Solving algebraic equations by using the bar model: Theoretical and empirical considerations

Anna-Katharina Roos; Leander Kempen

doi:10.29333/ejmste/15147

Full Text (PDF)

Anna-Katharina Roos ¹ , Leander Kempen ² ^*

More Detail

¹ Institute for Development and Research in Mathematics Education, TU Dortmund University, Dortmund, GERMANY² Institute of Mathematics and Computer Science, University of Greifswald, Greifswald, GERMANY^* Corresponding Author

Abstract

Solving equations is known to bear several challenges for learners. We discuss an approach based on conceptual understanding regarding the transformation of equations with the help of the so-called bar model in combination with the transposing strategy. First, we sketch shortly the main ideas that guided the development of the learning environment. Second, we discuss insights from the first design experiments with six students working with equation transformation in their regular school curriculum. These design experiments are embedded in a design research approach. In particular, we zoom into the semiotic processes of how learners connect several representations and emphasize a varying difficulty regarding single concept elements necessary to understand the concept of equivalent equations as a whole. Based on that, obstacles that come along with using the bar model are highlighted. Finally, we point to theoretical insights and implications for enhancing our learning environment.

Keywords

algebra
bar model
solving equations
multiple representations
conceptual understanding

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 20, Issue 9, September 2024, Article No: em2505

https://doi.org/10.29333/ejmste/15147

Publication date: 10 Sep 2024

Article Views: 377

Article Downloads: 265

Open Access References How to cite this article

References

Arcavi, A., Drijvers, P., & Stacey, K. (2016). The learning and teaching of Algebra: Ideas, insights, and activities. Routledge. https://doi.org/10.4324/9781315545189
Baysal, E., & Sevinc, S. (2022). The role of the Singapore bar model in reducing students’ errors on algebra word problems. International Journal of Mathematical Education in Science and Technology, 53(2), 289-310. https://doi.org/10.1080/0020739X.2021.1944683
Booth, J. L., Barbieri, C., Eyer, F., & Pare-Blagoev, E. J. (2014). Persistent and pernicious errors in algebraic problem solving. The Journal of Problem Solving, 7(1), 10-23. https://doi.org/10.7771/1932-6246.1161
Bruner, J. (1967). On knowing: Essays for the left hand. Harvard University Press.
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, 613-632. https://doi.org/10.1016/j.jmathb.2013.07.002
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 68-95). Routledge.
Drollinger-Vetter, B. (2011). Verstehenselemente und strukturelle Klarheit: Fachdidaktische Qualität der Anleitung von mathematischen Verstehensprozessen im Unterricht [Elements of understanding and structural clarity: Subject-didactic quality of the instruction of mathematical understanding processes in lessons]. Waxmann.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
Fagnant, A., & Vlassis, J. (2013). Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students. Educational Studies in Mathematics, 84(1), 149-168. https://doi.org/10.1007/s10649-013-9476-4
Fan, L., & Zhu, Y. (2007). From convergence to divergence: The development of mathematical problem solving in research, curriculum, and classroom practice in Singapore. ZDM, 39, 491-501. https://doi.org/10.1007/s11858-007-0044-1
Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the learning of Mathematics, 9(2), 19-25.
Flick, U. (2007). Managing the quality of qualitative research. SAGE. https://doi.org/10.4135/9781849209441
Goldin, G., & Shteingold, N. (2001). Systems of representation and the development of mathematical concepts. In A. A. Cuoco, & F. R. Curcio (Eds.), The role of representation in school mathematics (pp. 1-23). NCTM.
Gravemeijer, K. (2016). Mediating between concrete and abstract. In T. Nunes, & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 315-345). Psychology Press Ltd.
Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 17-51). Routledge.
Gravemeijer, K., & Prediger, S. (2019). Topic-specific design research: An introduction. In G. Kaiser, & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 33-57). Springer. https://doi.org/10.1007/978-3-030-15636-7_2
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
Ho, S. Y., & Lowrie, T. (2014). The model method: Students’ performance and its effectiveness. Journal of Mathematical Behavior, 35, 87-100. https://doi.org/10.1016/j.jmathb.2014.06.002
Hoven, J., & Garelick, B. (2007). Singapore math: Simple or complex? Educational Leadership, 65(3), 28-31.
Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner, & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 167-194). Lawrence Erlbaum. https://doi.org/10.4324/9781315044378-13
Kaur, B. (2019). The why, what and how of the ‘model’ method: A tool for representing and visualizing relationships when solving whole number arithmetic word problems. ZDM, 51, 151-168. https://doi.org/10.1007/s11858-018-1000-y
Kieran, C. (1988). Two different approaches among algebra learners. In A. F. Coxford (Ed.), The ideas of algebra (pp. 91-96). NCTM.
Kieran, C. (2006). Research on the learning and teaching of algebra: A broadening of sources of meaning. In Á. Gutiérrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 11-49). Brill. https://doi.org/10.1163/9789087901127_003
Knuth, E. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31, 500-507. https://doi.org/10.2307/749655
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2011). Middle-school students’ understanding of core algebraic concepts: Equivalence & variable. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 259-276). Springer. https://doi.org/10.1007/978-3-642-17735-4_15
Koleza, E. (2015). The bar model as a visual aid for developing complementary/variation problems. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the 9^th Congress of the European Society for Research in Mathematics Education (pp. 1940-1946). CERME.
Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20(1), 41-54. https://doi.org/10.1007/BF00356040
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64. https://doi.org/10.3102/00346543060001001
Malle, G. (1993). Didaktische Probleme der elementaren Algebra [Didactic problems in elementary algebra]. Vieweg & Sohn. https://doi.org/10.1007/978-3-322-89561-5
Marshall, A. M., Superfine, A. C., & Canty, R. S. (2010). Star students make connections. Teaching Children Mathematics, 17(1), 38-47. https://doi.org/10.5951/TCM.17.1.0038
Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 365-380). Springer. https://doi.org/10.1007/978-94-017-9181-6_13
Morin, L. L., Watson, S. M., Hester, P., & Raver, S. (2017). The use of a bar model drawing to teach word problem solving to students with mathematics difficulties. Learning Disability Quarterly, 40(2), 91-104. https://doi.org/10.1177/0731948717690116
Moschkovich, J. (2013). Principles and guidelines for equitable mathematics teaching practices and materials for English language learners. Journal of Urban Mathematics Education, 6(1), 45-57. https://doi.org/10.21423/jume-v6i1a204
Ng, S. F., & Lee, K. (2005). How primary five pupils use the model method to solve word problems. The Mathematics Educator, 9(1), 60-83.
Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282-313. https://doi.org/10.5951/jresematheduc.40.3.0282
Post, M., & Prediger, S. (2024). Teaching practices for unfolding information and connecting multiple representations: The case of conditional probability information. Mathematics Education Research Journal, 36(1), 97-129. https://doi.org/10.1007/s13394-022-00431-z
Prediger, S. (2008). The relevance of didactical categories for analyzing obstacles in conceptual change–Revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3-17. https://doi.org/10.1016/j.learninstruc.2006.08.001
Prediger, S., & Roos, A.-K. (2023). Deeper Learning in Mathematics: The case of algebraic equations. In D. Coyle, O. Meyer, & S. Staschen-Dielmann (eds.), A Deeper Learning Companion for CLIL: Putting Pluriliteracies into Practice (pp. 262-287). Cambridge University Press.
Renkl, A., Berthold, K., Große, C. S., & Schwonke, R. (2013). Making better use of multiple representations: How fostering metacognition can help. In R. Azevedo, & V. Aleven (Eds.), International handbook of metacognition and learning technologies (pp. 397-408). Springer. https://doi.org/10.1007/978-1-4419-5546-3_26
Schnell, S., & Prediger, S. (2014). Multiple representations as tools for discovering pattern and variability–Insights into the dynamics of learning processes. In T. Wassong, D. Frischemeier, P. R. Fischer, H. Hochmuth, & P. Bender (Eds.), Mit Werkzeugen Mathematik und Stochastik lernen [Learn mathematics and stochastics with tools] (pp. 179-192). Springer. https://doi.org/10.1007/978-3-658-03104-6
Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference–A longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389-408. https://doi.org/10.1007/s10649-011-9305-6
Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification–The case of algebra. Educational Studies in Mathematics, 26(2-3), 191-228. https://doi.org/10.1007/BF01273663
Tondorf, A., & Prediger (2022). Connecting characterizations of equivalence of expressions: Design research in grade 5 by bridging graphical and symbolic representations. Educational Studies in Mathematics, 111, 399-422. https://doi.org/10.1007/s10649-022-10158-0
Uribe, Á., & Prediger, S. (2021). Students’ multilingual repertoires-in-use for meaning-making: Contrasting case studies in three multilingual constellations. Journal of Mathematical Behavior, 62(100820), 100820. https://doi.org/10.1016/j.jmathb.2020.100820
Verschaffel, L., Torbeyns, J., De Smedt, B., Luwel, K., & Van Dooren, W. (2007). Strategy flexibility in children with low achievement in mathematics. Educational and Child Psychology, 24(2), 16-27. https://doi.org/10.53841/bpsecp.2007.24.2.16
Vicente, S., Verschaffel, L., Sanchez, R., & Munez, D. (2022). Arithmetic word problem solving: Analysis of Singaporean and Spanish textbooks. Educational Studies in Mathematics, 111(3), 375-397. https://doi.org/10.1007/s10649-022-10169-x
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
Weinberg, A., Dresen, J., & Slater, T. (2016) Students’ understanding of algebraic notation: A semiotic systems perspective. Journal of Mathematical Behavior, 43, 70-88. https://doi.org/10.1016/j.jmathb.2016.06.001
Wolters, M. A. (1983). The part-whole schema and arithmetical problems. Educational Studies in Mathematics, 14, 127-138. https://doi.org/10.1007/BF00303682
Yan, K. C. (2002). The model method in Singapore. The Mathematics Educator, 6(2), 47-64.

How to cite this article

APA

Roos, A.-K., & Kempen, L. (2024). Solving algebraic equations by using the bar model: Theoretical and empirical considerations. Eurasia Journal of Mathematics, Science and Technology Education, 20(9), em2505. https://doi.org/10.29333/ejmste/15147

Vancouver

Roos AK, Kempen L. Solving algebraic equations by using the bar model: Theoretical and empirical considerations. EURASIA J Math Sci Tech Ed. 2024;20(9):em2505. https://doi.org/10.29333/ejmste/15147

AMA

Roos AK, Kempen L. Solving algebraic equations by using the bar model: Theoretical and empirical considerations. EURASIA J Math Sci Tech Ed. 2024;20(9), em2505. https://doi.org/10.29333/ejmste/15147

Chicago

Roos, Anna-Katharina, and Leander Kempen. "Solving algebraic equations by using the bar model: Theoretical and empirical considerations". Eurasia Journal of Mathematics, Science and Technology Education 2024 20 no. 9 (2024): em2505. https://doi.org/10.29333/ejmste/15147

Harvard

Roos, A.-K., and Kempen, L. (2024). Solving algebraic equations by using the bar model: Theoretical and empirical considerations. Eurasia Journal of Mathematics, Science and Technology Education, 20(9), em2505. https://doi.org/10.29333/ejmste/15147

MLA

Roos, Anna-Katharina et al. "Solving algebraic equations by using the bar model: Theoretical and empirical considerations". Eurasia Journal of Mathematics, Science and Technology Education, vol. 20, no. 9, 2024, em2505. https://doi.org/10.29333/ejmste/15147