Swedish primary teacher education students’ perspectives on linear equations

Authors

  • Paul Andrews

Abstract

Linear equations, connecting arithmetic to the symbolism of formal mathematics, represent a key topic of mathematics. However, the understanding primary teacher education students bring to their studies has been rarely examined. In this study, students were invited to explain in writing how an unannotated solution to x + 5 = 4x – 1 had been conceptualised by the hidden solver. Data, coded against an iteratively derived framework, showed that most students were familiar with linear equations, able to articulate an objective for equation solving and offer solution strategies, typically based on either doing the same to both sides, swapping the side swapping the sign or both.

References

Alibali, M., Knuth, E., Hattikudur, S., McNeil, N. & Stephens, A. (2007). A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9 (3), 221-247. https://doi.org/10.1080/10986060701360902

Alvey, C., Hudson, R., Newton, J. & Males, L. M. (2016). Secondary pre-service teachers' algebraic reasoning about linear equation solving. Issues in the Undergraduate Mathematics Preparation of Teachers, 1 (1), 1-16.

Andrews, P. (2007). Negotiating meaning in cross-national studies of mathematics teaching: kissing frogs to find princes. Comparative Education, 43 (4), 489-509. https://doi.org/10.1080/03050060701611888

Andrews, P. & Sayers, J. (2012). Teaching linear equations: case studies from Finland, Flanders and Hungary. The Journal of Mathematical Behavior, 31 (4), 476-488. https://doi.org/10.1016/j.jmathb.2012.07.002

Andrews, P. & Larson, N. (2019). The development of a set of low-inference codes for uncovering students' understanding of linear equations: facilitating comparative analysis. Paper presented to the Seventh Conference on Research in Mathematics Education in Ireland, Dublin.

Andrews, P. & Xenofontos, C. (2017). Beginning teachers' perspectives on linear equations: a pilot quantitative comparison of Greek and Cypriot students. In T. Dooley & G. Gueudet (Eds.), Proceedings of CERME 10 (pp. 1594-1601). Institute of Education, Dublin City University.

Araya, R., Calfucura, P., Jiménez, A., Aguirre, C., Palavicino, A. M. et al. (2010). The effect of analogies on learning to solve algebraic equations. Pedagogies: An International Journal, 5 (3), 216-232. https://doi.org/10.1080/1554480X.2010.486160

Björklund, A., Clark, M., Edin, P.-A., Fredriksson, P. & Krueger, A. (2005). The Market comes to education in Sweden: an evaluation of Sweden's surprising school reforms. Russell Sage Foundation.

Buchbinder, O., Chazan, D. & Fleming, E. (2015). Insights into the school mathematics tradition from solving linear equations. For the Learning of Mathematics, 35 (2), 2-8.

Buchbinder, O., Chazan, D. & Capozzoli., M. (2019). Solving equations: exploring instructional exchanges as lenses to understand teaching and its resistance to reform. Journal for Research in Mathematics Education, 50 (1), 51-83. https://doi.org/10.5951/jresematheduc.50.1.0051

Caglayan, G. & Olive, J. (2010). Eighth grade students' representations of linear equations based on a cups and tiles model. Educational Studies in Mathematics, 74 (2), 143-162. https://doi.org/10.1007/s10649-010-9231-z

Capraro, 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

Casey, S., Lesseig, K., Monson, D. & Krupa, E. (2018). Examining preservice secondary mathematics teachers' responses to student work to solve linear equations. Mathematics Teacher Education and Development, 20 (1), 132-153.

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

Denancé, V. & Somat, A. (2015). Learning by explaining: impacts of explanations on the development of a competence. European Review of Applied Psychology, 65 (6), 307-315. https://doi.org/10.1016/j.erap.2015.10.005

Ellerton, N. & Clements, M. (2011). Prospective middle-school mathematics teachers' knowledge of equations and inequalities. In J. Cai & E. Knuth (Eds.), Early algebraization: a global dialogue from multiple perspectives (pp. 379-408). Springer. https://doi.org/10.1007/978-3-642-17735-4_21

Falkner, K., Levi, L. & Carpenter, T. (1999). Children's understanding of equality: a foundation for algebra. Mathematics Teaching in the Middle School, 6 (4), 232-236. https://doi.org/10.5951/TCM.6.4.0232

Filloy, E. & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9 (2), 19-25.

Fiorella, L. & Mayer, R. (2014). Role of expectations and explanations in learning by teaching. Contemporary Educational Psychology, 39 (2), 75-85. https://doi.org/10.1016/j.cedpsych.2014.01.001

Foster, C. (2018). Developing mathematical fluency: comparing exercises and rich tasks. Educational Studies in Mathematics, 97 (2), 121-141. https://doi.org/10.1007/s10649-017-9788-x

Foy, P., Arora, A. & Stanco, G. (2013). TIMSS 2011 user guide for the international database. Released items. Mathematics eighth grade. International Association for the Evaluation of Educational Achievement.

Herscovics, N. & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27 (1), 59-78. https://doi.org/10.1007/BF01284528

Huntley, M., Marcus, R., Kahan, J. & Miller, J. (2007). Investigating high-school students' reasoning strategies when they solve linear equations. The Journal of Mathematical Behavior, 26 (2), 115-139. https://doi.org/10.1016/j.jmathb.2007.05.005

Hästö, P., Palkki, R., Tuomela, D. & Star, J. (2019). Relationship between mathematical flexibility and success in national examinations. European Journal of Science and Mathematics Education, 7 (1), 1-13. https://doi.org/10.30935/scimath/9530

Isik, C. & Kar, T. (2012). The analysis of the problems posed by the pre-service teachers about equations. Australian Journal of Teacher Education, 37 (9), 93-113. https://doi.org/10.14221/ajte.2012v37n9.1

Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390- 419). Macmillan.

Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, 8 (1), 139-151.

Linsell, C. (2009). Students' knowledge and strategies for solving equations. In K. Hannah (Ed.), Findings from the Secondary Numeracy Project 2008 (pp. 29-43). New Zealand Ministry of Education.

Marschall, G. & Andrews, P. (2015). Polish teachers' conceptions of and approaches to the teaching of linear equations to grade six students: an exploratory case study. Research in Mathematics Education, 17 (3), 220-238. https://doi.org/10.1080/14794802.2015.1107498

McNeil, N., Grandau, L., Knuth, E., Alibali, M., Stephens, A. et al. (2006). Middle-school students' understanding of the equal sign: the books they read can't help. Cognition and Instruction, 24 (3), 367-385. https://doi.org/10.1207/s1532690xci2403_3

Ngu, B., Chung, S. & Yeung, A. (2015). Cognitive load in algebra: element interactivity in solving equations. Educational Psychology, 35 (3), 271-293. https://doi.org/10.1080/01443410.2013.878019

O'Neil, H., Chung, G., Kerr, D., Vendlinski, T., Buschang, R. & Mayer, R. (2014). Adding self-explanation prompts to an educational computer game. Computers in Human Behavior, 30 (1), 23-28. https://doi.org/10.1016/j.chb.2013.07.025

Pawley, D., Ayres, P., Cooper, M. & Sweller, J. (2005). Translating words into equations: a cognitive load theory approach. Educational Psychology, 25 (1), 75-97. https://doi.org/10.1080/0144341042000294903

Pirie, S. & Martin, L. (1997). The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations. Educational Studies in Mathematics, 34 (2), 159-181. https://doi.org/10.1023/A:1003051829991

Sfard, A. (1995). The development of algebra: confronting historical and psychological perspectives. The Journal of Mathematical Behavior, 14 (1), 15-39. https://doi.org/10.1016/0732-3123(95)90022-5

Skolverket. (2011). Curriculum for the compulsory school, preschool class and the recreation centre 2011. Skolverket.

Star, J. & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31 (3), 280-300. https://doi.org/10.1016/j.cedpsych.2005.08.001

Stephens, A. (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

Tall, D., Lima, R. de & 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), 1-13. https://doi.org/10.1016/j.jmathb.2013.12.003

Tanisli, D. & Kose, N. (2013). Pre-service mathematic teachers' knowledge of students about the algebraic concepts. Australian Journal of Teacher Education, 38 (2), 1-18. https://doi.org/10.14221/ajte.2013v38n2.1

Vaiyavutjamai, P. & Clements, M. (2006). Effects of classroom instruction on student performance on, and understanding of, linear equations and linear inequalities. Mathematical Thinking and Learning, 8 (2), 113-147. https://doi.org/10.1207/s15327833mtl0802_2

Vlassis, J. (2002). The balance model: hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49 (3), 341-359. https://doi.org/10.1023/A:1020229023965

Warren, E. & Cooper, T. (2005). Young children's ability to use the balance strategy to solve for unknowns. Mathematics Education Research Journal, 17 (1), 58-72. https://doi.org/10.1007/BF03217409

Wasserman, N. (2014). Introducing algebraic structures through solving equations: vertical content knowledge for K-12 mathematics teachers. PRIMUS, 24 (3), 191-214. https://doi.org/10.1080/10511970.2013.857374

Wetzstein, A. & Hacker, W. (2004). Reflective verbalization improves solutions - the effects of question-based reflection in design problem solving. Applied Cognitive Psychology, 18 (2), 145-156. https://doi.org/10.1002/acp.949

Xu, L., Liu, R.-D., Star, J., Wang, J., Liu, Y. & Zhen, R. (2017). Measures of potential flexibility and practical flexibility in equation solving. Frontiers in Psychology, 8, 1368. https://doi.org/10.3389/fpsyg.2017.01368

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Published

2024-11-19

How to Cite

Andrews, P. (2024). Swedish primary teacher education students’ perspectives on linear equations. NOMAD Nordic Studies in Mathematics Education, 25(2), 29–48. Retrieved from https://tidsskrift.dk/NOMAD/article/view/149050

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