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Some multiplicative structures in elementary education: a view from relational paradigm

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Abstract

The multiplicative reasoning that students should develop in elementary school is a key area of research in contemporary mathematics education. Researchers employ various views including multiplication as arithmetic operation, multiplicative structures, and multiplicative relationships. They also propose various classifications of multiplicative structures to support students’ development of multiplicative reasoning and problem solving. Our work contributes to this conversation by focusing on simple multiplicative relationships. Drawing on Davydov’s Theory of Developmental Instruction, we employ the relational view to analyze some multiplicative structures identified by researchers and practitioners. We propose a typology of basic multiplicative relationships as well as their specific graphical representations as a set of mental tools allowing for a holistic flexible understanding of the multiplicative situations, simple and complex, primary school students usually encounter (in grades 1–6). We suggest that the new approach may better support students’ multiplicative and relational reasoning and problem solving.

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Notes

  1. This very general definition can be applied to ill-defined problems (ex. By how much is the world’s climate getting hotter?) or even to numerical problems (ex. Is eight times two equal to sixteen?). Both examples can be interpreted as presenting multiplicative relationships.

  2. Theoretically speaking, quantitative relationships can be established between two (equality), three (basic additive or multiplicative relationships) or even more elements.

  3. Davydov’s unit of measurement refers to any unit (such as 2 apples or 1/2 an apple), not only standardized units such as kilometer or hour.

References

  • Artigue, M. (2011). Les défis de l’enseignement des mathématiques dans l’éducation de base. Paris, France: Organisation des Nations Unies pour l’éducation, la science et la culture (UNESCO).

  • Bednarz, N., & Janvier, B. (1996). Algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 115–136). Dordrecht, the Netherlands: Kluwer Academic Publishers.

  • Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Fong Ng, S., & Schmittau, J. (2005). The development of studients’ algebraic thinking in earlier grades: Lessons from China and Singapore. Zentralblatt Für Didaktik Der Mathematik, 37(1), 5–15.

    Article  Google Scholar 

  • Davis, B., & Renert, M. (2013). The math teachers know: Profound understanding of emergent mathematics. London, UK: Routledge.

  • Davydov, V. V. (1982). Psychological characteristics of the formation of mathematical operations in children. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: Cognitive perspective (pp. 225–238). Hillsdale, NJ: Lawrence Erlbaum Associates.

  • Davydov, V. V. (1992). The psychological analysis of multiplication procedures. Focus on Learning Problems in Mathematics, 14(1), 3–67.

    Google Scholar 

  • Davydov, V. V. (2008). Problems of developmental instruction: A theoretical and experimental psychological study. Hauppauge, NY: Nova Science Publishers.

    Google Scholar 

  • DeBlois, L. (2011). Enseigner les mathématiques. Des intentions à préciser pour planifier, guider et interpréter. St-Foy, Qc: Les Presses de l’Université Laval.

    Book  Google Scholar 

  • Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York, NY: NCTM.

  • Greer, B. (1994). Extending the meaning of multiplication and division. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). New York, NY: State University of New York Press.

  • Greer, B. (2012). Inversion in mathematical thinking and learning. Educational Studies in Mathematics, 79, 429–438.

    Article  Google Scholar 

  • Izsák, A., & Beckmann, S. (2019). Developing a coherent approach to multiplication and measurement. Educational Studies in Mathematics, 101, 83–103.

    Article  Google Scholar 

  • Kaur, B. (2018). The why, what and how of the ‘model’ method: A tool for representing and visualising relationships when solving whole number arithmetic word problems. ZDM - Mathematics Education, 51(1), 151–168.

  • Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., & von Aster, M. (2011). Mental number line training in children with developmental dyscalculia. NeuroImage, 57(3), 782–795. https://doi.org/10.1016/j.neuroimage.2011.01.070

    Article  Google Scholar 

  • Maffia, A., & Mariotti, M. A. (2018). Intuitive and formal models of whole number multiplication: Relations and emerging structures. For the Learning of Mathematics, 38(3), 30–36.

  • Mancl, D. B. (2011). Investigating the effects of a combined problem-solving strategy for students with learning difficulties in mathematics (Unpublished doctoral dissertation). University of Nevada, Las Vegas, NV.

  • Marshall, S. (1995). Schemes in problem solving. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • Moyer-Packenham, P. S., Ulmer, L. A., & Anderson, K. L. (2012). Examining pictorial models and virtual manipulatives for third-grade fraction instruction. Journal of Interactive Online Learning, 11(3), 103–120.

    Google Scholar 

  • 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.

    Google Scholar 

  • Nunes, T., & Csapó, B. (2011). Developing and assessing mathematical reasoning. In Csapó & M. Szendrei (Eds.), Framework for diagnostic assessment of mathematics (pp. 17–56). Budapest, Hungary: Nemzeti Tankönyvkiadó.

  • Pape, S. J. (2003). Compare word problems: Consistency hypothesis revisited. Contemporary Educational Psychology, 28(3), 396–421. https://doi.org/10.1016/S0361-476X(02)00046-2

    Article  Google Scholar 

  • Polotskaia, E., & Savard, A. (2018). Using the relational paradigm: Effects on pupils’ reasoning in solving additive word problems. Research in Mathematics Education, 20(1), 70–90.

    Article  Google Scholar 

  • Quebec Ministry of Education Leisure and Sport (2009). Document d’accompagnement, Progression des apprentissages, Mathématique. Ministère de l’Éducation, du Loisir et du Sport.

  • Robertson, S. I. (2017). Problem solving: Perspectives from cognition and neuroscience. London, New York: Routledge, Taylor & Francis Group.

    Google Scholar 

  • Rockwell, S. B. (2012). Teaching students with autism to solve additive word problems using schema-based strategy instruction (Unpublished doctoral dissertation). University of Florida, Gainesville, FL.

  • 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.

  • Schmidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems. Educational Studies in Mathematics, 28(28), 55–72.

  • Schmittau, J. (2010). The relevance of Russian elementary mathematics education. In A. Karp & B. R. Vogeli (Eds.), Russian mathematics education: History and world significance (pp. 253–278). New York, NY: World scientific.

  • Schwartz, J. L. (1996). Semantic aspects of quantity. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Sun, X. H., Xin, Y. P., & Huang, R. (2019). A complementary survey on the current state of teaching and learning of whole number arithmetic and connections to later mathematical content. ZDM - Mathematics Education, 51, 1–12. https://doi.org/10.1007/s11858-019-01041-z

  • Thompson, P. W. (1989). A cognitive model of quantity-based reasoning in algebra. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

  • Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208.

    Article  Google Scholar 

  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany, NY: SUNY Press.

    Google Scholar 

  • Van de Walle, J. A., & Lovin, L. H. (2008). L’enseignement des mathématiques : L’élève au centre de son apprentissage. Canada: ERPI.

    Google Scholar 

  • Venenciano, L., Yagi, S. L., & Zenigami, F. K. (2020). The development of relational thinking: A study of measure up first-grade students and their symbolic understandings. Educational Studies in Mathematics. This issue.

  • Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 124–127). New York, NY: Academic Press.

  • Verschaffel, L., Dooren, W., Greer, B., & Mukhopadhyay, S. (2010). Reconceptualising word problems as exercises in mathematical modelling. Journal für Mathematik-Didaktik, 31(1), 9–29.

    Article  Google Scholar 

  • Xin, Y. P., Zhang, D., Park, J. Y., Tom, K., Whipple, A., & Si, L. (2011). A comparison of two mathematics problem-solving strategies: Facilitate algebra-readiness. The Journal of Educational Research, 104(6), 381–395.

    Article  Google Scholar 

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Funding

This project was supported by the Social Sciences and Humanities Research Council of Canada (application N 430-2015-00884) and the Quebec ministry of Education and Higher Education.

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Authors and Affiliations

  1. Université du Québec en Outaouais, Gatineau, Canada

    Elena Polotskaia

  2. McGill University, Montreal, Canada

    Annie Savard

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  1. Elena Polotskaia
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  2. Annie Savard
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Correspondence to Elena Polotskaia.

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Appendix 1 Multiplicative situations modeled with basic relationships

Appendix 1 Multiplicative situations modeled with basic relationships

Table 3 The example of situations in the table are equivalent to those described in various sources, but they are formulated as situations, not problems. It means that all quantities are known and there is no question. All types of models used in this table accept real numbers as values for quantities.
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Polotskaia, E., Savard, A. Some multiplicative structures in elementary education: a view from relational paradigm. Educ Stud Math 106, 447–469 (2021). https://doi.org/10.1007/s10649-020-09979-8

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