# (Fish) food for thought: Authority shifts in the interaction between mathematics and reality

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## Abstract

This theoretical paper explores the decision-making process involved in modelling and mathematizing situations during problem solving. Specifically, it focuses on the authority behind these choices (i.e., what or who determines the chosen mathematical models). We show that different types of situations involve different sources of authority, thereby creating different degrees of freedom for the problem solver engaged in the modelling process. It also means that mathematics plays different roles in these problems and situations. This epistemological analysis on the meaning of modelling implies that we should reconsider the mathematical status of realistic solutions and raises questions on the validity of some traditional choices of mathematical models and their use in diagnosing children’s conceptions. It also suggests constructing modelling tasks by choosing a certain variety of situations that might lead to a better understanding of the roles of mathematics.

## Keywords

Mathematical Concept Problem Solver Modelling Task Coalitional Game Proportion Model## Preview

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## References

- Aumann, R. (1999). On the matter of the man with three wives.
*Moriah, 22*(3/4), 98–107 (in Hebrew).Google Scholar - Aumann, R., & Maschler, M. (1985). Game theoretic analysis of a bankruptcy problem from the Talmud.
*Journal of Economic Theory*, 36, 195–213.CrossRefGoogle Scholar - Bonotto, C. (2007). How to replace word problems with activities of realistic mathematical modelling. In W. Blum, P. Galbraith, M. Niss, & H-W Henn (Eds.),
*Modelling and applications in mathematics education. The 14*^{th}*ICMI Study (New ICMI Studies Series)*(Vol. 10, pp. 185–192). New York:Springer.Google Scholar - De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and irresistibility of secondary school students’ errors.
*Educational Studies in Mathematics, 50*(3), 311–334.CrossRefGoogle Scholar - Department of Education and Early Childhood Development, Victoria (2007).
*Mathematics Developmental Continuum P-10*. Retrieved April 16, 2009 from www.education.VIC.gov.au/studentlearning/studentresources/maths/mathscontinuum/number/n55004p.htmGoogle Scholar - Galbraith, P. (2007). Beyond the low hanging fruit. In W. Blum, P. Galbraith, M. Niss, & H-W Henn (Eds.),
*Modelling and applications in mathematics education: The 14*^{th}*ICMI Study (New ICMI Studies Series)*(Vol. 10, pp. 79–88). New York:Springer.Google Scholar - Greer, B. (1993). The mathematical modelling perspective on wor(l)d problems.
*Journal of Mathematical Behavior, 12*(2), 239–250.Google Scholar - Hart, K. (1981).
*Children’s understanding of mathematics: 11–16*. London:John Murray.Google Scholar - Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs.
*For the Learning of Mathematics, 11*(1), 26–32.Google Scholar - Koirala, H. P. (1999). Teaching mathematics using everyday context: What if academic mathematics is lost? In O. Zaslavsky (Ed.),
*Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 161–168). Haifa, Israel:PME.Google Scholar - Lo, J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader.
*Journal for Research in Mathematics Education, 28*(2), 216–236.CrossRefGoogle Scholar - Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning.
*Journal of Mathematical Behavior, 22*(3), 335–368.CrossRefGoogle Scholar - Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. Galbraith, M. Niss, & H-W Henn (Eds.)
*Modelling and applications in mathematics education: The 14*^{th}*ICMI Study (New ICMI Studies Series)*(Vol. 10, pp. 3–33). New York:Springer.Google Scholar - Piaget, J., Grize, J. B., Szeminska, A., & Bang, V. (1977).
*Epistemology and psychology of functions*. Dordrecht, The Netherlands:Reidel.Google Scholar - Rivest, R. L., Shamir, A., & Adleman, L. (1978). A method for obtaining digital signatures and public-key cryptosystems.
*Communications of the ACM, 21*(2), 120–126.CrossRefGoogle Scholar - Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 41–52). Hillsdale, NJ:Erlbaum.Google Scholar - Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint.
*Journal of the Learning Sciences, 16*(4), 565–613.Google Scholar - Stillman, G., Brown, J., & Galbraith, P, (2008). Research into the teaching and learning of applications and modelling in Australia. In H. Forgasz, A. Barkatas, & A. Bishop (Eds.), Research in mathematics education in Australia 2004–2007 (pp. 141–164). Rotterdam, The Netherlands:Sense Publishers.Google Scholar
- Tourniaire, F. (1986). Proportions in elementary school.
*Educational Studies in Mathematics, 17*(4), 401–412.CrossRefGoogle Scholar