# Distinguishing Two Stages of Mathematics Conceptual Learning

Article

## Abstract

In this theoretical article, we distinguish two stages of learning a new mathematical concept – participatory and anticipatory. We use a recently developed mechanism for explaining mathematical conceptual learning – reflection on activity-effect relationship – as well as von Glasersfeld’s tripartite model of a scheme, to explain qualitative distinctions between the two stages. We use this distinction to explain why instructional interventions (including inquiry-based approaches) may not bring about the intended instructional goals.

### Keywords

abstraction concept development conceptual learning conceptualization epistemology learning mathematics learning stages of learning## Preview

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