Knowing, Doing, and Teaching Multiplication

This investigation analyzes the structure and process of multidigit multiplica-tion. It includes a review of recent theories of mathematical knowledge and a description of several fourth-grade math lessons conducted in a regular class-room 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. Illustra-tions from instructional sessions show children generating and testing hypo-theses when salient connections are made between concrete materials and prin-cipled, computational practices. Implications for teaching are discussed along with suggestions for future research. Ever since there have been schools in this country, Americans have been de-bating what children should be learning in them. A significant part of that de-bate has addressed the subject of mathematics and pits the proponents of teaching computational skill against the advocates of fostering conceptual understanding. Computation is an aspect of mathematical knowledge that is familiar to most teachers and parents; hence, they are likely to support its place in the school curriculum. Most mathematicians, in contrast, see com-putation as an almost insignificant branch of their subject and thus are likely to believe that it is less important to be skilled in computation than in under-standing how abstract mathematical principles can be used to analyze and solve problems. Two different views of what it means to know mathematics underlie this disagreement. Developments in curriculum and instruction have exacerbated the conflicts between these views, and teachers are often left to figure out ac-Requests for reprints should be sent to Magdalene Lampert, Department of Teacher Educa-tion