Identifying patterns is an important part of mathematical investigation, but many students struggle to justify their pattern-based generalizations. These findings have led some to argue for a de-emphasis on patterning, but others argue that it can support insight into a problem’s structure. We introduce a phenomenon, empirical re-conceptualization, in which learners generalize based on an empirical pattern, and then re-interpret it from a structural perspective. We elaborate this construct by providing a representative example of empirical reconceptualization from two secondary students. Our findings indicate that developing empirical results can foster subsequent insights, which can in turn lead to justification and proof.
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Empirical Re-Conceptualization: Bridging from Empirical Patterns to Insight and Understanding
Identifying patterns is an important part of mathematical investigation, but many students struggle to explain or justify their pattern-based generalizations or conjectures. These findings have led some researchers to argue for a de-emphasis on pattern-based activities, but others argue that empirical investigation can support the discovery of insight into a problem’s structure. We introduce a phenomenon we call empirical re-conceptualization, in which learners identify a conjecture based on an empirical pattern, and then re-interpret that conjecture from a structural perspective. We elaborate this construct by drawing on interview data from undergraduate calculus students and research mathematicians, providing a representative example of empirical re-conceptualization from each participant group. Our findings indicate that developing empirical results can foster subsequent insights, which can in turn lead to justification and proof.
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- Award ID(s):
- 1920538
- PAR ID:
- 10192432
- Date Published:
- Journal Name:
- Proceedings of the twenty-third Annual Conference on Research in Undergraduate Mathematics Education
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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