# Designing opportunities to learn mathematics theory-building practices

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## 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.

### Keywords

Theory building Problem solving Abstraction Instructional design## Notes

### Acknowledgements

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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