What is Mathematical Ability and How do Ability Differences Emerge in Mathematics Education?

Abstract

The problem I would like to address concerns the ability differences in mathematics education. My presentation is divided into two parts: First, I will analyze conventional ideas about ability in mathematics education. In this part of my remarks I would like to try and show why I consider it necessary to reexamine ability in school mathematics. In the second part of my lecture I will proceed to outline an alternative. I will present some of the basic assumptions on the nature of thought, especially on those thought processes on which mathematical ability is based. And I will try to demonstrate that the emergence of ability differences can be more fully explained and that the mathematics teacher will gain new perspectives on teaching, if these assumptions are taken into account.

The manuscript published here is the text of a lecture given in April, 1979, at the colloquium on education of the mathematics department of the Technische Hochschule (Institute of Technology) in Darmstadt. I hesitated to agree to its publication because the theoretical approach outlined in the sections “The Concept of Cognitive Structure” and “Learning and the Development of Cognitive Structures” has implications that go beyond those discussed in the context of the present topic. A more comprehensive exposition and explanation is actually needed than can be provided within the confines of a discussion of a partial aspect of the topic. On the other hand, I could not very well do without at least presenting an outline of this more general approach if I wanted to show that the radical nature of my criticism of the conventional patterns of assessing ability is not the result of a one-sided consideration of a single aspect of learning processes taking place in the school context, but, in my judgment, is the necessary consequence of an interpretation of cognitive processes guided by “cognitive” principles. I am aware of the shortcomings of this presentation and must ask the reader to examine the seriousness of the deliberations outlined in what follows making use of the references listed in note 9.