ABSTRACT
In the realm of mathematics, as in other fields, the school curriculum has classically followed a path from simple to complex—from adding and subtracting single digits to computing complex multidigit arithmetic problems to solving algebraic equations. In Chapter 2 we showed how developers of arithmetic drill programs have tried to build instruction on the basis of a progression from easier to harder problems. We also pointed out, however, that the early theories underlying drill were unable to explain why some bonds should be practiced before others or, in general, why any particular order of teaching should be better than any other. The result was an intuitive and empirical approach to deciding which problems should come where in a sequence—testing hundreds of problems on hundreds of children and then using speed or accuracy rates to order the content of instruction. Studies on problem difficulty, such as those of Brownell and Suppes (see discussion in Chapter 2, this volume), identified specific characteristics of problems—for example the number of computational steps or the size of the numbers involved. But even these studies were still essentially empirical in their approach. There was no theory to explain the psychological difficulty of various problem characteristics; thus, there was no explanation for why learning the easy problems first ought to make it easier to learn the harder ones.
