# 5) The centrality of metaphor in the teaching of mathematics

## Resumo

**Inglês:**One of the challenges of teaching mathematics is that it is not about anything, literally. Mathematical objects (triangles, groups, surds, etc) do not exist in the real world. It is impossible to pick up a circle, although it is fairly easy to pick up a small piece of metal in the shape of a circle. The task of a (constructivist) teacher is to present to the student experiences from which the student may abstract the various aspects of the generalised mathematical CONCEPT and construct them into his own personal concept. In this paper it is argued that metaphoris the principal—perhaps the only—tool at the teacher‘s disposal to achieve this, and that the most important job of the teacher is to select the metaphor for presentation to the student which will most readily help her to construct herown concept. Examples will be presented to show that any given mathematical CONCEPT typically has several metaphors from which the teacher may choose, and also that no single metaphor can ever be robust enough to faithfully represent all the characteristics of the CONCEPT. The teacher therefore is faced with the task of selecting not a single metaphor, but a sequence of metaphors, the union of which will beable to represent all the characteristics of the CONCEPT.

**metaphor; teaching; mathematics**

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