What does it mean to understand mathematics?

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Abstract
The paper discusses some of the factors which have to be considered in constructing an adequate description of the understanding of mathematics. It examines three models of understanding which have appeared during the last three years. The evolution of the concept of understanding mathematics is traced from early ‘all‐or‐none’ dichotomies to models based on a classification of mathematical understanding into multiple categories. While it is recognized that the ultimate test for a model of understanding is its usefulness in the classroom, it is suggested that categorizations based directly on educational practice are fraught with conceptual difficulties. Reference is made in this connection to the role of language and of mathematical concepts; limitations of the ‘levels’ approach are indicated. Understanding is characterized as a state of cognition; the distinction between understanding and knowledge is emphasized.

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