Abstract
Linear (proportional) functions are undoubtedly one of the most common models for representing and solving both pure and applied problems in elementary mathematics education. But according to several authors, different aspects of the current culture and practice of school mathematics develop in students a tendency to use these linear models also in situations in which they are not applicable. This article reports two closely related studies about this phenomenon in 12–13- and 15–16-year old students working on word problems involving lengths and areas of similar plane figures of different kinds of shapes, as well as about the influence of drawings in breaking this improper use of linearity. Generally speaking, the results provide a convincing demonstration of the predominance of the linear model in secondary students' solutions of this kind of mensurational problem.
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REFERENCES
Behr, M. J., Harel, G., Post, T., and Lesh, R.: 1992, ‘Rational number, ratio, and proportion’, in D.A. Grouws (ed.), Handbook of Research on Mathematical Teaching and Learning, Macmillan, New York, pp. 276-295.
Berté, A.: 1993, Mathématique dynamique, Nathan, Paris.
De Bock, D., Verschaffel, L., and Janssens, D.: 1996, De lineariteitsillusie: een exploratief onderzoek, Internal report, Center for Instructional Psychology and Technology, University of Leuven, Leuven.
De Corte, E., Greer, B., and Verschaffel, L.: 1996, ‘Mathematics teaching and learning’, in D. Berliner and R. Calfee (eds.), Handbook of Educational Psychology, Macmillan, New York, pp. 491-549.
Feys, R.: 1995, ‘Meten en metend rekenen’, in L. Verschaffel and E. De Corte (eds.), Naar een nieuwe reken/wiskundedidactiek voor de basisschool en de basiseducatie. Deel 3: Verder bouwen aan gecijferdheid, Studiecentrum Open Hoger Onderwijs (StOHO)/Acco, Brussel/Leuven, pp. 99-135.
Freudenthal, H.: 1973, Mathematics as an Educational Task, D. Reidel, Dordrecht.
Freudenthal, H.: 1983, Didactical Phenomenology of Mathematical Structures, D. Reidel, Dordrecht.
Groupe d' Enseignement Mathématique: 1994, De question en question. Mathématiques 2, Didier Hatier, Brussels.
Hart, K.M.: 1981, Children's Understanding of Mathematics: 11-16, Murray, London.
Karplus, R., Pulos, S., and Stage, E.K.: 1983, ‘Early adolescents proportional reasoning on ‘;rate’ problems’, Educational Studies in Mathematics 14, 219-233.
Lin, F.L.: 1991, ‘Characteristics of ‘adders’ in proportional reasoning’, Proceedings of the National Science ROC(D) 1(1), 1-13.
National Council of Teachers of Mathematics: 1989, Curriculum and Evaluation Standards for School Mathematics, NCTM, Reston, VA.
National Council of Teachers of Mathematics: 1994, Curriculum and Evaluation Standards for School Mathematics Addenda Series Grades 5-8. Understanding Rational Numbers and Proportions, NCTM, Reston, VA.
Pólya, G.: 1945, How to Solve It, Princeton University Press, Princeton.
Rogalski, J.: 1982, ‘Acquisition de notions relatives à la dimensionalité des mesures spatiales (longueur, surface)’, Recherches en didactique des mathématiques 3(3), 343-396.
Rouche, N.: 1989, ‘Prouver: amener à l’évidence ou contrÔler des implications?’, in Commission inter-IREM Histoire et Epistémologie des Mathématiques, La démonstration dans lhistoire, IREM de Besançon et IREM de Lyon, Lyon, pp. 8-38.
Schoenfeld, A.: 1992, ‘Learning to think mathematically: problem solving, metacognition, and sense making in mathematics’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 334-370.
Streefland, L.: 1984, ‘Search for the roots of ratio: Some thoughts on the long term learning process (Towards::: a theory). Part I: Reflections on a teaching experiment’, Educational Studies in Mathematics 15, 327-348.
Tourniaire, F. and Pulos, S.: 1985, ‘Proportional reasoning: A review of the literature’, Educational Studies in Mathematics 16, 181-204.
Treffers, A.: 1987, Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction. The Wiskobas Project, D. Reidel, Dordrecht.
Vanotterdijk, R.: 1995, ‘Toelichting bij de visie achter het voorstel van eindtermen 1ste graad SO’, Wiskunde en Onderwijs 21, 4-53.
Verschaffel, L.: 1995, ‘Visies op reken/wiskundeonderwijs op de basisschool’, in L. Verschaffel and E. De Corte (eds.), Naar een nieuwe reken/wiskundedidactiek voor de basisschool en de basiseducatie. Deel 1: Achtergronden, Studiecentrum Open Hoger Onderwijs (StOHO)/Acco, Brussel/Leuven, pp. 95-128.
Verschaffel, L. and De Corte, E.: 1996, ‘Number and arithmetic’, in A. Bishop, K. Clements, C. Keitel and C. Laborde (eds.), International Handbook of Mathematics Education. Part I, Kluwer, Dordrecht, pp. 99-138.
Verschaffel, L. and De Corte, E.: 1997, ‘Word problems. A vehicle for promoting authentic mathematical understanding and problem solving in the primary school’, in T. Nunes and P. Bryant (eds.), Learning and Teaching Mathematics: An International Perspective, Psychology Press, Hove, UK, pp. 69-98.
Verschaffel, L. and De Corte, E.: in press, ‘Teaching realistic mathematical modelling in the elementary school. A teaching experiment with fifth graders’, Journal for Research in Mathematics Education.
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De Bock, D., Verschaffel, L. & Janssens, D. The Predominance of the Linear Model in Secondary School Students' Solutions of Word Problems Involving Length and Area of Similar Plane Figures. Educational Studies in Mathematics 35, 65–83 (1998). https://doi.org/10.1023/A:1003151011999
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DOI: https://doi.org/10.1023/A:1003151011999