Publication 300

Thompson P. W. (2008) Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In: Figueras O., Cortina J. L., Alatorre S., Rojano T. & Sépulveda A. (eds.) Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education. Volume 1. PME, Morélia (Mexico): 45–64. Fulltext at https://cepa.info/300
Mathematics during the late 18th century through the early 20th century experienced a period of turmoil and renewal that was rooted in a variety of attempts to put mathematics on solid conceptual footing. Taken-for-granted meanings of concept after concept, from number to function to system, came under increasing scrutiny because they could not carry the weight of new ways of thinking. In a very real sense, that period of time can be characterized as mathematicians’ search for broad, encompassing coherence among foundational mathematical meanings. Part of the resolution of this quest was the realization that meanings can be designed. We can decide what an idea will mean according to how well it coheres with other meanings to which we have also committed, and we can adjust meanings systematically to produce the desired coherence. Mathematics education is in the early stages of a similar period. Competing curricula and standards can be seen as expressions of competing systems of meanings--but the meanings themselves remain tacit and therefore competing systems of meanings cannot be compared objectively. I propose a method by which mathematics educators can make tacit meanings explicit and thereby address problems of instruction and curricula in a new light.

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