SS_ARM - POST_SS_ARM -12.8 14.83597 -5.104 34 .000 PRE_PSS_MC - POST_PSS_ARM -22.000 15.60354 -8.341 34 .000 Based on table 6, the sig (2 tailed) p-values of PRE_SS_ARM - POST_SS_ARM is lower than 0.025. Hence, H0 is accepted. In other words, the hypothesis "there is any significant difference between students' mean score in spatial skills pre-test and post- test in the class that is taught using ARM" is accepted. Furthermore, based on the neg- ative mean difference value, 𝜇2 > 𝜇1, while 𝜇2 is the post-test, and 𝜇1 is the pre-test. Hence, it can be concluded that there is an improvement in students' spatial skills taught using the Augmented Reality Module. Besides, based on table 6, the sig (2 tailed) p-values of PRE_PSS_ARM - POST_PSS_ARM is lower than 0.05. Hence, H0 is accepted. It can be concluded that the hypothesis "there is any significant difference between students' mean score in iJIM ‒ Vol. 15, No. 05, 2021 165 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… problem-solving skills pre-test and post-test in the class that is taught using ARM" is accepted. Furthermore, based on the negative mean difference value, 𝜇2 > 𝜇1, while 𝜇2 is the post-test, and 𝜇1 is the pre-test. Hence, it can be concluded that there is an improvement in students' problem-solving skills taught using the Augmented Reality Module. Table 7. Paired samples t-test in the control class Mean Std. Dev t Df sig PRE_SS_MC - POST_SS_MC -8.2857 18.47028 -2.654 34 0.012 PRE_PSS_MC - POST_PSS_MC -12.971 12.68275 -6.051 34 0.000 Based on table 7, the sig (2 tailed) p-values of PRE_PSS_MC - POST_PSS_MC are lower than 0.025. Hence, H0 is accepted. It can be concluded that the hypothesis "there is any significant difference between students’ mean score in spatial skills pre-test and post-test in the control class" is accepted. Furthermore, based on the negative mean difference value, 𝜇2 > 𝜇1, while 𝜇2 is the post-test and 𝜇1 is the pre-test. Hence, it can be concluded that there is an improvement in students’ spatial skills in the control class. In addition, based on table 7, the sig (2 tailed) p-values of PRE_PSS_MC - POST_PSS_MC are lower than 0.025. Hence, H0 is accepted. It can be concluded that the hypothesis "there is any significant difference between students’ mean score in problem-solving skills pre-test and post-test in the control class" is accepted. Further- more, based on the negative mean difference value, 𝜇2 > 𝜇1, while 𝜇2 is the post-test, and 𝜇1 is the pre-test. Hence, it can be concluded that there is an improvement in stu- dents’ problem-solving skills in the control class. Since it has been known that there is a difference in students' mean scores based on pre-test and post-test results, an independent t-test was conducted in order to identify which class has better learning outcomes, assessed from the post-test results on prob- lem-solving skills and spatial skills. The results of the independent t-test are shown in Table 8. Table 8. Independent T-test between ARM post-test and MC post-test Mean Std. Deviation t df sig Spatial Skill 9.0285 2.87419 3.141 68 .002 Problem-solving skills 6.9142 1.87528 3.687 68 .000 Based on Table 8, the sig (2 tailed) p-value on spatial skills is lower than 0.025. Hence, H0 is accepted. In other words, it can be concluded that the hypothesis “there is any significant difference between students’ mean score in spatial skills post-test result in ARM class and MC class” is accepted. Furthermore, based on the positive mean difference value, µ1 is better than µ2, while 𝜇2 is the control class post-test, and 𝜇1 is the ARM class post-test. Based on table 8, the sig (2 tailed) p-value on problem-solving skills is lower than 0.025. Hence, H0 is accepted. In other words, it can be concluded that the hypothesis “there is any significant difference between students’ mean score in problem-solving skills post-test result in ARM class and MC class” is accepted. Furthermore, based on 166 http://www.i-jim.org Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… the positive mean difference value, µ1 is better than µ2, while 𝜇2 is the post-test for the control class, and 𝜇1 is the post-test for the ARM class. 4.2 Discussion Based on Table 6 and Table 7, it can be concluded that there is no improvement in both classes, and both on the spatial skills and problem-solving skills. The hypothesis test result reveals that conventional approaches effectively improve students' problem- solving skills and spatial skills. In identifying students' problem-solving skills in conventional classes, it can be con- cluded that the improvement is due to the use of a module published by the government since the competencies related to problem-solving skills contained in Indonesia's cur- riculum. Meanwhile, Augmented Reality Module has been designed for improving stu- dents' problem-solving skills. The differences between both modules are in the indica- tors of problem-solving skills, such as understanding the problem, making plans, and conducting evaluation in each previous stage [32]. Besides, in the learning process, the AR module plays a role in stimulating and attracting students' interest in solving prob- lems since the AR module's appearance is more exciting and challenging (Wasko, 2013). Figure 1 indicates the comparison between the Augmented Reality Module (MAR) test and the module used by students of the control class (MC). Fig. 1. The Comparison of the Test Meanwhile, in identifying students’ spatial skills improvement in the control class, the teacher’s strategy in using Geogebra application in the learning process is due to the teacher's strategy. Geogebra application is the latest version, and it presents three dimensions objects that can stimulate students' spatial skills [34]. The difference be- tween these two technologies is: even though the first application (Geogebra) is in the iJIM ‒ Vol. 15, No. 05, 2021 167 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… three-dimensional form (3D), but the students cannot see the whole objects from vari- ous sides, and the student can only play with the object on the computer by using a mouse. Meanwhile, the students can access the Augmented Reality Module directly by using their smartphones, and it seems real. These two applications are free of charge since these applications' development is limited to specific spaces, but the researchers' module is the Augmented Reality Module customized based on the user's requests. A picture is a feature that can only be found in AR, and it distinguishes AR and Geogebra 3D applications. Fig. 2. The differences between 3D AR and Geogebra. However, even though there is an improvement in students' problem-solving skills and spatial skills in the control class, based on table 8, the improvement of students who are taught using ARM is immense. This is due to the differences between the Aug- mented Reality Module and the teacher's conventional module. It is in line with the results of previous studies that ARM can improve students' problem-solving skills [35][36][37][38][39], and ARM is effective in attracting students' interest in answering the test, which consists of several questions related to problem-solving -skills [40] Based on Table 8, the Augmented Reality Module's effectiveness in improving stu- dents' spatial skills is higher than the control class since Vector material is difficult to be explained by using a whiteboard, or it can only be displayed simultaneously using the Augmented Reality Module. Students can see the vector shape in various 3D views, and it has a direct effect on students' spatial skills based on several aspects, such as Spatial Perception, Spatial AR Visualization, Mental Rotation, Spatial Relation, Spatial Orientation, that are provided and developed in Augmented Reality learning [41-49]. There is a correlation between problem-solving and spatial skills since spatial skills play a role in developing problem-solving skills [50]. In line with Guntur's previous research, spatial skills can stimulate students' problem-solving skills since they are cor- related with each other [51]. The relationship between Augmented Reality, problem- solving skills, spatial skills, and Augmented Reality improves both of the skills shown in Figure 3. 168 http://www.i-jim.org Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… Fig. 3. The relationship between AR, spatial skills, and problem-solving skills 5 Conclusion Based on the results of the analyses, it can be concluded that the use of the Augmented Reality Module is effective in improving problem-solving skills and spatial skills, as proved by the improvement in students' mean score in pre-test and post-test. Besides, besides comparing the post-test scores in each class, it can also be concluded that using the Augmented Reality Module in the learning process is more effective than conventional learning since the students are usually taught using a conventional module from the government and Geogebra application. This study's results can be a reference for teachers and other researchers to use Augmented Reality Module to improve students' spatial skills and problem-solving skills in learning Mathematics. This study's limitations are that the module's material is limited to Vector material for the tenth-grade students, and the program can only be accessed on Android. For further research, developing other Augmented Reality modules for other materials and using a larger population is suggested. 6 Acknowledgement We would thank DRPM Kemenristekdikti of Indonesia, who has funded this re- search, thanks to SMAN 1 Ngemplak for the permission given to be done, thanks to Yogyakarta State University. iJIM ‒ Vol. 15, No. 05, 2021 169 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… 7 References [1] S. Papadakis, M. Kalogiannakis, and N. Zaranis, “Improving Mathematics Teaching in Kin- dergarten with Realistic Mathematical Education,” Early Childhood Education Journal, vol. 45, no. 3, pp. 369–378, 2017, https://doi.org/10.1007/s10643-015-0768-4 [2] S. Papadakis, M. Kalogiannakis, and N. Zaranis, “The effectiveness of computer and tablet assisted intervention in early childhood students’ understanding of numbers. An empirical study conducted in Greece,” Education and Information Technologies, vol. 23, no. 5, pp. 1849–1871, 2018, https://doi.org/10.1007/s10643-015-0768-4 [3] H. Hedberg, J. Nouri, P. Hansen, and R. Rahmani, “A Systematic Review of Learning through Mobile Augmented Reality,” International Journal of Interactive Mobile Technolo- gies (iJIM), vol. 12, no. 3, pp. 75–85, 2018. https://doi.org/10.3991/ijim.v12i3.8404 [4] Zaranis, N., Kalogiannakis, M., & Papadakis, “Using mobile devices for teaching realistic mathematics in kindergarten education. Creative Education, 4(07), 1.,” SciRes, vol. 4, no. 7, p. 1, 2013, https://doi.org/10.4236/ce.2013.47a1001 [5] Hejab Ma’azer Al Fawareh and S. Jusoh, “The Use and Effects of Smartphones in Higher Education,” International Journal of Interactive Mobile Technologies (iJIM), vol. 11, no. 6, pp. 103–111, 2017. https://doi.org/10.3991/ijim.v11i6.7453 [6] S. Papadakis, M. Kalogiannakis, and N. Zaranis, “Comparing Tablets and PCs in teaching Mathematics: An attempt to improve Mathematics Competence in Early Childhood Educa- tion,” Preschool and Primary Education, vol. 4, no. 2, p. 241, 2016, https://doi.org/10.12681/ ppej.8779 [7] M. I. S. Guntur, W. Setyaningrum, H. Retnawati, and M. Marsigit, “Assessing the Potential of Augmented Reality in Education,” in The 11th International Conference on E-Education, E-Business, E-Management and E-Learning (IC4E 2020)--EI & Scopus, 2020, pp. 93–97, https://doi.org/10.1145/3377571.3377621 [8] R. Azuma, Y. Baillot, R. Behringer, S. Feiner, S. Julier, and B. MacIntyre, “Recent advances in augmented reality,” IEEE Computer Graphics and Applications, 2001, https://doi.org/10.1109/38.963459 [9] M. Akçayır and G. Akçayır, “Advantages and challenges associated with augmented reality for education: A systematic review of the literature,” Educational Research Review, vol. 20, pp. 1–11, 2017, https://doi.org/10.1016/j.edurev.2016.11.002 [10] P. Menezes, “An Augmented Reality U-Academy Module : From Basic Principles to Con- nected Subjects,” International Journal of Interactive Mobile Technologies (iJIM), vol. 11, no. 5, pp. 105–117, 2017. https://doi.org/10.3991/ijim.v11i5.7074 [11] H. K. Wu, S. W. Y. Lee, H. Y. Chang, and J. C. Liang, “Current status, opportunities and challenges of augmented reality in education,” Computers and Education, vol. 62, pp. 41– 49, 2013, https://doi.org/10.1016/j.compedu.2012.10.024 [12] K. R. Bujak, I. Radu, R. Catrambone, B. MacIntyre, R. Zheng, and G. Golubski, “A psycho- logical perspective on augmented reality in the mathematics classroom,” Computers and Education, vol. 68, pp. 536–544, 2013, https://doi.org/10.1016/j.compedu. 2013.02.017 [13] G. Liestøl, “Augmented Reality Storytelling Narrative Design and Reconstruction of a His- torical Event in situ,” International Journal of Interactive Mobile Technologies (iJIM), vol. 13, no. 10, pp. 196–209, 2019. https://doi.org/10.3991/ijim.v13i12.11560 [14] K. Lee, "Augmented Reality in Education and Training, 56(2), 13-21. doi," Tech Trends, vol. 56, no. 2, pp. 13–21, 2012, https://doi.org/10.1007/s11528-012-0559-3. [15] NCTM, The principles, and standards for school mathematics. Reston: National Council of Teachers of Mathematics., 2000. 170 http://www.i-jim.org https://doi.org/10.1007/s10643-015-0768-4 https://doi.org/10.1007/s10643-015-0768-4 https://doi.org/10.3991/ijim.v12i3.8404 https://doi.org/10.4236/ce.2013.47a1001 https://doi.org/10.3991/ijim.v11i6.7453 https://doi.org/10.12681/ppej.8779 https://doi.org/10.12681/ppej.8779 https://doi.org/10.1145/3377571.3377621 https://doi.org/10.1109/38.963459 https://doi.org/10.1016/j.edurev.2016.11.002 https://doi.org/10.3991/ijim.v11i5.7074 https://doi.org/10.1016/j.compedu.2012.10.024 https://doi.org/10.1016/j.compedu.2013.02.017 https://doi.org/10.1016/j.compedu.2013.02.017 https://doi.org/10.3991/ijim.v13i12.11560 https://doi.org/10.1007/s11528-012-0559-3 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… [16] G. Butterworth, J., & Thwaites, Thinking Skills: Critical Thinking and Problem Solving, Second Edi. Cambridge.: Cambridge University Press, 2013. [17] H. Kaufmann, K. Steinbügl, A. Dünser, and J. Glück, “Improving Spatial Abilities by Ge- ometry Education in Augmented Reality - Application and Evaluation Design,” in Proceed- ings of the Virtual Reality International Conference (VRIC), 2005, pp. 25–34. https://doi.org/10.1145/1152760.1152776 [18] Coimbra, “Augmented reality: an enhancer for higher education students in maths learn- ing?,” in 6th International Conference on Software Development and Technologies for En- hancingAccessibility and Fighting Infoexclusion, 2015, pp. 332 – 339, doi: 10.1016/j.procs.2015.09.277. [19] J. A. Van de Walle, K. S. Karp, and J. M. Bay-Williams, Elementary and middle school mathematics teaching developmentally, 7th ed. Boston, New York San Francisco: Allyn & Bacon, 2010. [20] Erkki Pehkonen, Liisa Näveri, and Anu Laine, "On teaching problem-solving in school mathematics On Teaching Problem Solving in School Mathematics," c.e.p.s Journal, vol. 3, no. 4, pp. 9–24, 2013. https://doi.org/10.37626/ga9783959870641.0.25 [21] Jarrett, D., Stepanek, and A. Sutton, "Problem-solving: Getting to the heart of mathematics.," A Math and Science Journal, vol. 1, pp. 1–24, 2000. [22] R. Killen, Effective teaching strategies: a lesson from research and practice., 7th ed. South Melbourne: Cengage Learning, 2016. [23] S. E. and M. Swan, "students ' strategies for problem-solving in mathematics : ' Sample Stu- dent Work,'" Journal Of The International Society for Design and Development in Educa- tion, vol. 2, no. 7, pp. 1–31, 2014. [24] E. Ersoy, "Problem-solving and its teaching in mathematics," The Online Journal of New Horizons in Education, vol. 6, no. 2, pp. 79–87, 2016. [25] M. Wena, Strategi pembelajaran inovatif kontemporer, 6th ed. Jakarta: Bumi Aksara, 2011. [26] J. Kirkley and R. Foshay, “Principles for Teaching Problem Solving,” in Technical Paper, The Roach Organization, inc, 2003. [27] M. C. Linn and A. C. Petersen, “Emergence and characterization of sex differences in spatial ability,” Child Development, vol. 56, no. 6, pp. 1479–1498, 1985, https://doi.org/10. 2307/1130467 [28] J. Mulligan, “Looking within and beyond the geometry curriculum: connecting spatial rea- soning to mathematics learning.,” ZDM Mathematics Education, vol. 47, pp. 511–517, 2015, https://doi.org/10.1007/s11858-015-0696-1 [29] A. Bosnyak and R. N. Kondor, "The spatial geometry and spatial geometrical knowledge of university students, majored in mathematics," Acta Dicdactica Universitatis Comeniunde, vol. 8, no. 1, pp. 1–25, 2008. [30] J. Sarama and D. H. Clements, Early childhood mathematics education research (Learning trajectory for young children). New York, NY: Routledge Taylor and Francis Group., 2009. [31] R. L. Ebel and D. A. Frisbie, Essentials of Educational Measurement, 5th Ed. USA: Prentice- Hall, Inc, 1991. [32] Polya, How to Solve It: A new aspect of mathematical methods. USA: Princeton University Press, 2004. [33] C. Wasko, “What Teachers Need to Know About Augmented Reality Enhanced Learning Environments,” TechTrends, vol. 57, no. 4, pp. 17–21, 2013, doi: 10.1007/s11528-013- 0672-y. https://doi.org/10.1007/s11528-013-0672-y [34] R. Vágová and M. Kmetová, "Geogebra, a Tool to Improve Students ' Visual Imaging," ACTA Didactica Napocensia, vol. 12, no. 2, pp. 225–237, 2019, https://doi.org/10. 24193/adn.12.2.18 iJIM ‒ Vol. 15, No. 05, 2021 171 https://doi.org/10.1145/1152760.1152776 https://doi.org/10.37626/ga9783959870641.0.25 https://doi.org/10.2307/1130467 https://doi.org/10.2307/1130467 https://doi.org/10.1007/s11858-015-0696-1 https://doi.org/10.1007/s11528-013-0672-y https://doi.org/10.24193/adn.12.2.18 https://doi.org/10.24193/adn.12.2.18 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… [35] D. N. Eh Phon, M. B. Ali, and N. D. A. Halim, “Collaborative augmented reality in educa- tion: A review,” Proceedings - 2014 International Conference on Teaching and Learning in Computing and Engineering, LATTICE 2014, pp. 78–83, 2014, https://doi.org/10.1109/ latice.2014.23 [36] F. Ke and Y. C. Hsu, “Mobile augmented-reality artifact creation as a component of mobile computer-supported collaborative learning,” Internet and Higher Education, vol. 26, pp. 33– 41, 2015, https://doi.org/10.1016/j.iheduc.2015.04.003 [37] S. Fleck, M. Hachet, and J. M. Christian Bastien, “Marker-based Augmented Reality: In- structional-design to improve children interactions with astronomical concepts,” Proceed- ings of IDC 2015: The 14th International Conference on Interaction Design and Children, pp. 21–28, 2015, https://doi.org/10.1145/2771839.2771842 [38] M. Dunleavy, C. Dede, and R. Mitchell, "Affordances and limitations of immersive, partic- ipatory augmented reality simulations for teaching and learning," Journal of Science Educa- tion and Technology, vol. 18, no. 1, pp. 7–22, 2009, https://doi.org/10.1007/s10956-008- 9119-1 [39] D. Karagozlu, “Determination of the impact of augmented reality application on the success and problem-solving skills of students,” Quality and Quantity, vol. 52, no. 5, pp. 2393–2402, 2018, https://doi.org/10.1007/s11135-017-0674-5 [40] R. Hodhod, H. Fleenor, and S. Nabi, “Adaptive Augmented Reality Serious Game to Foster Problem Solving Skills,” 2014. [41] E. T. Gün and B. Atasoy, “The effects of augmented reality on elementary school students’ spatial ability and academic achievement,” Egitim ve Bilim, vol. 42, no. 191, pp. 31–51, 2017, https://doi.org/10.15390/eb.2017.7140 [42] T. V. Do and J. W. Lee, “A multiple-level 3D-LEGO game in augmented reality for improv- ing spatial ability,” Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5613 LNCS, no. PART 4, pp. 296–303, 2009, https://doi.org/10.1007/978-3-642-02583-9_33 [43] Y. T. Liao, C. H. Yu, and C. C. Wu, “Learning geometry with augmented reality to enhance spatial ability,” Proceedings - 2015 International Conference on Learning and Teaching in Computing and Engineering, LaTiCE 2015, pp. 221–222, 2015, https://doi.org/10. 1109/latice.2015.40. [44] K.-H. C. C.-C. Tsai, “Affordances of Augmented Reality in Science Learning: Suggestions for Future Research,” Journal of Science Education and Technology, vol. 22, no. 4, pp. 449– 462, 2013, https://doi.org/10.1007/s10956-012-9405-9 [45] H. Kaufmann and B. Meyer, "Simulating physical, educational experiments in augmented reality," ACM SIGGRAPH Asia 2008 Educators Programme, SIGGRAPH Asia’08, 2008, https://doi.org/10.1145/1507713.1507717 [46] E. Quintero, P. Salinas, E. González-Mendívil, and H. Ramírez, “Augmented reality app for calculus: a proposal for the development of spatial visualization,” Procedia Computer Sci- ence, vol. 75, no. Vare, pp. 301–305, 2015, doi: 10.1016/j.procs.2015.12.251. https://doi.org/10.1016/j.procs.2015.12.251 [47] Papadakis, St. (2020). Robots and Robotics Kits for Early Childhood and First School Age. International Journal of Interactive Mobile Technologies (iJIM), 14 (18), 34-56. https://doi.org/10.3991/ijim.v14i18.16631 [48] Papadakis, S. (2020). Apps to Promote Computational Thinking Concepts and Coding Skills in Children of Preschool and Pre-Primary School Age. In Mobile Learning Applications in Early Childhood Education (pp. 101-121). IGI Global. https://doi.org/10.4018/978-1-7998- 1486-3.ch006 172 http://www.i-jim.org https://doi.org/10.1109/latice.2014.23 https://doi.org/10.1109/latice.2014.23 https://doi.org/10.1016/j.iheduc.2015.04.003 https://doi.org/10.1145/2771839.2771842 https://doi.org/10.1007/s10956-008-9119-1 https://doi.org/10.1007/s10956-008-9119-1 https://doi.org/10.1007/s11135-017-0674-5 https://doi.org/10.15390/eb.2017.7140 https://doi.org/10.1007/978-3-642-02583-9_33 https://doi.org/10.1109/latice.2015.40 https://doi.org/10.1109/latice.2015.40 https://doi.org/10.1007/s10956-012-9405-9 https://doi.org/10.1145/1507713.1507717 https://doi.org/10.1016/j.procs.2015.12.251 https://doi.org/10.3991/ijim.v14i18.16631 https://doi.org/10.4018/978-1-7998-1486-3.ch006 https://doi.org/10.4018/978-1-7998-1486-3.ch006 Paper—The Effectiveness of Augmented Reality in Learning Vector to Improve Students’ Spatial… [49] Kalogiannakis, M., & Papadakis, S. (2020). The Use of Developmentally Mobile Applica- tions for Preparing Pre-Service Teachers to Promote STEM Activities in Preschool Class- rooms. In Mobile Learning Applications in Early Childhood Education (pp. 82-100). IGI Global. https://doi.org/10.4018/978-1-7998-1486-3.ch006 [50] S. Krulik and Robert E. Reys, Problem Mathematics. Virginia.: NCTM, 1980. [51] [51] M. I. S. Guntur, W. Setyaningrum, H. Retnawati, and M. Marsigit, "Can augmented reality improve a problem-solving and spatial skill ?," in International Seminar on Innova- tion in Mathematics and Mathematics Education, 2020, p. 1581, https://doi.org/10. 1088/1742-6596/1581/1/012063 8 Authors Muhamad Ikhsan Sahal Guntur (Scopus ID: 57218398738) is a Master's Mathe- matics Education student at Yogyakarta State University (ikhsan.guntur@gmail.com). Wahyu Setianingrum (Scopus ID: 57195479725) is a lecturer at the Faculty of Mathematics Education at Yogyakarta State University. She has concerns about using technology in education. She is an editorial at mathematics education research journal Indonesia (JRPM) and Journal Universitas Mercu Buana Yogyakarta. Article submitted 2020-10-03. Resubmitted 2020-12-24. Final acceptance 2020-12-24. Final version pub- lished as submitted by the authors. iJIM ‒ Vol. 15, No. 05, 2021 173 https://doi.org/10.4018/978-1-7998-1486-3.ch006 https://doi.org/10.1088/1742-6596/1581/1/012063 https://doi.org/10.1088/1742-6596/1581/1/012063 ikhsan.guntur@gmail.com).