Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay

Elena Polotskaia; Annie Savard; Viktor Freiman

doi:10.12973/eurasia.2015.1325a

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Elena Polotskaia ¹ ^* , Annie Savard ², Viktor Freiman ³

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¹ Université du Québec en Outaouais² McGill University³ Université de Moncton^* Corresponding Author

Abstract

This essay proposes a reflection on the learning difficulties and teaching approaches associated with word problem solving. We question the development of word problem solving skills in the early grades of elementary school. We are trying to revive the discussion for two reasons. First, the knowledge in question—reversibility of arithmetic operations and flexibility of mathematical thinking—is the key element in elementary mathematics. Second, we hope to create a shift in the understanding of this knowledge development in students. Using the folk tale “The Three Little Pigs” as a metaphor, we analyze difficulties students experience while learning to solve word problems involving addition and subtraction. We formulate a hypothesis about the cognitive duality of word problem solving. This hypothesis explains a number of well-known learning difficulties and suggests teaching principles that could help avoid developmental obstacles and pitfalls within the teaching/learning process.

Keywords

additive word problems
elementary mathematics
knowledge development
thinking duality
problem solving

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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 11, Issue 2, 2015, 251-261

https://doi.org/10.12973/eurasia.2015.1325a

Publication date: 02 Apr 2015

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References

Barrouillet, P., & Camos, V. (2002). Savoirs, savoir-faire arithmétiques, et leurs déficiences (version longue). Ministère de la Recherche, programme cognitique, école et sciences cognitives.
Brousseau, G. (1988). Théorie des situations didactiques. Grenoble: La Pensée Sauvage.
Bruno, A., & Martinon, A. (2004). The importance of the statement in addition and subtracion word problems. Focus on Learning Problems in Mathematics, 26(1), 1–22.
Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children‘s problem-solving processes. Journal for Research in Mathematics Education, 24(5), 428 – 441.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Carpenter, T. P., Moser, J. M., & Bebout, H. C. (1988). Representation of addition and word problems. Journal for Research in Mathematics Education, 19(4), 345– 357.
Cummins, D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405–438.
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, New Jersey: Lawrence Erlbaum Associates.
Davydov, V. V. (2008). Problems of developmental instruction: a theoretical and experimental psychological study. Hauppauge, NY: Nova Science Publishers.
DeBlois, L. (1997). Quand additionner ou soustraire implique comparer. Éducation et Francophonie, 25(1), 102–120.
DeBlois, L. (2006). Influence des interprétations des productions des élèves sur les stratégies d‘intervention en classe de mathématique. Educational Studies in Mathematics, 62(3), 307–329.
DeBlois, L. (2011). Enseigner les mathématiques. Des intentions à préciser pour planifier, guider et interpréter. St-Foy, Qc: Les Press de l‘Université Laval.
Erdniev, P. M. (1979). Obuchenie matematike v nachal’nykh klassakh: Opyt obuchenia metodom ukrupnenia didacticheskih edinic [Teaching mathematics in elementary school: Method of desegmentation of didactic units]. Moscow: Pedagogika.
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.
Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Hanbook of research on mathematics teaching and learning (pp. 243–275). Reston, VA: National Council of Teachers of Mathematics.
Gamo, S., Sander, E., & Richard, J.-F. (2009). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20(5), 400–410.
Hatano, G. (1982). Cognitive consequences of practice in culture specific procedural skills. The Quartely Newsletter of the Laboratory of Comparative Human Cognition, 4, 15–18.
Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. xi–xiii). Mahwah, NJ, US: Lawrence Erlbaum Associates.
Hatano, G., & Oura, Y. (2003). Reconceptualizing school learning using insight from expertise research. Educational Researcher, 32(8), 26–29.
Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87(1), 18–32.
Iannece, D., Mellone, M., & Tortora, R. (2010). Early multiplicative thought: a kindergarten path. In M. F. Pinto & T. F. Kawasaki (Eds.), 34th Conference of the international groupe for the psychology of mathematics education (pp. 121–127). Belo Horizonte, Brazil.
Kilpatrick, J., Swafford, J., & Bradford, F. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National academy press.
Kintsch, W. (2005). An overview of top-down and bottom-up effects in comprehension: The CI perspective. Discourse Processes, 39(2), 125–128.
Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109–29.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. Chicago: University Of Chicago Press.
Lewis, A. B., & Mayer, R. E. (1987). Students‘ miscomprehension of relational statements in arithmetic word problems. Journal of Educational Psychology, 79(4), 363–371.
Masson, S., Potvin, P., Riopel, M., Foisy, L. B., & Lafortune, S. (2012). Using fMRI to study conceptual change : Why and how ? International Journal of Environmental & Science Education, 7(1), 19–35.
Mikhakov, S. (2010). Tri porosenka [Three little pigs]. Moscow: Samovar.
Neef, N. A., Nelles, D. E., Iwata, B. A., & Page, T. J. (2003). Analysis of precurrent skills in solving mathematics story problems. Journal of Applied Behavior Analysis, 36(1), 21–33.
Nesher, P. (1980). The Stereotyped nature of school word problems. For the Learning of Mathematics, 1(1), 41–48.
Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373– 394.
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.
Okamoto, Y. (1996). Modeling Children‘s Understanding of Quantitative Relations in Texts : A Developmental Perspective. Cognition and Instruction, 14(4), 37–41.
Pape, S. J. (2003). Compare word problems: Consistency hypothesis revisited. Contemporary Educational Psychology, 28(3), 396–421.
Polotskaia, E., (2014). How elementary students learn to mathematically analyze word problems: The case of addition and subtraction. Ph.D. dissertation, DISE, McGill University, Montreal, Canada.
Polotskaia, E., Freiman, V., & Savard, A. (2013). Developing reasoning about simple additive structures: one task for elementary students. In A. M. Lindmeier & A. Heinze (Eds.), Mathematics learning across the life span. The 37th conference of the international group for the psychology of mathematics education (Vol. 5, p. 253). Kiel, Germany.
Riley, M. S., Greeno, J. G., & Heller, J. L. (1984). Development of children‘s problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). Orlando, FL: Academic Press, Inc.
Rodriguez, M. C. (2004). Effects of differential arithmetic practice on children’s solution of mathematical word problems. Experimental Psychology. Cornell University.
Savard, A., & Polotskaia, E. (2014). Gérer l‘accès aux mathématiques dans la résolution de problèmes textuels : une exploration du côté de l‘enseignement primaire. Éducation et Francophonie, XLII(2), 138–157.
Savard, A., Polotskaia, E., Freiman, V., & Gervais, C. (2013). Tasks to promote holistic flexible reasoning about simple additive structures. In C. Margolinas (Ed.), Proceedings of ICMI Study 22 Task Design in Mathematics Education (Vol. 1, pp. 271–280). Oxford, England.
Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov. The Mathematics Educator, 8(1), 60–87.
Sfard, A. (1987). Two conceptions of mathematical notions: Operational and structural. In Proceedings of the Eleventh International Conference of PME (Vol. 3, p. 162). Montreal, Qc.
Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.
Skemp, R. R. (1987). Relational understanding and instrumental understanding. In Psychology of Learning mathematics (pp. 152–163). Lawrence Erlbaum Associates.
Stavy, R., & Babai, R. (2009). Problem solving in mathematics: insights from reaction time and brain imaging studies. In EARLI Advanced Study Colloquium: Cognitive Neuroscience Meets Mathematics Education (pp. 26– 27). Brugge, Hof van Watervliet.
Thevenot, C. (2010). Arithmetic word problem solving: Evidence for the construction of a mental model. Acta Psychologica, 133(1), 90–5.
Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 39–59). Hillsdale, New Jersey: Lawrence Erlbaum Associates.
Verschaffel, L., & Corte, E. (1993). A decade of research on word problem solving in Leuven: Theoretical, methodological, and practical outcomes. Educational Psychology Review, 5(3), 239–256.
Verschaffel, L., Luwel, K., Torbeyns, J., & Dooren, W. Van. (2009). Conceptualizing , investigating , and enhancing adaptive expertise in elementary mathematics education, XXIV(2003), 335–359.
Vygotsky, L. S. (1996). Lekcii po pedologii (Lectures about pedology). Igevsk: Udmurtskii universitet (Udmourt University).
Vygotsky, L. S., Luria, A. R., & Knox, J. E. (1993). Studies on the history of behavior: Ape, primitive, and child. (J. E. Knox & V. I. Golod, Eds.) (p. 249). Hillsdale, NJ: Lawrence Erlbaum Associates.
Wachsmuth, I. (1981). Two modes of thinking - also relevent for the learning of mathematics? For the Learning of Mathematics, 2(2), 38–45.
Xin, Y. P. (2007). Word problem solving tasks in textbooks and their relation to student performance. The Journal of Educational Research, 100(6), 347–360.
Xin, Y. P. (2008). The effect of schema-based instruction in solving mathematics word problems: an emphasis on prealgebraic conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39(5), 526–551.
Xin, Y. P., Wiles, B., & Lin, Y.-Y. (2008). Teaching conceptual model-based word problem story grammar to enhance mathematics problem solving. The Journal of Special Education, 42(3), 163–178.
Zaitseva, S. A., & Tselischeva, I. I. (2010). Organizatsia raboty po proverke reshenia prostyh textovyh zadach [Methods of the solution verification for simple word problems]. Nachal`naia Shkola plus Do I Posle [Elementary School + Before and After], 7, 13–17.
Znamenskaia, O., Ostroverh, O., Riabina, L., & Hassan, B. (2009). Monitoring individualnogo progressa uchebnykh deistvii shkolnikov [Monitoring of students‘ individual progress in learning activities]. Voprosy Obrazovania, 3, 5– 75.

How to cite this article

APA

Polotskaia, E., Savard, A., & Freiman, V. (2015). Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), 251-261. https://doi.org/10.12973/eurasia.2015.1325a

Vancouver

Polotskaia E, Savard A, Freiman V. Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay. EURASIA J Math Sci Tech Ed. 2015;11(2):251-61. https://doi.org/10.12973/eurasia.2015.1325a

AMA

Polotskaia E, Savard A, Freiman V. Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay. EURASIA J Math Sci Tech Ed. 2015;11(2), 251-261. https://doi.org/10.12973/eurasia.2015.1325a

Chicago

Polotskaia, Elena, Annie Savard, and Viktor Freiman. "Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay". Eurasia Journal of Mathematics, Science and Technology Education 2015 11 no. 2 (2015): 251-261. https://doi.org/10.12973/eurasia.2015.1325a

Harvard

Polotskaia, E., Savard, A., and Freiman, V. (2015). Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), pp. 251-261. https://doi.org/10.12973/eurasia.2015.1325a

MLA

Polotskaia, Elena et al. "Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay". Eurasia Journal of Mathematics, Science and Technology Education, vol. 11, no. 2, 2015, pp. 251-261. https://doi.org/10.12973/eurasia.2015.1325a