Problem solving as modelling: a case of augmented‐quotient division problem

There are three main aspects in mathematical problem solving, especially when the problem includes a real situation: #opa#cp formulating the problem in mathematical terms; #opb#cp operating within the mathematical world; #opc#cp translating the mathematical results into the original situation. In research on problem solving, researchers’ attentions have been mainly paid to the first aspect #opa#cp, and the third aspect #opc#cp has not been deliberated. The purpose of this paper is making this aspect clearer by applying the theory of mathematical modelling to problem solving. Taking problem solving as a kind of mathematical modelling, the mathematical results can be seen as a basis for forecasts, decisions, or actions, and the third aspect of problem solving can be seen as the process in which solvers would make forecasts or decisions and take an action on the problem situation. The example, the case of augmented‐quotient division problems, is presented to illustrate this aspect in detail, and it shows the...

[1]  W. Kintsch.  Learning From Text , 1986, Knowing, Learning, and Instruction.

[2]  E. Silver,et al.  Examinations of Situation-Based Reasoning and Sense-Making in Students’ Interpretations of Solutions to a Mathematics Story Problem , 1992 .

[3]  Henk van der Kooij.  Assessment of Mathematical Modelling and Applications , 1992 .

[4]  T. Mormann,et al.  The Space of Mathematics: Philosophical, Epistemological, and Historical Explorations , 1992 .

[5]  I. Lakatos,et al.  Proofs and Refutations: Frontmatter , 1976 .

[6]  K. Nunokawa.  Solver's structures of a problem situation and their global restructing , 1994 .

[7]  Werner Blum,et al.  Applied mathematical problem solving, modelling, applications, and links to other subjects — State, trends and issues in mathematics instruction , 1991 .

[8]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[9]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[10]  W. Blum,et al.  Modelling, applications and applied problem solving , edited by Werner Blum, Mogens Niss and Ian Huntley. Pp 250. £30. 1989. ISBN 0-7458-0633-3 (Ellis Horwood) , 1990, The Mathematical Gazette.

[11]  J. Hiebert.  Conceptual and procedural knowledge : the case of mathematics , 1987 .

[12]  P. V. Bendegem.  Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics , 1993 .

[13]  A. Schoenfeld.  Cognitive Science and Mathematics Education , 1987 .

[14]  João Pedro da Ponte,et al.  Mathematical Problem Solving and New Information Technologies , 1992, NATO ASI Series.