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 Problem- Solving 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’ problem- solving 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 problem- solving 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 Problem- Solving 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 Problem- Solving 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. (2016). How to improve your problem-solving skills. QS Top Universities, 1(1), 1-3. Iilonga & Ogbonnaya, Grade 10 Namibian learners' problem-solving skills … 286 Akyüz, G. (2020). Non-routine problem solving performances of mathematics teacher candidates. Educational Research and Reviews, 15(5), 214-224. https://doi.org/10.5897/ERR2020.3907 Al-Mutawah, M. A., Thomas, R., Eid, A., Mahmoud, E. Y., & Fateel, M. J. (2019). Conceptual understanding, procedural knowledge and problem-solving skills in mathematics: High school graduates work analysis and standpoints. International journal of education and practice, 7(3), 258-273. https://doi.org/10.18488/journal.61.2019.73.258.273 Albin, S., & von Watzdorf, A. (2019). A comparative analysis of word problems in six junior primary Mathematics textbooks in Namibia. Journal of Engineering Research and Application, 9(6), 47-55. Aljaberi, N. M. (2015). University students' learning styles and their ability to solve mathematical problems. International Journal of Business and Social Science, 6(4), 152-165. Berlinghoff, W. P., & Gouvêa, F. Q. (2021). Math through the ages: A gentle history for teachers and others expanded second edition (Vol. 32). American Mathematical Soc. Brijlall, D. (2015). Exploring the stages of Polya’s problem-solving model during collaborative learning: A case of fractions. International Journal of Educational Sciences, 11(3), 291-299. https://doi.org/10.1080/09751122.2015.11890401 Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3), 613-632. https://doi.org/10.1016/j.jmathb.2013.07.002 Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27(2-3), 147-164. https://doi.org/10.1080/02702710600642467 Chang, B. (2019). Reflection in learning. Online Learning, 23(1), 95-110. https://doi.org/10.24059/olj.v23i1.1447 Charles, R., Lester, F., & O’Daffer, P. (1987). How to evaluate progress in problem solving. National Council of Teachers of Mathematics. Daulay, K. R., & Ruhaimah, I. (2019). Polya theory to improve problem-solving skills. Journal of Physics: Conference Series, 1188(1), 012070. https://doi.org/10.1088/1742-6596/1188/1/012070 Directorate of National Examinations and Assessment. (2014). Examiners’ reports. DNEA Printers. Directorate of National Examinations and Assessment. (2015). Examiners’ reports. DNEA Printers. Directorate of National Examinations and Assessment. (2016). Examiners’ reports. DNEA Printers. Directorate of National Examinations and Assessment. (2017). Examiners’ reports. DNEA Printers. Directorate of National Examinations and Assessment. (2018). Examiners’ reports. DNEA Printers. https://doi.org/10.5897/ERR2020.3907 https://doi.org/10.18488/journal.61.2019.73.258.273 https://doi.org/10.1080/09751122.2015.11890401 https://doi.org/10.1016/j.jmathb.2013.07.002 https://doi.org/10.1080/02702710600642467 https://doi.org/10.24059/olj.v23i1.1447 https://doi.org/10.1088/1742-6596/1188/1/012070 Volume 12, No 2, September 2023, pp. 275-290 287 Ersoy, E. (2016). Problem solving and its teaching in mathematics. The Online Journal of New Horizons in Education, 6(2), 79-87. Franestian, I. D., Suyanta, S., & Wiyono, A. (2020). Analysis problem solving skills of student in Junior High School. Journal of Physics: Conference Series, 1440(1), 012089. https://doi.org/10.1088/1742-6596/1440/1/012089 Gray, C. A. (2018). The impact of applying the first two steps of Polya’s four problem- solving steps in an advanced mathematics classroom. Doctoral dissertation. University of South Carolina - Columbia. Retrieved from https://scholarcommons.sc.edu/etd/5037 Greer, B., Verschaffel, L., & De Corte, E. (2002). “The Answer is Really 4.5”: Beliefs About Word Problems. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A Hidden Variable in Mathematics Education? (pp. 271-292). Springer Netherlands. https://doi.org/10.1007/0-306-47958-3_16 Gunawan, G., Harjono, A., Nisyah, M. a., Kusdiastuti, M., & Herayanti, L. (2020). Improving students' problem-solving skills using inquiry learning model combined with advance organizer. International Journal of Instruction, 13(4), 427-442. https://doi.org/10.29333/iji.2020.13427a Hendriana, H., Purwasih, R., Triawan, E., Prasetio, Y., & Satria, T. (2018). Analysis of student problem solving skill and activity concentration on a senior high school. Al- Jabar: Jurnal Pendidikan Matematika, 9(1), 1-8. https://doi.org/10.24042/ajpm.v9i1.2068 Hijada Jr, M. V., & Cruz, M. L. D. (2022). The gap between comprehension level and problem-solving skills in learning mathematics. Universal Journal of Educational Research, 1(1), 35-43. Hooshyar, D., Pedaste, M., Yang, Y., Malva, L., Hwang, G.-J., Wang, M., Lim, H., & Delev, D. (2021). From gaming to computational thinking: An adaptive educational computer game-based learning approach. Journal of Educational Computing Research, 59(3), 383-409. https://doi.org/10.1177/0735633120965919 In'am, A. (2014). The implementation of the Polya method in solving euclidean geometry problems. International Education Studies, 7(7), 149-158. https://doi.org/10.5539/ies.v7n7p149 İncebacak, B. B., & Ersoy, E. (2016). Problem solving skills of secondary school students. China-USA Business Review, 15(6), 275-285. https://doi.org/10.17265/1537- 1514/2016.06.002 Jupri, A., & Drijvers, P. (2016). Student difficulties in mathematizing word problems in algebra. Eurasia Journal of Mathematics, Science and Technology Education, 12(9), 2481-2502. https://doi.org/10.12973/eurasia.2016.1299a Khabibah, S., Manuharawati, M., & Agung, L. (2018). Problem solving ability: A case study in postgraduate mathematics student. In Proceedings of the Mathematics, Informatics, Science, and Education International Conference. https://doi.org/10.2991/miseic-18.2018.54 Kivunja, C. (2015). Exploring the pedagogical meaning and implications of the 4Cs" super skills" for the 21st century through Bruner's 5E lenses of knowledge construction to https://doi.org/10.1088/1742-6596/1440/1/012089 https://scholarcommons.sc.edu/etd/5037 https://doi.org/10.1007/0-306-47958-3_16 https://doi.org/10.29333/iji.2020.13427a https://doi.org/10.24042/ajpm.v9i1.2068 https://doi.org/10.1177/0735633120965919 https://doi.org/10.5539/ies.v7n7p149 https://doi.org/10.17265/1537-1514/2016.06.002 https://doi.org/10.17265/1537-1514/2016.06.002 https://doi.org/10.12973/eurasia.2016.1299a https://doi.org/10.2991/miseic-18.2018.54 Iilonga & Ogbonnaya, Grade 10 Namibian learners' problem-solving skills … 288 improve pedagogies of the new learning paradigm. Creative Education. https://doi.org/10.4236/ce.2015.62021 Krawitz, J., Schukajlow, S., & Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: what are important barriers? Educational Psychology, 38(10), 1221-1238. https://doi.org/10.1080/01443410.2018.1502413 Kusumadewi, C. A., & Retnawati, H. (2020). Identification of elementary school students’ difficulties in mathematical problem-solving. Journal of Physics: Conference Series, 1511(1), 012031. https://doi.org/10.1088/1742-6596/1511/1/012031 Lee, C. I. (2016). An appropriate prompts system based on the Polya method for mathematical problem-solving. Eurasia Journal of Mathematics, Science and Technology Education, 13(3), 893-910. https://doi.org/10.12973/eurasia.2017.00649a Lubis, J. N., Panjaitan, A., Surya, E., & Syahputra, E. (2017). Analysis mathematical problem solving skills of student of the grade VIII-2 junior high school Bilah Hulu Labuhan Batu. International Journal of Novel Research in Education and Learning, 4(2), 131-137. Lupahla, N. (2014). Assessing the algebraic problem solving skills of Grade 12 learners in Oshana Region, Namibia. Master’s dissertation. UNISA Institutional Repository. Retrieved from http://hdl.handle.net/10500/19225 Maulyda, M. A., Hidayati, V. R., Rosyidah, A. N. K., & Nurmawanti, I. (2019). Problem- solving ability of primary school teachers based on Polya’s method in Mataram City. Pythagoras, 14(2), 139-149. https://doi.org/10.21831/pg.v14i2.28686 Meutia, C. I., Ikhsan, M., & Saminan, S. (2020). Mathematical problem-solving skills of junior high school students. Journal of Physics: Conference Series, 1460(1), 012010. https://doi.org/10.1088/1742-6596/1460/1/012010 National Institute for Educational Development. (2016). Namibia junior secondary certificate mathematics guide. NIED. Nikmah, I. L., Juandi, D., & Prabawanto, S. (2019). Students’ difficulties on solving mathematical problem based on ESD objectives. Journal of Physics: Conference Series, 1157(3), 032116. https://doi.org/10.1088/1742-6596/1157/3/032116 Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102(1), 9-28. https://doi.org/10.1007/s10649-019-09903-9 Nurkaeti, N. (2018). Polya’s strategy: an analysis of mathematical problem solving difficulty in 5th grade elementary school. EduHumaniora: Jurnal Pendidikan Dasar, 10(2), 140-147. https://doi.org/10.17509/eh.v10i2.10868 Palm, T. (2006). Word problems as simulations of real-world situations: A proposed framework. For the learning of Mathematics, 26(1), 42-47. Pentang, J., Ibañez, E. D., Subia, G., Domingo, J. G., Gamit, A. M., & Pascual, L. E. (2021). Problem-solving performance and skills of prospective elementary teachers in Northern Philippines. Hunan Daxue Xuebao, 48(1), 122-132. Peterson, R. L., Boada, R., McGrath, L. M., Willcutt, E. G., Olson, R. K., & Pennington, B. F. (2017). Cognitive prediction of reading, math, and attention: Shared and unique https://doi.org/10.4236/ce.2015.62021 https://doi.org/10.1080/01443410.2018.1502413 https://doi.org/10.1088/1742-6596/1511/1/012031 https://doi.org/10.12973/eurasia.2017.00649a http://hdl.handle.net/10500/19225 https://doi.org/10.21831/pg.v14i2.28686 https://doi.org/10.1088/1742-6596/1460/1/012010 https://doi.org/10.1088/1742-6596/1157/3/032116 https://doi.org/10.1007/s10649-019-09903-9 https://doi.org/10.17509/eh.v10i2.10868 Volume 12, No 2, September 2023, pp. 275-290 289 influences. Journal of Learning Disabilities, 50(4), 408-421. https://doi.org/10.1177/0022219415618500 Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton university press. Polya, G. (1957). How to solve it: A new aspect of mathematical method. Prentice university press. Polya, G. (1973). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton university press. Polya, G. (1981). Mathematical discovery on understanding learning and teaching problem solving. John Wiley and Sons. Rahman, M. M. (2019). 21st century skill "problem solving": Defining the concept. Asian Journal of Interdisciplinary Research, 2(1), 64-74. https://doi.org/10.34256/ajir1917 Riyadi, R., Syarifah, T. J., & Nikmaturrohmah, P. (2021). Profile of students’ problem- solving skills viewed from Polya's four-steps approach and elementary school students. European Journal of Educational Research, 10(4), 1625-1638. https://doi.org/10.12973/eu-jer.10.4.1625 Schoenfeld, A. H. (2014). Mathematical problem solving. Elsevier. Sikukumwa, E. (2017). Types of strategies used to solve algebraic word problems by grade 12 ordinary level Mathematics learners of Kavango East region of Namibia. Doctoral dissertation. University of Namibia. Retrieved from http://hdl.handle.net/11070/2023 Sipayung, T. N., & Anzelina, D. (2019). An analysis of students problem solving skills using a realistic mathematics approach on integers materials. Journal of Physics: Conference Series, 1211(1), 012083. https://doi.org/10.1088/1742- 6596/1211/1/012083 Steinbrink, G. (2009). Reading mathematic. Math in the Middle Institute Partnership, Action Research Project Report. Master’s dissertation. University of Nebraska-Lincoln. Sumaryanta, D. (2015). Pedoman Penskoran [Scoring rubric]. Idealmathedu: Indonesian Digital Journal of Mathematics and Education, 2(3), 181-190. Tambychik, T., & Meerah, T. S. M. (2010). Students’ difficulties in mathematics problem- solving: What do they say? Procedia - Social and Behavioral Sciences, 8, 142-151. https://doi.org/10.1016/j.sbspro.2010.12.020 Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54(1), 63-75. https://doi.org/10.1023/B:EDUC.0000005237.72281.bf Van Dooren, W., Lem, S., De Wortelaer, H., & Verschaffel, L. (2019). Improving realistic word problem solving by using humor. The Journal of Mathematical Behavior, 53, 96-104. https://doi.org/10.1016/j.jmathb.2018.06.008 Wickramasinghe, I., & Valles, J. (2015). Can we use Polya’s method to improve students’ performance in the statistics classes? Numeracy, 8(1), 12. https://doi.org/10.5038/1936-4660.8.1.12 Yayuk, E., & Husamah, H. (2020). The difficulties of prospective elementary school teachers in item problem solving for mathematics: Polya’s steps. Journal for the https://doi.org/10.1177/0022219415618500 https://doi.org/10.34256/ajir1917 https://doi.org/10.12973/eu-jer.10.4.1625 http://hdl.handle.net/11070/2023 https://doi.org/10.1088/1742-6596/1211/1/012083 https://doi.org/10.1088/1742-6596/1211/1/012083 https://doi.org/10.1016/j.sbspro.2010.12.020 https://doi.org/10.1023/B:EDUC.0000005237.72281.bf https://doi.org/10.1016/j.jmathb.2018.06.008 https://doi.org/10.5038/1936-4660.8.1.12 Iilonga & Ogbonnaya, Grade 10 Namibian learners' problem-solving skills … 290 Education of Gifted Young Scientists, 8(1), 361-378. https://doi.org/10.17478/jegys.665833 Yeh, C. Y. C., Cheng, H. N. H., Chen, Z.-H., Liao, C. C. Y., & Chan, T.-W. (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