ABSTRACT
A recurring theme of the last several chapters has been that students of mathematics should be taught in ways that focus on an understanding of mathematical concepts—by making clear either the structure of the subject matter or the inter relations among elements of a stated problem. Bruner, Wertheimer, and Piaget would certainly agree that conceptually based representations of mathematical principles and problems facilitate mathematics performance. Studies of children's computational performances, too, pointed to the importance of conceptual understanding (see Chapter 4). Research on algebra word problems suggested that different problem representations had an impact on the success of solution efforts. Work on children's addition and subtraction algorithms clearly implied that even very simple arithmetic tasks, when efficiently performed, are rooted in a conceptual understanding of basic mathematical principles. Researchers showed that even errors are often signs of intelligent, although partial, under standing of basic principles. But none of the research we have discussed thus far has explored in detail the nature of conceptual representations. Nor has it explored the development of the understanding necessary to building and using them.
