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Eddie M. Gray and David O. Tall

In this paper we consider the duality between process and concept in mathematics, in particular, using the same symbolism to represent both a process (such as the addition of two numbers 3 + 2) and the product of that process (the sum 3 + 2). The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema. Symbolism that inherently represents the amalgam of process/concept ambiguity we call a “procept.” We hypothesize that the successful mathematical thinker uses a mental structure that is manifest in the ability to think proceptually. We give empirical evidence from simple arithmetic to support the hypothesis that there is a qualitatively different kind of mathematical thought displayed by the more able thinker compared to that of the less able one. The less able are doing a more difficult form of mathematics, which eventually causes a divergence in performance between them and their more successful peers.

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Werner Liedtke, Eddie Gray and David Tall

I enjoyed reading the article about the proceptual view of arithmetic by Gray and Tall (March 1994). The vivid “snapshot” presented by the authors gives the reader a glance at a few of the arithmetical abilities of young students in the UK. The article included a few interesting comments about how the difference between students classified as “less able” might be explained. Some of the data collected about the Jailer group led the authors to make the intriguing suggestion that “slow learner” may be a misnomer. since students who belong to this category do not learn techniques more s lowly—they develop different techniques. This suggestion can be of value to those of us who keep s truggling against the notion that any kind of a “deficit” seems to automatically imply that there exists a need for direct (procedurally oriented) instruction and practice. as opposed to instruction that is conceptually focused. I think the suggestion can also provide valuable food for thought to those who use the hypothetical construct “learning rate” to draw conclusions about, and design special programs for, groups of students.