Fostering students’ mathematical understanding: Toward a theoretical framework

The development of mathematical understanding is widely recognized as a critical goal of mathematics education. Mathematical understanding can be both a product and a process. However, there are varied perspectives on what mathematical understanding is and how it may be fostered. This paper describes a framework that organizes and explains the ways that mathematics educators conceive of research and practice regarding mathematical understanding. Specifically, the framework describes five aspects of mathematical understanding (factual, procedural, conceptual, metacognitive, and affective) and five lenses (mathematical, cognitive, social, diagnostic, and instructional) that mathematics educators use when focusing on fostering students’ mathematical understanding. We illustrate the framework with examples drawn from work published in leading mathematics education research journals. We hope that researchers and practitioners can use this framework to shape their perspectives of different aspects and different lenses of understanding. The paper concludes with a discussion of future directions for research and practice on fostering mathematical understanding.