Theories of Mathematical Learning

By Leslie P. Steffe; Pearla Nesher et al. | Go to book overview

26
Is the Metaphor of Mental Object Appropriate for a Theory of Learning Mathematics?

Willi Dörfler
University of Klagenfurt, Austria


MIND AS A SPACE FOR MENTAL OBJECTS

In many papers, talks, and discussions about mathematical thinking, terms are used like mental objects, mental entities, cognitive constructions, and others. What is the use of such notions? They are used to model and explain thinking and problem solving in mathematics (but also elsewhere). Therefore, they are to be taken as theoretical terms within a psychological theory about how the human mind works and proceeds to produce a behavior that might be judged by an expert observer as mathematical (and correct, i.e., in accordance with established mathematical norms).

According to my interpretation of those papers, the basic tenets and assumptions of that theory are the following points. First, the mind (or cognition) is viewed metaphorically as a kind of space that can contain something and that can be structured. As the product of so-called cognitive or mental constructions in that mental space, mental objects originate or are produced. These mental objects then can be manipulated, transformed, combined, and so on with a kind of mental operation. And, even more importantly, the mental objects are representatives or replicas of so-called mathematical objects. This means they have properties and behave as the mathematical objects do. It is postulated that for an adequate understanding of a part of mathematics, the learner has to construct mentally the mental objects corresponding to the entities of the mathematics in such a way that a kind of isomorphism holds between those two realms. This is viewed to guarantee that the (mental) manipulation of the mental objects produces correct results. Other ways of expressing this model are in the discourse about mental representation of abstract objects (and the mathematical objects are of course abstract objects).

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