Assessing What Students Think About Solving Word Problems: The Case of Reversible Reasoning in Students
Abstract
Although reversible reasoning is an important strategy in solving mathematical word problems, few studies have examined the link between reversible reasoning and word problem solving. The present study therefore aims to examine students’ thought processes in translating statements within word problems by integrating reversible reasoning. A qualitative approach was adopted involving 71 students with diverse backgrounds, genders, educational institutions and achievement levels. A task designed to stimulate reversible reasoning was developed and supporting stimulus questions were formulated. Data were collected through Google Form submissions, think-aloud protocols and interviews. Twelve students who demonstrated indications of reversible reasonin were interviewed to gain in-depth insights. The collected data were analysed using a case study approach comprising three stages: preliminary analysis, open coding and axial coding. Six research themes were identified related to the students’ mental activities in reversing temporal sequences, reversing rate relationships, reversing residual perspectives, reversing variable roles, reversing mathematical operations and reversing rate-time concepts. The findings highlight the importance of designing learning experiences that stimulate mental flexibility, conceptual understanding and metacognition as frameworks for fostering reversible reasoning.
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References
Boonen, A. J. H., de Koning, B. B., Jolles, J., & van der Schoot, M. (2016). Word problem solving in contemporary math education: A plea for reading comprehension skills training. Frontiers in Psychology, 7. https://doi.org/10.3389/fpsyg.2016.00191
Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (4th ed.). Pearson.
Csíkos, C., & Szitányi, J. (2020). Teachers’ pedagogical content knowledge in teaching word problem solving strategies. ZDM – Mathematics Education, 52(1), 165–178. https://doi.org/10.1007/s11858-019-01115-y
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
Fuchs, L., Fuchs, D., Seethaler, P. M., & Barnes, M. A. (2020). Addressing the role of working memory in mathematical word-problem solving when designing intervention for struggling learners. ZDM – Mathematics Education, 52(1), 87–96. https://doi.org/10.1007/s11858-019-01070-8
Fujita, T., Kondo, Y., Kumakura, H., & Kunimune, S. (2017). Students’ geometric thinking with cube representations: Assessment framework and empirical evidence. Journal of Mathematical Behavior, 46, 96–111. https://doi.org/10.1016/j.jmathb.2017.03.003
Goulet-Lyle, M. P., Voyer, D., & Verschaffel, L. (2020). How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving? ZDM – Mathematics Education, 52(1), 139–149. https://doi.org/10.1007/s11858-019-01098-w
Greer, B. (2012). Inversion in mathematical thinking and learning. Educational Studies in Mathematics, 79(3), 429–438. https://doi.org/10.1007/s10649-011-9317-2
Haciomeroglu, E. S. (2015). The role of cognitive ability and preferred mode of processing in students’ calculus performance. Eurasia Journal of Mathematics, Science and Technology Education, 11(5), 1165–1179. https://doi.org/10.12973/eurasia.2015.1400a
Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432. https://doi.org/10.1080/07370008.2010.511565
Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196
Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results. ZDM – Mathematics Education, 43(3), 359–372. https://doi.org/10.1007/s11858-011-0322-9
Hong, Y. Y., & Thomas, M. O. J. (2015). Graphical construction of a local perspective on differentiation and integration. Mathematics Education Research Journal, 27(2), 183–200. https://doi.org/10.1007/s13394-014-0135-6
Hord, C., & Xin, Y. P. (2013). Intervention research for helping elementary school students with math learning difficulties understand and solve word problems: 1996–2010. Learning Disabilities: A Multidisciplinary Journal, 19(1). https://doi.org/10.18666/ldmj-2013-v19-i1-4789
Ikram, M., Purwanto, Nengah Parta, I., & Susanto, H. (2020). Mathematical reasoning required when students seek the original graph from a derivative graph. Acta Scientiae, 22(6), 45–64. https://doi.org/10.17648/acta.scientiae.5933
Ikram, M., Purwanto, Parta, I. N., & Susanto, H. (2020). Relationship between reversible reasoning and conceptual knowledge in composition of function. Journal of Physics: Conference Series, 1521(3). https://doi.org/10.1088/1742-6596/1521/3/032004
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. Basic Books, Inc.
Kar, T. (2015). Analysis of problems posed by sixth-grade middle school students for the addition of fractions in terms of semantic structures. International Journal of Mathematical Education in Science and Technology, 46(6), 879–894. https://doi.org/10.1080/0020739X.2015.1021394
Lee, K., Ng, S. F., & Bull, R. (2018). Learning and solving algebra word problems: The roles of relational skills, arithmetic, and executive functioning. Developmental Psychology, 54(9). https://doi.org/10.1037/dev0000561
Leong, Y. H., Toh, T. L., Tay, E. G., Quek, K. S., & Dindyal, J. (2012). Relooking “Look Back”: A student’s attempt at problem solving using Polya’s model. International Journal of Mathematical Education in Science and Technology, 43(3), 357–369. https://doi.org/10.1080/0020739X.2011.618558
Mamolo, A., Ruttenberg-Rozen, R., & Whiteley, W. (2015). Developing a network of and for geometric reasoning. ZDM - Mathematics Education, 47(3), 483–496. https://doi.org/10.1007/s11858-014-0654-3
Miles, M. B., Huberman, A. M., & Saldana, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.). SAGE Publications, Inc.
Natsheh, I., & Karsenty, R. (2014). Exploring the potential role of visual reasoning tasks among inexperienced solvers. ZDM - Mathematics Education, 46(1), 109–122. https://doi.org/10.1007/s11858-013-0551-1
Öçal, M. F., Şen, C., Güler, G., & Kar, T. (2020). The investigation of prospective mathematics teachers’ non-algebraic solution strategies for word problems. International Journal of Mathematical Education in Science and Technology, 51(4), 563–584. https://doi.org/10.1080/0020739X.2019.1597936
Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an Extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11, 4–9. http://math.coe.uga.edu/TME/Issues/v11n1/2olive.pdf
Paoletti, T. (2020). Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya. Journal of Mathematical Behavior, 57, Article 100741. https://doi.org/10.1016/j.jmathb.2019.100741
Powell, S. R., Berry, K. A., & Benz, S. A. (2020). Analyzing the word-problem performance and strategies of students experiencing mathematics difficulty. Journal of Mathematical Behavior, 58, Article 100759. https://doi.org/10.1016/j.jmathb.2020.100759
Ramful, A. (2014). Reversible reasoning in fractional situations: Theorems-in-action and constraints. Journal of Mathematical Behavior, 33, 119–130. https://doi.org/10.1016/j.jmathb.2013.11.002
Ramful, A. (2015). Reversible reasoning and the working backwards problem solving strategy. Australian Mathematics Teacher, 71(4), 28–32.
Savard, A., & Polotskaia, E. (2017). Who’s wrong? Tasks fostering understanding of mathematical relationships in word problems in elementary students. ZDM - Mathematics Education, 49(6), 823–833. https://doi.org/10.1007/s11858-017-0865-5
Seah, R., & Horne, M. (2019). The construction and validation of a geometric reasoning test item to support the development of learning progression. Mathematics Education Research Journal, 32, 607–628. https://doi.org/10.1007/s13394-019-00273-2
Swidan, O., & Fried, M. (2021). Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies. Journal of Mathematical Behavior, 62, Article 100847. https://doi.org/10.1016/j.jmathb.2021.100847
Verschaffel, L., Schukajlow, S., Star, J., & Van Dooren, W. (2020). Word problems in mathematics education: a survey. ZDM - Mathematics Education, 52(1). https://doi.org/10.1007/s11858-020-01130-4
Vondrová, N., Novotná, J., & Havlíčková, R. (2019). The influence of situational information on pupils’ achievement in additive word problems with several states and transformations. ZDM - Mathematics Education, 51(1), 183–197. https://doi.org/10.1007/s11858-018-0991-8
Xin, Y. P. (2019). The effect of a conceptual model-based approach on ‘additive’ word problem solving of elementary students struggling in mathematics. ZDM - Mathematics Education, 51(1), 139–150. https://doi.org/10.1007/s11858-018-1002-9
Yao, X. (2020). Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames. Journal of Mathematical Behavior, 60, Article 100803. https://doi.org/10.1016/j.jmathb.2020.100803
Yin, R. K. (2011). Qualitative research from start to finish (2nd ed.). The Guilford Press.
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