Abstract
Mathematicians commonly distinguish two modes of work in the discipline: Problem solving, and theory building. Mathematics education offers many opportunities to learn problem solving. This paper explores the possibility, and value, of designing instructional activities that provide supported opportunities for students to learn mathematics theory-building practices. It begins by providing a definition of these theory-building practices on the basis of which to formulate principles for the design of such instructional activities. The paper offers theoretical arguments that theory-building practices serve not only the synthesizing role that they play in disciplinary mathematics, but they also have the potential to enrich learners’ reasoning powers and enhance their problem solving skills. Examples of problem sets designed for this purpose are provided and analyzed.
Notes
In science, in contrast, Bereiter (2016, p. 1) writes, “Theory building starts when an explanatory idea is modified or further developed to produce a better explanation.”
Example: A. Which is more, the surface area of an open cylinder of height 5 cm. and radius 2 cm., or the area of a 5 cm. by 4π cm. rectangle? B. Show that the surface area of an open cylinder of height 5 cm. and radius 2 cm. equals the area of a 5 cm. by 4π cm. rectangle. Problems A and B involve the same mathematical structure, related to surface area measurement, but have different logical structures.
It is also informed by the author’s many years of university mathematics teaching at many levels, plus years of close study of records of K-12 teaching.
Here, for subsets X, Y of M, X\Y = the part of X not in Y = {x ∈ X | x ∉ Y} = X\(X∩Y).
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The author thanks Deborah Ball for help in framing the main argument of the paper, and the referees for many valuable suggestions.
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Bass, H. Designing opportunities to learn mathematics theory-building practices. Educ Stud Math 95, 229–244 (2017). https://doi.org/10.1007/s10649-016-9747-y
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DOI: https://doi.org/10.1007/s10649-016-9747-y