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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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A Case of a Computational Setting Facilitating Empirical Re-Conceptualization
The use of computer programming is ubiquitous for research mathematicians, and two common practices are facilitation of experimentation and testing of conjectures. These practices align with empirical re-conceptualization, which is the process where empirical data is used to identify patterns, form related conjectures, and then re-interpret the conjectures from a structural perspective. However, literature on empirical re-conceptualization has only examined student work done by hand. We present case study data of one student, Allen, using the programming language Python to facilitate empirical re-conceptualization as he found the closed-form solution for binomial coefficient C(n,k). We discuss why the computational environment was productive for empirical re-conceptualization and provide avenues for future research.
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- Award ID(s):
- 1650943
- PAR ID:
- 10518112
- Publisher / Repository:
- SIG/MAA on RUME
- Date Published:
- Journal Name:
- RUME XXV Conference Proceedings
- Volume:
- 1
- Issue:
- 1
- ISSN:
- 2474-9346
- Page Range / eLocation ID:
- 494-501
- Format(s):
- Medium: X
- Location:
- Omaha, NE
- Sponsoring Org:
- National Science Foundation
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