NORTHEASTERN SECTION OF THE MAA SPRING 1999 MEETING June 11-12, 1999 Colby College, Waterville, ME Abstracts Ed Dubinsky, Georgia State University Title: Meaning and Formalism in Mathematics A question that is important in the philosophy of mathematics and mathematics education is, What is the source of mathematical ideas, of meaning, in the mind of an individual? It is surely the case that this question does not have a single answer applicable to all mathematics and all people, so it is unsurprising and appropriate that many answers have been proposed. One possible source that may not have been given sufficient attention is formalism in the sense of expressing mathematical concepts in a formal language. In this talk, after discussing briefly some sources of mathematical ideas that have been considered in mathematics education such as direct physical experiences, metaphors, analogies, and intuition, I will consider two ways in which formalism can relate to mathematical meaning. One way is to capture an idea, perhaps held vaguely and intuitively and then use formal analyses to make it more precise and to investigate its properties. An even less considered use of formalism is to begin the development of an idea from its formal expression and to use the syntax to construct meaning for the concept. Both of these uses of formalism are skills that mathematicians generally possess and students need to learn. Although this is a major responsibility of mathematics education, it is not one which we are very successful at for more than a very small percentage of students. A major point of the talk is to describe some pedagogical approaches that can help students develop these skills so as to incorporate formalism as a powerful tool in understanding difficult mathematical concepts. | |
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