INVESTIGATING PRE-SERVICE TEACHERS’ (PST) MATHEMATICAL THINKING TRAJECTORIES (MTT) IN SOLVING SYSTEM OF LINEAR EQUATIONS PROBLEMS

Sudirman Sudirman(1*), Ahmad Hasan Saifurrisal(2), Pradina Parameswari(3), Nego Linuhung(4),

(1) Universitas Negeri Malang
(2) Universitas Negeri Malang
(3) Universitas Negeri Malang
(4) Universitas Negeri Malang
(*) Corresponding Author


Abstract


Linear algebra is an area of mathematics that is widely considered challenging. One of the subjects covered is the system of linear equations in two variables. During their first year of college, Pre-Service Teachers (PST) study this subject using increasingly sophisticated approaches and theories. The objective of this qualitative study is to classify and describe the Mathematical Thinking Trajectories (MTT) of PST in solving a system of linear equations in two variables problems. Thirty-three PST majoring in mathematics education who enrolled in the Errors and Misconceptions in Mathematics course participated in this study, which used written answers as research data. The Three Worlds of Mathematics theory framework is used in this study. Seven MTT classifications were identified by the research findings: (1) embodied, (2) symbolic, (3) embodied-symbolic, (4) symbolic-embodied, (5) formal, (6) embodied-symbolic-embodied, and (7) symbolic-embodied-symbolic. “Embodied” refers to how PST applies mathematical ideas that focus on actual objects, “symbolic” focuses on symbols, operations, and their properties, and “formal” focuses on definitions, theorems, and logical procedures. In learning practices, the results of this study can be applied to create learning that is meaningful and that fosters the growth of algebraic thinking abilities.

Keywords


Algebraic thinking; mathematical thinking trajectory; meaningful learning; system of linear equations; three worlds of mathematics theory.

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References


Aarto-Pesonen, L., & Piirainen, A. (2020). Teacher Students’ Meaningful Learning in Widening Learning Worlds. Teaching Education, 31(3), 323–342. https://doi.org/10.1080/10476210.2018.1561662

Ahmed, A. Y., Miller, V. W., Gebremeskel, H. H., & Ebessa, A. D. (2019). Mapping Inequality in Access to Meaningful Learning in Secondary Education in Ethiopia: Implications for Sustainable Development. Educational Studies, 45(5), 554–581. https://doi.org/10.1080/03055698.2018.1509777

Alreshidi, N. A. K. (2023). Enhancing Topic-Specific Prior Knowledge of Students Impacts Their Outcomes in Mathematics. Frontiers in Education, 8. https://doi.org/10.3389/feduc.2023.1050468

Binder, T., Sandmann, A., Sures, B., Friege, G., Theyssen, H., & Schmiemann, P. (2019). Assessing Prior Knowledge Types as Predictors of Academic Achievement in The Introductory Phase of Biology and Physics Study Programmes Using Logistic regression. International Journal of STEM Education, 6(1). https://doi.org/10.1186/s40594-019-0189-9

Bittermann, A., McNamara, D., Simonsmeier, B. A., & Schneider, M. (2023). The Landscape of Research on Prior Knowledge and Learning: a Bibliometric Analysis. In Educational Psychology Review (Vol. 35, Issue 2). Springer. https://doi.org/10.1007/s10648-023-09775-9

Chin, K. E., Jiew, F. F., & Tall, D. (2022). The Articulation Principle for making long-term sense of mathematical expressions by how they are spoken and heard: Two case studies. Mathematics Enthusiast, 19(2), 657–676. https://doi.org/10.54870/1551-3440.1567

Constantinou, A. C., Guo, Z., & Kitson, N. K. (2023). The impact of prior knowledge on causal structure learning. Knowledge and Information Systems, 65(8), 3385–3434. https://doi.org/10.1007/s10115-023-01858-x

De Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18. https://doi.org/10.1007/s10649-007-9086-0

Fazio, L. K., DeWolf, M., & Siegler, R. S. (2016). Strategy Use and Strategy Choice in Fraction Magnitude Comparison. Journal of Experimental Psychology: Learning Memory and Cognition, 42(1), 1–16. https://doi.org/10.1037/xlm0000153

Filloy, E., & Rojano, T. (1989). Solving Equations: the Transition from Arithmetic to Algebra. For the Learning of Mathematics, 9(2), 19–25. http://www.jstor.org/stable/40247950 .

Greve, A., Cooper, E., Tibon, R., & Henson, R. N. (2019). Knowledge Is Power: Prior Knowledge Aids Memory for Both Congruent and Incongruent Events, but in Different Ways. Journal of Experimental Psychology: General, 148(2), 325–341. https://doi.org/10.1037/xge0000498.supp

Grogan, J., Innes, P., Carter, J., & Raciti, M. (2023). Troubling Knowledge in Australian Indigenous Studies: How Prior Knowledge Affects Undergraduate Student Learning. Higher Education Research and Development, 42(8), 1920–1935. https://doi.org/10.1080/07294360.2023.2209512

Hailikari, T. (2009). Assessing University Students’ Prior Knowledge: Implications for Theory and Practice [University of Helsinki]. https://helda.helsinki.fi/server/api/core/bitstreams/37035d88-f392-4cb0-beb9-3b9765b2eb0a/content

Heitzmann, N., Stadler, M., Richters, C., Radkowitsch, A., Schmidmaier, R., Weidenbusch, M., & Fischer, M. R. (2023). Learners’ Adjustment Strategies Following Impasses in Simulations - Effects of Prior Knowledge. Learning and Instruction, 83. https://doi.org/10.1016/j.learninstruc.2022.101632

Kärki, T., Keinänen, H., Tuominen, A., Hoikkala, M., Matikainen, E., & Maijala, H. (2018). Meaningful Learning with Mobile Devices: Pre-service Class Teachers’ Experiences of Mobile Learning in The Outdoors. Technology, Pedagogy and Education, 27(2), 251–263. https://doi.org/10.1080/1475939X.2018.1430061

Kidron, I., & Tall, D. (2015). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics, 88(2), 183–199. https://doi.org/10.1007/s10649-014-9567-x

Kieran, C. (2022). The Multi-dimensionality of Early Algebraic Thinking: Background, Overarching Dimensions, and New Directions. ZDM - Mathematics Education, 54(6), 1131–1150. https://doi.org/10.1007/s11858-022-01435-6

Kosiol, T., Rach, S., & Ufer, S. (2019). (Which) Mathematics Interest is Important for a Successful Transition to a University Study Program? International Journal of Science and Mathematics Education, 17(7), 1359–1380. https://doi.org/10.1007/s10763-018-9925-8

Lee, J. Y., Donkers, J., Jarodzka, H., & van Merriënboer, J. J. G. (2019). How Prior Knowledge Affects Problem-Solving Performance in a Medical Simulation Game: Using Game-logs and Eye-tracking. Computers in Human Behavior, 99, 268–277. https://doi.org/10.1016/j.chb.2019.05.035

Levin, M., & Walkoe, J. (2022). Seeds of Algebraic Thinking: a Knowledge in Pieces Perspective on The Development of Algebraic Thinking. ZDM - Mathematics Education, 54(6), 1303–1314. https://doi.org/10.1007/s11858-022-01374-2

Mayer, R. E. (2002). Rote Versus Meaningful Learning. Theory into Practice, 41(4), 226–232. https://doi.org/10.1207/s15430421tip4104_4

Pitta-Pantazi, D., Chimoni, M., & Christou, C. (2020). Different Types of Algebraic Thinking: an Empirical Study Focusing on Middle School Students. International Journal of Science and Mathematics Education, 18(5), 965–984. https://doi.org/10.1007/s10763-019-10003-6

Sibgatullin, I. R., Korzhuev, A. V., Khairullina, E. R., Sadykova, A. R., Baturina, R. V., & Chauzova, V. (2022). A Systematic Review on Algebraic Thinking in Education. Eurasia Journal of Mathematics, Science and Technology Education, 18(1), 1–15. https://doi.org/10.29333/EJMSTE/11486

Simonsmeier, B. A., Flaig, M., Deiglmayr, A., Schalk, L., & Schneider, M. (2022). Domain-Specific Prior Knowledge and Learning: A Meta-Analysis. Educational Psychologist, 57(1), 31–54. https://doi.org/10.1080/00461520.2021.1939700

Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. Reprinted from the December 1976 Issue of Mathematics Teaching, the Journal of the Association of Teachers of Mathematics, Great Britain. www.nctm.org.

Södervik, I., Vilppu, H., Boshuizen, H., & Murtonen, M. (2022). Development of University Teachers’ Professional Vision of Students’ Prior Knowledge During a Short Pedagogical Training. International Journal of Teaching and Learning in Higher Education, 34(1), 7–24. https://files.eric.ed.gov/fulltext/EJ1363721.pdf

Song, H. S., Kalet, A. L., & Plass, J. L. (2016). Interplay of Prior Knowledge, Self-Regulation and Motivation in Complex Multimedia Learning Environments. Journal of Computer Assisted Learning, 32(1), 31–50. https://doi.org/10.1111/jcal.12117

Stewart, S., Andrews-Larson, C., & Zandieh, M. (2019). Linear Algebra Teaching and Learning: Themes from Recent Research and Evolving Research Priorities. ZDM - Mathematics Education, 51(7), 1017–1030. https://doi.org/10.1007/s11858-019-01104-1

Tall, D. (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 20(2), 5–24. https://doi.org/10.1007/BF03217474

Tall D. (2016). Long Term Effect of Sense-making and Anxiety in Algebra. In Stewart, S. (ed): And the Rest Is Just Algebra, (Springer, New York). https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2016b-long-term-sense-algebra.pdf

Tall, D. (2019). Long-term Principles for Meaningful Teaching and Learning of Mathematics. To appear in Sepideh Stewart (ed.): Mathematicians’ Reflections on Teaching: A Symbiosis with Mathematics Education Theories. Springer. https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2019c-long-term-framework.pdf

Tall, D. (2020a). Building Long-term Meaning in Mathematical Thinking: Aha! and Uh-Huh!. In Bronislaw Czarnocha and William Baker (Eds): Creativity of an Aha! Moment and Mathematics Education, 226–259. https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2020b-aha-uhu.pdf

Tall, D. (2020b). Making Sense of Mathematical Thinking over the Long Term: The Framework of Three Worlds of Mathematics and New Developments. Draft. To appear in Tall, D. & Witzke, I. (Eds.): MINTUS: Beiträge Zur Mathematischen, Naturwissenschaftlichen Und Technischen Bildung. Wiesbaden: Springer. https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2020a-3worlds-extension.pdf

Tall, D., de Lima, R. N., & Healy, L. (2014). Evolving a Three-World Framework For Solving Algebraic Equations in The Light of What a Student Has Met-Before. The Journal of Mathematical Behavior, 34, 1–13. https://doi.org/10.1016/J.JMATHB.2013.12.003

Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97–124. https://doi.org/10.1007/s10649-014-9531-9

Thurn, C., Nussbaumer, D., Schumacher, R., & Stern, E. (2022). The Role of Prior Knowledge and Intelligence in Gaining from a Training on Proportional Reasoning. Journal of Intelligence, 10(2). https://doi.org/10.3390/jintelligence10020031

Vargas-Hernández, J. G., & Vargas-González, O. C. (2022). Strategies for Meaningful Learning in Higher Education. Journal of Research in Instructional, 2(1), 47–64. https://doi.org/10.30862/jri.v2i1.41

Wettergren, S. (2022). Identifying and Promoting Young Students’ Early Algebraic Thinking. LUMAT: International Journal on Math, Science and Technology Education, 10(2), 190–214. https://doi.org/10.31129/10.2.1617

Witherby, A. E., Carpenter, S. K., & Smith, A. M. (2023). Exploring The Relationship between Prior Knowledge and Metacognitive Monitoring Accuracy. Metacognition and Learning, 18(2), 591–621. https://doi.org/10.1007/s11409-023-09344-z

Wu, Z. (2017). Effects of Using Problem of The Week in Teaching on Teacher Learning and Change in Algebraic Thinking and Algebra. ZDM - Mathematics Education, 49(2), 203–221. https://doi.org/10.1007/s11858-017-0844-x

Zambrano R., J., Kirschner, F., Sweller, J., & Kirschner, P. A. (2019). Effects of Prior Knowledge on Collaborative and Individual Learning. Learning and Instruction, 63. https://doi.org/10.1016/j.learninstruc.2019.05.011




DOI: http://dx.doi.org/10.24127/ajpm.v13i3.10388

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