Title: Future mathematics teachers' ideas about proof: an explorative study using the ISIS-problem Authors: De Bock, Dirk ×Greer, BrianVan Dooren, Wim # Editors: Tzekaki, MariannaKaldrimidou, MariaSakonidis, Haralambos Issue Date: 2009 Publisher: PME Host Document: Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education. In Search for Theories in Mathematics Education. vol:1 pages:364 Conference: Conference of the International Group for the Psychology of Mathematics Education. In search for theories in Mathematics education edition:33 location:Thessaloniki, Greece date:19-24 July 2009 Abstract: The Isis problem, which has a link with the Isis cult of ancient Egypt (Davis & Hersh, 1981), asks: 'Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result.' The problem can be initially approached using routine expertise but then requires (for almost all school and college students) adaptive expertise, yet relies on the most rudimentary technical mathematics. It can be extended in numerous ways, for example by asking which triangles with integral sides have the corresponding property (a significantly more difficult problem) or by shifting up dimensionally to ask which cuboids with integer sides have volume and surface area numerically equal. Interesting questions then arise as to which proofs for the original problem are extendible. The problem is notable for the multiplicity and variety of proofs (empirically grounded, algebraic, geometrical) and associated representations. A selection of such proofs provides an instrument for probing students' ideas about proof. A group of 39 Flemish pre-service mathematics teachers was confronted with the Isis problem. More specifically, we first asked them to solve the problem and to look for more than one solution. Second, we invited them to study five given proofs (factorization, tiles, unit fractions, graph, table) and to rank these proofs from best to worst. We will present different self-found proofs in this group of Flemish pre-service mathematics teachers, as well as their rankings of and comments on the five given proofs. The results highlight a preference of many students for algebraic proofs (factorization and unit fractions) as well as their rejection of experimentation. Because the Isis problem relates two quantities of different dimensionality, it also connects with the considerable body of research showing that students do not understand the basic principle that linear enlargements by factor k result in 2-dimensional quantities, such as area, being enlarged by a factor of k2, and 3-dimensional quantities, such as volume, by a factor of k3 (De Bock, Van Dooren, Janssens, & Verschaffel, 2007), a principle that explains many phenomena in biology and engineering. References Davis, P., & Hersh, R. (1981). The mathematical experience. Boston: Birkhauser. De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2007). The illusion of linearity: From analysis to improvement. New York: Springer. Description: Also available on CD-ROM ISBN: 978-960-243-653-0 Publication status: published KU Leuven publication type: IC Appears in Collections: Education and TrainingInstructional Psychology and Technology Specific Teacher Training Programme in MathematicsLeuven Academic Centre for Professional Development in EducationResearch Centre for Educational Research & Development, Campus Brussels (-)Faculty of Economics and Business (FEB) - miscellaneous
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