Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics

The growing recognition that imagination, far from being a peripheral adjunct plays a central role in reasoning has important implications for the teaching and learning of mathematics. This paper has two main parts. In the first part, examples from high school case studies are used to illustrate prototypical mathematical images and the use of imagery in metaphoric and metonymic ways in mathematics. In the second part, pattern imagery and other types of imagery are discussed as central components in a model of mathematical reasoning. It is suggested that different forms of imagery may be used in ways that make abstraction and generalization possible in mathematics. The more abstract forms are idiosyncratically constructed by individuals. However, it is suggested that the activity of the imagination in the model presented here is central to meaning-making, and that the process makes shared meanings possible. Some social consequences for mathematics classrooms are explored briefly.

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