Getting to Know: Tracing Students' Mathematical Knowledge From Intuition to Competence

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Abstract
The purpose of this article is to describe the evolution of four different classes of students' mathematical knowledge, namely, intuitive, concrete, computational, and conceptual. The method for studying these classes of knowledge involved research in the classrooms of two expert elementary mathematics teachers. In one classroom, a second-grade teacher and 8 of her students were observed and interviewed before, during, and after a series of lessons on subtraction with regrouping. In the other classroom, a fourth-grade teacher and 11 of her students were similarly studied throughout a series of lessons on fractions. A detailed portrait of how the different types of knowledge interacted and evolved over time in these children is provided, and suggestions for future research are presented.

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