Abstract
This paper discusses the relationship between arithmetical concepts and solution procedures or methods. A case study is presented to illustrate the contention that some children with relatively sophisticated concepts express those concepts by using primitive methods. The process of distinguishing between these children and those with primitive concepts involves inferring the children's purposes, intentions, and anticipations about the results of their potential problem solving activity. The case study of the child's learning in the context of direct instruction also indicates that her awareness of the limitations of her current methods played a crucial role in her development. The generality of this analysis is then illustrated by drawing analogies from the philosophy of science and mathematics, and from an analysis of the activity of professional practitioners.
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