A Dialectical Approach to the Formation of Mathematical Abstractions

ion is an important construct for mathematics educators. Empiricist accounts of abstraction, which associate abstractions with generalisations stemming from the recognition of common features amongst a set of particular instances, dominated Western mathematics education literature in the last century but have been challenged in recent years by ideas inspired by the dialectical materialist and situated accounts of human cognition. This paper outlines a dialectical materialist1 approach to abstraction and broadly addresses the following questions: • How are abstractions mediated and what kind of mediation are we talking about? • What is the role of tutor interventions in guiding students? • What are the implications of a dialectic view of abstraction on student development? • What kind of things are abstracted? In this paper we first attend to empiricist accounts of abstraction and discuss difficulties involved in these accounts. We do this because these accounts are so pervasive. We then focus on dialectical materialist accounts of abstraction, paying particular attention to the works of Davydov (1972/1990) and Hershkowitz, Schwarz, and Dreyfus (2001), two accounts that have had a great impact on our thoughts on abstraction. We then describe and provide data from empirical research designed to investigate student abstraction within the framework of the model of Hershkowitz et al. Finally, we return to the four questions above. Mathematics Education Research Journal 2007, Vol. 19, No. 2, 89–112 1 Dialectical materialism is used in the sense explained in Chapter 7 of Davydov (1972/1990): “An individual person’s thought is the functioning of historically developed forms of society’s activity which have been conferred on him” (p. 232). 20-9 MERJ v.19 n.2 final 2/11/07 9:46 AM Page 89