Teaching children to use schematic drawings to solve addition and subtraction word problems.

Two classes of second graders of average and above-average mathematics ability were taught to use differing schematic drawings to represent differing categories of addition and subtraction word problems. Children entered the three-digit numbers used in the problems into the schematic drawings and then were to use the drawings to facilitate the choice of the solution procedure. The children were able to make the correct drawing for a given category, usually inserted the numbers from the problem into a schematic drawing correctly, and usually selected the correct solution strategy for the problem. There was little support for the hypotheses that children use a single part-part-whole schema to solve either all categories of problems or the more difficult Change problems. The most difficult problems were those in which the underlying semantic subtractive problem category (Change-Get-Less and Compare) conflicted with the addition solution strategy required to solve the problem. The good-to-excellent posttest performance on most of the possible kinds of addition and subtraction word problems indicates that most of these problems are within the zone of proximal development of second graders of average and above-average mathematics ability. Thus American textbooks can include many of the more difficult word problems, as do textbooks in the Soviet Union. Solving addition and subtraction word problems involves at least three aspects: representing the word problem situation, selecting a solution strategy, and using the solution strategy to find the answer. Initially, children solve word problems by representing the problem with concrete objects and then using these objects to carry out the solution strategy (Briars Carpenter Fuson, 1988; Riley, Greeno, Fuson, 1988). Finally, children solve problems by choosing an arithmetic operation (addition or subtraction) and then using some particular method of adding or subtracting such as thinking strategies, known facts, or the multidigit addition or subtraction algorithms (Carpenter Fuson, 1988). Thus the first two aspects of problem solving may be merged for small numbers or simple types of problems, but they are separate for large numbers because these require the choice of an algorithm. The most common method of teaching addition and subtraction word problems ignores children's need to represent the problem situation and instead focuses only on the solution strategy: Children are taught to write a solution addition or subtraction sentence (e.g., 8 + 5 = ? or 8 - 5 = ?) for a problem and then are to write the answer for the sentence. The disadvantage of this approach is particularly strong for the more complex kinds of word problems, for these require not only that children represent a problem but also that they