Does mathematical learning occur in going from concrete to abstract or in going from abstract to concrete?

The notions of abstract and concrete are central to the conceptualization of mathematical knowing and learning. It is generally accepted that development goes from concrete toward the abstract; but dialectical theorists maintain just the opposite: development consists of an ascension from the abstract to the concrete. In this article, we reformulate the relationship of abstract and concrete consistent with a dialectical materialist approach to conscious human activity, as it was developed in the line of cultural-historical psychology. Our reformulation of development in and through interpretation shows that rather than being a movement from concrete to abstract or from abstract to concrete, development occurs in a double ascension that simultaneously moves in both direction: it is a passage of one in the other. In the proposed approach, the theoretical contradictions of earlier approaches to the issue of abstract have been eliminated. (C) 2006 Elsevier Inc. All rights reserved.