Knowing, Doing, and Teaching Multiplication

TLDR

Four types of mathematical knowledge—intuitive, concrete, computational, and principled—are identified, with implications for teaching and future research. The study analyzes multidigit multiplication and argues that instruction should strengthen connections among the four types of mathematical knowledge. The authors review recent theories of mathematical knowledge and describe fourth‑grade lessons that illustrate these concepts. Classroom observations show children generating and testing hypotheses when concrete materials are linked to principled and computational practices.

Abstract

This investigation analyzes the structure and process of multidigit multiplication. It includes a review of recent theories of mathematical knowledge and a description of several fourth-grade math lessons conducted in a regular classroom setting. Four types of mathematical knowledge are identified: intuitive, concrete, computational, and principled knowledge. The author considers each type in terms of its relation to instructional issues and suggests that instruction should focus on strengthening the connections among the four types. Illustrations from instructional sessions show children generating and testing hypotheses when salient connections are made between concrete materials and principled, computational practices. Implications for teaching are discussed along with suggestions for future research.