Empirical re-conceptualization: From empirical generalizations to insight and understanding

Abstract

Identifying patterns is an important part of mathematical reasoning, but many students struggle to justify their pattern-based generalizations. Some researchers argue for a de-emphasis on patterning activities, but empirical investigation has also been shown to support discovery and insight into problem structures. We introduce the phenomenon of empirical re-conceptualization, which is the development of an empirical generalization that is subsequently re-interpreted from a structural perspective. We define and elaborate empirical re-conceptualization by drawing on data from secondary and undergraduate students. We also identify four major instructional supports we found for empirical re-conceptualization: (1) experiencing need for verification, (2) fostering contextual interpretation, (3) fostering reflection and justification, and (4) fostering pattern exploration, as well as three processes facilitating the transition from empirical to deductive reasoning: (a) verification, (b) justification, and (c) creation / interpretation.