# New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies

## Abstract

The initial assumption of this article is that there is an overemphasis on abstraction-from-actions theoretical approaches in research on knowing and learning mathematics. This article uses a critical reflection on research on students’ ways of constructing mathematical concepts to distinguish between abstraction-from-actions theoretical approaches and abstraction-from-objects theoretical approaches. Acknowledging and building on research on knowing and learning processes in mathematics, this article presents a theoretical framework that provides a new perspective on the underlying abstraction processes and a new approach for interpreting individuals’ ways of constructing concepts on the background of their strategies to make sense of a mathematical concept. The view taken here is that the abstraction-from-actions and abstraction-from-objects approaches (although different) are complementary (rather than opposing) frameworks. The article is concerned with the theoretical description of the framework rather than with its use in empirical investigations. This article addresses the need for more advanced theoretical work in research on mathematical learning and knowledge construction.

### Keywords

Cognition Learner types Reflective abstraction Reflectural abstraction Sense making strategies Structural abstraction Theory development## Notes

### Acknowledgments

This article is a restructured and deeply extended version of an invited presentation given at the Federal University of Rio de Janeiro (Brazil) in November 2013. I express my gratitude for the comments given by Márcia M. F. Pinto. Discussions with David O. Tall have been particularly helpful and insightful in the elaboration of several key ideas put forward in this article. Special thanks to Gabriele Kaiser and Klaus Hasemann for their encouragement and ongoing advice. I am also grateful to the anonymous reviewers for their suggestions for improvement. The views expressed in this article do not necessarily reflect those of the researchers mentioned.

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