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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. 

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.