24. Mathematics Education Research
Paper
Exploring Children’s Reasoning Process in Strategy Games
Yasin Memis1, Belma Türker Biber2
1Ministry of National Education, Turkey,; 2Aksaray University, Turkey
Presenting Author: Memis, Yasin
Numerous curricula aim to develop students’ mathematical reasoning, which is an essential aspect of their education. Students can develop reasoning skills through various tasks that go beyond the formal curriculum (McFeetors & Palfy, 2018). However, appropriate tasks are essential to support students’ mathematical reasoning in the classroom (Brodie, 2010; Jeannotte & Kieran, 2017). Mathematical reasoning can be developed and elicited through meaningful and challenging learning experiences (Stein et al., 1996). Moreover, it has been suggested that tasks should encourage students to make conjectures and generalisations, search for similarities and differences between objects, and use their prior knowledge and other generalisations with which they are already familiar (Jeannotte & Kieran, 2017). Games play an essential role in these educational tasks because they provide students with an appropriate environment for presenting and defending their arguments (Mousoulides & Sriraman, 2014). Furthermore, strategic games have a large number of potential strategies that include a number of different components. Therefore, while playing these games, students use various reasoning skills without realising it (McFeetors & Palfy, 2018).
The role of mathematical games in the teaching and learning of mathematics has been recognised for decades (Dienes, 1963; Brousseau & Gibel, 2005). According to Ernest (1986), games have the potential to positively influence the development of students’ conceptual reasoning, higher-order thinking, and motivation to learn mathematics. Moreover, carefully designed mathematical games can help students develop problem-solving skills (Pintér, 2010) and effectively apply the critical actions of mathematical reasoning (McFeetors & Palfy, 2018). Although reasoning is a significant component of student achievement in mathematics, few studies have reported how students demonstrate reasoning skills while playing games. Our research aims to examine the reasoning process of fifth-grade students while they play a strategy game.
The ability to reason is essential to understanding mathematics (The Programme for International Student Assessment [PISA], 2022). Lithner (2000, p. 166) defines reasoning as a way of thinking that is adopted to make claims and reach conclusions. It is critical for educators to develop students' reasoning skills to prepare them for more advanced learning (Vale et al., 2017). Many elements of reasoning are closely related to elementary school mathematics, such as forming hypotheses, sampling, comparing, recognising patterns, justifying, and generalising (Lampert, 2001). According to reSolve: Assessing Mathematical Reasoning (Australian Academy of Science [AAS] and Australian Association of Mathematics Teachers [AAMT], 2017) these actions of reasoning can be classified into three main categories: analysing, verifying, and generalising.
This paper focuses on the actions students display during the process of analysis, including the first steps of mathematical reasoning. We expect this study to provide new insights into the different types of reasoning students use when playing strategic games and the function of these games in early grades.
Methodology, Methods, Research Instruments or Sources UsedThis research was designed to investigate mathematical reasoning as a process that can emerge from playing appropriate games rather than as a directly taught skill (Jeannotte & Kieran, 2017). We conducted the study with 5th graders (ages 10–11) and evaluated student responses based on analysing which is highlighted in the literature relating to mathematical reasoning as a first process. Mathematical reasoning framework developed by reSolve: assessing reasoning (AAS & AAMT, 2017) was utilized in this study. This framework was utilized because it provides insight into different types of reasoning processes and is a comprehensive guide to their function in early years classrooms.
Mathematical games without a chance factor allow students to develop strategies and thus can be an effective tool for the reasoning process. The Chomp game is a type of nim game, and it was chosen to provide students with an opportunity to use reasoning skills. During the game, two players take turns removing different rectangular areas from a particular rectangular area (e.g. 3x3, 4x5), and the person who gets the last piece loses. Nim games require only a limited background in mathematics, so they can be practised by individuals of all ages. They pose a series of problems that allow the students to demonstrate their reasoning abilities. Furthermore, winning depends on the development of strategies because there are no chance factors in these games. The students were asked questions about their strategies during the game, and their reasoning skills were assessed based on their answers. Both written and verbal data were collected while playing the games in pairs.
Data analysis was conducted by using a coding tool based on reSolve: assessing reasoning (AAS & AAMT, 2017) reasoning framework. In this study, only the act of analysing was addressed, and this process was evaluated within the scope of three basic understandings: 1) Exploring the problem and connecting it with known facts and properties, 2) Comparing and contrasting cases, and 3) Sorting and categorizing cases (AAS & AAMT, 2017). Based on these understandings, indicators and examples were created, and student actions were analysed with the coding tool.
Conclusions, Expected Outcomes or FindingsAs a result of the research, it was observed that students initially made random movements while becoming familiar with the game. With time, students began to observe their own movements and those of their opponents in order to make more informed decisions. Throughout the game, students had to use a variety of reasoning processes since there was no chance factor involved. It is common for these activities to begin with the discovery of patterns and the prediction of future events, which involve the analysis process.
The findings indicate that students performed different types of analysis while playing games. The following quotes from students’ arguments exemplify the analysing process of reasoning:
S1 – 'The most difficult of the three games was 4x5. 3x3 and 4x4 were similar. One was different because they were both squares. The other was different because it was not a square’. (Distinguishing/comparing similarities and differences)
S2 – ‘If I begin the game first and consistently get two squares, I will win 95% of the games’. (Create claims from data/experiences)
On the other hand, it was observed that as they played the game, they were able to present deeper mathematical arguments and support them systematically. Additionally, researchers encouraged students by asking prompt questions that acted as catalysts for them to articulate their reasoning in this process. The preliminary findings indicate that mathematical games presented in a supportive environment allow students to experience a variety of reasoning processes, including analysis. Moreover, our findings support the idea that all students can provide informal justifications and that strategically designed games assist pupils’ progress in reasoning (McFeetors & Palfy, 2018).
ReferencesAustralian Academy of Science and Australian Association of Mathematics Teachers. (2017). reSolve: Mathematics by Inquiry. Retrieved from http://www.resolve.edu.au/
Brodie, K. (2010). Teaching Mathematical Reasoning: A Challenging Task. In: Brodie, K. (eds) Teaching Mathematical Reasoning in Secondary School Classrooms. (pp. 7-22) Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09742-8_1.
Brousseau, G., & Gibel, P. (2005). Didactical handling of students’ reasoning processes in problem solving situations. Educational Studies in Mathematics. 59, 13-58. doi:10.1007/s10649-005-2532-y.
Diénès, Z. P. (1963). An experimental study of mathematics learning. London: Hutchinson.
Ernest, P. (1986). Games. A rationale for their use in the teaching of mathematics in school. Mathematics in school, 15(1), 2-5.
Herbert, S., & Williams, G. (2021). Eliciting mathematical reasoning during early primary problem solving. Mathematics Education Research Journal, 1-27.
Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16. https://doi.org/10.1007/s10649-017-9761-8.
Kollosche, D. (2021). Styles of reasoning for mathematics education. Educational Studies in Mathematics, 107(3), 471-486.
Lithner, J. (2000). Mathematical reasoning in school tasks. Educational Studies in Mathematics, 41(2), 165-190.
McFeetors, P. J., & Palfy, K. (2018). Educative experiences in a games context: Supporting emerging reasoning in elementary school mathematics. The Journal of Mathematical Behavior, 50, 103-125.
Mousoulides, N., Sriraman, B. (2014). Mathematical Games in Learning and Teaching. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4978-8_97.
Pintér, K. (2010). Creating games from mathematical problems. Primus, 21(1), 73-90. https://doi. org/10.1080/10511970902889919
Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488. https://doi.org/http://dx.doi.org/10.3102/00028312033002455.
Vale, C., Bragg, L. A., Widjaja, W., Herbert, S., & Loong, E. Y.-K. (2017). Children's Mathematical Reasoning: Opportunities for Developing Understanding and Creative Thinking. Australian Primary Mathematics Classroom, 22, 3-8.
24. Mathematics Education Research
Paper
Middle School Students’ Mathematics Achievement: Do Test Anxiety and Metacognition Matter?
Utkun Aydin1, Meriç Özgeldi2
1University of Glasgow, United Kingdom; 2Mersin University, Turkey
Presenting Author: Aydin, Utkun
Mathematics is typically conceived of as being a core discipline in curricula at all levels of education. For this reason, mathematics achievement is crucial to student placement, selection, and admission across most educational systems around the world. This information is also highly relevant in view of the fact that testing is a common practice in contemporary society, which is widely used to make important decisions about an individual’s status across primary, secondary, and higher education (Zeidner, 1998). Test anxiety, however, is a key affective variable can impede both achievement in general (Cassady & Johnson, 2002) and mathematics achievement in particular (Higbee & Thomas, 1999). It is defined as a subjective emotional state that includes a set of cognitive, physiological, and behavioral responses to concerns about possible fear of failure, experienced before or during an evaluative situation (Bodas et al., 2008). It has detrimental effects on schooling, occupational, and overall life outcomes (von der Embse et al., 2015). It is widely acknowledged that high level of test anxiety is associated with lower level of performance (Ng & Lee, 2015). Since students have to cope with constant mathematics pressure at school, it is of major interest for teachers and researchers to identify and strengthen/weaken those factors in students, which primarily influence mathematics achievement negatively. Additionally, metacognition plays an important role in mathematics achievement, as it shapes students’ conscious use and control of their own cognitive functions in educational settings (Brown, 1987). Metacognition is causally referred to as one’s awareness and regulation of own cognitive processes consisting of two components: knowledge of cognition and regulation of cognition (Flavell, 1979). Although research on metacognition has made it quite clear that highly metacognitive students perform better than their less metacognitive counterparts on most performance assessments including mathematics tests, the role of test anxiety in activating metacognitive knowledge and regulatory processes, are much less clear (Hacker et al., 1998). It has been suggested that the impact of affective factors such as test anxiety on performance is also related to metacognition (Zeidner, 1998). However, previous studies have predominantly examined the effect of each variable in isolation, and mainly in relation to general performance that place little demand on specifically mathematics achievement.
There is a vast amount of empirical evidence for the influence of test anxiety on metacognition and mathematics achievement as well as the impact of metacognition on mathematics achievement (e.g., Miesner & Maki, 2007; Sherman & Wither, 2003). Despite this, the combined importance of test anxiety and metacognition in mathematics achievement has been largely ignored, as most studies approach test anxiety in the form of math anxiety whereas others neglected metacognition. To the best of our knowledge based on a comprehensive review of literature undertaken, only two studies (Tok, 2013; Veenman et al., 2000) were conducted that in a sense harmonizes with our emphasis on the differential effects of test anxiety, metacognition, and mathematics achievement on one another.
More specifically, little is known about how these affective and cognitive factors differentially contribute to individual differences in mathematics achievement. This question added significance in light of the researchers, who suggested that “the relationship between test anxiety and metacognition may be a worthwhile field for research, while simultaneously helping to establish links between affect and cognition more generally” (e.g., Zeidner, 1998). Specifically, we hypothesized that: (a) test anxiety would significantly effect metacognition and mathematics achievement (H1); (b) metacognition would significantly effect mathematics achievement (H2); and (c) metacognition would mediate the relationship between test anxiety and mathematics achievement (H3).
Methodology, Methods, Research Instruments or Sources UsedParticipants
943 students (442 males and 501 females) from five public middle schools (477 seventh and 466 eighth graders) in Turkey participated in the present study.
Measurements
Children’s Test Anxiety Scale (CTAS). The Turkish adaption (Aydın & Bulgan, 2017) of the CTAS (α= .88), which was originally developed by Wren and Benson (2004) was used to measure students’ test anxiety. The 30-item scale comprised three subscales – Thoughts (α= .82); Off-Task Behaviors (α= .72); and Autonomic Reactions (α= .75). Students responded to each statement of the CTAS on a 4-point scale: (1) almost never, (2) some of the time, (3) most of the time, and (4) almost always. The possible scores on the CTAS ranged from 30 to 120.
Junior Metacognitive Awareness Inventory (Jr. MAI). The Turkish adaption (Aydın & Ubuz, 2010) of the Jr. MAI (α= .85), which was originally developed by Sperling et al. (2002) was used to measure students’ metacognition. The 17-item inventory comprised two subdimensions – Knowledge of Cognition (α= .75) and Regulation of Cognition (α= .79). Students responded to each statement of the Jr. MAI on a 5-point scale: (1) never, (2) seldom, (3) sometimes, (4) often, and (5) always. The possible scores on the Jr. MAI ranged from 18 to 90.
Mathematics Achievement Test (MAT). The researcher developed MAT was used to assess students’ mathematics achievement. The test was composed of 18 multiple-choice items originally released by the Trends in International Mathematics and Science Study (TIMSS) from those used in TIMSS 2007, 2011, and 2015. The items, adapted into Turkish, were released by the Ministry of National Education (available from http://timss.meb.gov.tr/www/aciklanan-sorular/icerik/1). These items were reviewed in terms of their content domains (i.e., Number, Algebra, Geometry, and Data and Chance) and cognitive domains (i.e., Knowing, Applying, and Reasoning) by two middle school teachers with over 20 years of experience, and a staff member in mathematics education, who had expertise in cross-cultural comparisons in international assessments. Possible scores on the test ranged from 0 to 18.
Procedure
The data were collected during the spring semester of the 2018/2019 academic year. Students completed the CTAS, Jr. MAI, and MAT in two consecutive mathematics classes (each 40 minutes).
Data Analysis
The first two hypotheses (H1 and H2) were tested by performing a one-way multivariate analysis of variance (One-Way MANOVA), whereas the last hypothesis (H3) was tested using a one-way analysis of covariance (One-Way ANCOVA) via SPSS version 21.0.
Conclusions, Expected Outcomes or FindingsDescriptive Statistics
Results showed that students reported moderate test anxiety (M= 63.82, SD= 14.99), metacognition (M= 63.14, SD= 11.18), and moderate-to-high mathematics achievement (M= 10.15, SD= 4.24).
Inferential Statistics
The Effect of Test Anxiety on Metacognition and Mathematics Achievement
A one-way MANOVA was conducted to explore the impact of test anxiety on metacognition and mathematics achievement (H1). There was a statistically significant difference among low-, moderate-, and high-test anxious students on the combined dependent variables, metacognition and mathematics achievement, F(4, 1836)= 3.75, p= .005; Wilks’ Lambda= .98; partial η^2= .008.
The Effect of Metacognition on Mathematics Achievement
A one-way ANOVA was conducted to compare mean scores of the low-, moderate-, and high-metacognitive students on their mathematics achievement (H2). The analysis yielded significant differences, F(2, 940)= 29.61, p= .000, partial η^2 = .05, among the students who are low-, moderate-, and high-metacognitive in performing mathematics.
Metacognition as a Mediator
A one-way ANCOVA was used to test whether metacognition can mediate the effect of test anxiety on mathematics achievement (H3). Results revealed that the effect of test anxiety on mathematics achievement became nonsignificant, F(2, 937)= .98, p= .373, partial η^2 = .02 when controlling for the effect of metacognition, indicating that metacognition plays a significant role in the effect of test anxiety on mathematics achievement.
These findings supported previous research indicating the differential effects of test anxiety on metacognition (Miesner & Maki, 2007) as well as the role of metacognition in prompting students’ mathematics performance (Bond & Ellis, 2013). While the research context is Turkey, the findings of the present study can be valuable both for European contexts and for international context considering that the national characteristics have an impact for rendering more precise information about the cognitive and affective factors affecting mathematics achievement, as proposed by Higbee and Thomas (1999).
ReferencesAydın, U., & Bulgan, G. (2017). Çocuklarda Sınav Kaygısı Ölçeği’nin Türkce uyarlaması [Adaptation of Children’s Test Anxiety Scale to Turkish].Elementary Education Online, 16(2), 887-899.
Aydın, U., & Ubuz, B. (2010). Turkish version of the junior metacognitive awareness inventory: An exploratory and confirmatory factor analysis. Education and Science, 35(157), 30-47.
Bodas, J., Ollendick, T. H., & Sovani, A. V. (2008). Test anxiety in Indian children: A cross-cultural perspective. Anxiety, Stress, & Coping, 21(4), 387-404.
Bond, J. B., & Ellis, A. K. (2013). The effects of metacognitive reflective assessment on fifth and sixth graders' mathematics achievement. School Science and Mathematics, 113(5), 227-234.
Brown, A. (1987). Metacognition, executive control, self-regulation, and other more mysterious mechanisms. In F. Weinert & R. Kluwe (Eds.), Metacognition, motivation, and Understanding (pp. 65-116). Erlbaum.
Cassady, J. C., & Johnson, R. E. (2002). Cognitive test anxiety and academic performance. Contemporary Educational Psychology, 27(2), 270-295.
Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive developmental inquiry. American Psychologist, 34, 906-911.
Hacker, D. J., Dunlosky, J., & Graesser, A. C. (Eds.) (1998). Metacognition in educational theory and practice. Lawrence Erlbaum Associates.
Higbee, J. L., & Thomas, P. V. (1999). Affective and cognitive factors related to mathematics achievement. Journal of Developmental Education, 23(1), 8-24.
Miesner, M. T., & Maki, R. H. (2007). The role of test anxiety in absolute and relative metacomprehension accuracy. European Journal of Cognitive Psychology, 19(4-5), 650-670.
Ng, E., & Lee, K. (2015). Effects of trait test anxiety and state anxiety on children's working memory task performance. Learning and Individual Differences, 40, 141-148.
Sperling, R. A., Howard, B. C., Miller, L. A., & Murphy, C. (2002). Measures of children’s knowledge and regulation of cognition. Contemporary Educational Psychology, 27(1), 51-79.
Tok, Ş. (2013). Effects of the know-want-learn strategy on students’ mathematics achievement, anxiety and metacognitive skills. Metacognition and Learning, 8(2), 193-212.
Veenman, M. V., Kerseboom, L., & Imthorn, C. (2000). Test anxiety and metacognitive skillfulness: Availability versus production deficiencies. Anxiety, Stress and Coping, 13(4), 391-412.
von der Embse, N. P., Schultz, B. K., & Draughn, J. D. (2015). Readying students to test: The influence of fear and efficacy appeals on anxiety and test performance. School Psychology International, 36(6), 620-637.
Wren, D. G., & Benson, J. (2004). Measuring test anxiety in children: Scale development and internal construct validation. Anxiety, Stress, and Coping, 17(3), 227 – 240.
Zeidner, M. (1998). Test anxiety: The state of the art. Plenum.
24. Mathematics Education Research
Paper
An Investigation of Eighth-Grade Students' Algebraic Thinking
Nurbanu Yılmaz-Tığlı1, Erdinç Çakıroğlu2
1Zonguldak Bülent Ecevit University, Turkiye; 2Middle East Technical University,Türkiye
Presenting Author: Yılmaz-Tığlı, Nurbanu
Algebra is described as a “mathematical language that combines operations, variables, and numbers to express mathematical structure and relationships in succinct forms” (Blanton et al., 2011, p. 67). It is one of the crucial branches of mathematics which constitutes a gateway between arithmetic reasoning in elementary school and advanced mathematics of higher grades (Blanton & Kaput, 2005). Researchers have agreed on the importance of algebraic thinking in learning mathematics (Asquith et al., 2007; Kieran, 2004). Stephens (2008) suggested that algebra in K-12 refers to “a way of thinking instead of something we simply do (e.g., collect like terms, isolate the variable, change signs when we change sides)” (p. 35). Researchers emphasized that the focus should not be on understanding the rules to manipulate symbols and use algebraic procedures excellently but on developing algebraic thinking. Hence, identifying students’ conceptions, difficulties, and errors in algebra might be a good step in determining these standards.
In Third International Mathematics and Science Study (TIMMS), students were asked real-world problems to use algebraic models and explain the relationships. Despite Turkish eighth-grade students performed gradually increasing performance in algebra tests from year to year (MoNE, 2014; MoNE, 2016; MoNE, 2020), the scores of eight grade students in algebra items in TIMMS 2019 presented that Turkish eighth-grade students’ algebra scores were under the average mathematics score (MoNE, 2020). The difficulties students faced while learning algebra resulted in them becoming isolated from mathematics and stopping learning mathematics early in high school (Kaput, 2002). Thus, a nationwide movement, algebra for all, was called by U.S. educators and researchers to get all students to attain algebra (Moses & Cobb, 2001). In response to these concerns, the National Council of Teachers of Mathematics (NCTM) proposed instructional programs that enable learners “to understand patterns, relations, and functions,” “to represent and analyze mathematical situations and structures using algebraic symbols,” “to use mathematical models to represent and understand quantitative relationships,” and “to analyze the change in various contexts” (NCTM, 2000, p. 37). Thus, it might be beneficial to explore the algebraic thinking of Turkish eighth-grade students to improve their algebra performance. This study investigates the research questions:
- What is the nature of eighth-grade students’ algebraic thinking around the issues of equivalence and equations, generalized arithmetic, variable, and functional thinking?
- Which difficulties and errors do eighth-grade students have around the issues of equivalence and equations, generalized arithmetic, variable, and functional thinking?
Methodology, Methods, Research Instruments or Sources UsedThis case study explores eighth-grade students’ conceptions and difficulties in algebra. Participants of the study are 267 eighth-grade students in a public middle school in Turkey. To investigate students’ conceptions and difficulties in algebra, first, the researchers prepared an Algebra Diagnostic Test (ADT) based on informal classroom observations in algebra classes, interviews with middle school mathematics teachers, and studies in the related literature. The test was prepared considering the big ideas in algebra (Blanton et al., 2015) and Turkey’s middle school mathematics curriculum (MoNE, 2018). Before conducting ADT on students, Pilot Testing I and Pilot Testing II processes were held to ensure validity and reliability issues. 140 students participated in Pilot Testing I in the spring semester of the 2017-2018 academic year, and 136 students participated in Pilot Testing II in the fall semester of the 2018-2019 academic year. Finally, ADT included 17 open-ended items in the scope of the big ideas of equivalence, expressions, equations, and inequalities, generalized arithmetic, variable, and functional thinking. ADT was administered to eighth-grade students in the spring semester of the 2018-2019 academic year. Students’ responses to the items in ADT were analyzed by coding their conceptions, solution strategies, and difficulties. Thus, students’ responses were explored in frequencies and percentages, considering their strategies and errors.
Conclusions, Expected Outcomes or FindingsResults indicated that students successfully did simple arithmetic and symbolic manipulations and solved algebra story or word problems. However, students were unsuccessful in the tasks such as comparing two algebraic expressions, writing the general rule of an algebra story problem, and interpreting the covariation in functions. Students generally focused on using x to manipulate the symbols instead of considering the relational understanding of x. Although students could write an algebraic expression based on a word problem, most students, who could write the symbolic expression, struggled to identify what x refers to in the algebraic expressions they wrote. Asquith et al. (2007) found that more than half of the students hold a multiple-values interpretation. Conversely, 25% of the students could express a multiple-values interpretation to answer the task comparing 3n and n+6. In functional thinking items, students were asked to find the answer for a specific value and write the general rule based on the algebra story problem. Results showed that most students were unsuccessful in writing the general rule of a given problem, although they could solve the problem using arithmetic (e.g., doing a substitution, guess-and-test, modeling, & unwinding). Also, it was observed that students mainly prefer to solve the problems using arithmetic even if they could write the algebraic expression symbolically.
ReferencesAsquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272. https://doi.org/10.1080/10986060701360910
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal For Research in Mathematics Education, 36(5), 412-446. https://doi.org/10.2307/30034944
Blanton, M., Levi, L., Crites, T., & Dougherty, B. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3–5. In R. M. Zbiek (Series Ed.), Essential understanding series. National Council of Teachers of Mathematics.
Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J. S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39-87.
Kaput, J. J. (2002). Research on the development of algebraic reasoning in the context of elementary mathematics: A brief historical overview. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L.
Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the international group for the psychology of mathematics education (pp. 120–122). ERIC.
Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, M. & M. Kendal (Eds), The Future of the Teaching and Learning of Algebra The 12thICMI Study (pp. 21-33). Springer.
Ministry of National Education [MoNE] (2018). Matematik dersi öğretim programı ilkokul ve ortaokul 1-8 sınıflar [Mathematics curriculum primary and middle school grades 1-8]. Retrieved on July 10, 2020 from http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329
Ministry of National Education. [MoNE]. (2014). TIMSS 2011 Ulusal Matematik ve Fen Raporu: 8. Sınıflar [TIMMS 2011 National Mathematics and Science Report: 8th Grade]. Retrieved November 01, 2022, from https://timss.meb.gov.tr/meb_iys_dosyalar/2022_03/07135958_TIMSS-2011-8-Sinif.pdf
Ministry of National Education. [MoNE]. (2016). TIMSS 2015 Ulusal Matematik ve Fen Ön Raporu [TIMMS 2015 National Mathematics and Science Preliminary Report]. Retrieved November 01, 2022, from https://timss.meb.gov.tr/meb_iys_dosyalar/2022_03/07135609_TIMSS_2015_Ulusal_Rapor.pdf
Ministry of National Education. [MoNE]. (2020). TIMSS 2019 Türkiye Ön Raporu [TIMMS 2019 Turkey Preliminary Report]. Retrieved November 01, 2022, from http://www.meb.gov.tr/meb_iys_dosyalar/2020_12/10173505_No15_TIMSS_2019_Turkiye_On_Raporu_Guncel.pdf
Moses, R. P., & Cobb, C. E. (2001). Radical equations. Beacon Press.
National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Author.
Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers?. The Journal of Mathematical Behavior, 27(1), 33-47. https://doi.org/10.1016/j.jmathb.2007.12.002
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