This project concerns conceptual development in childhood, focusing on the processes whereby children develop new problem-solving strategies. When children solve problems, they `encode` - that is, extract and mentally represent -- perceptual regularities about the problems. The central theoretical premise of this project is that children use this encoded knowledge when they need to construct new strategies. This theoretical framework leads to predictions about when children will generate new strategies, and about the types of strategies they will generate. The proposed experiments examine problem encoding and strategy change in children learning the concept of mathematical equivalence, which is the notion that the two sides of an equation (e.g., 6 + 4 = 10) represent the same quantity. Understanding of this concept will be probed with problems of the form 3 + 4 + 5 = ? + 5; children are asked to supply the missing number (in this case, 7). Study 1 tests two hypotheses: (a) that feedback and instruction lead to changes in problem encoding; and (b) that children who encode problem features they do not use in applying their current strategies are especially likely to create new strategies. Study 2 explores two additional hypotheses: (a) that modifications in children's encoding will lead them to construct new strategies; an (b) that the types of new strategies children generate will depend on the specific problem features they encode. In addition to testing these substantive hypotheses the research will develop a set of new methodological tools for assessing children's problem encoding. Two novel behavioral measures will be developed: (a) assessments of performance on problem recognition tests (e.g., Which problem did you see earlier?, with choices including 3 + 4 + 5 = ?, 3 + 4 + 5 + 5 = ?, etc.); and (b) assessments of spontaneous hand gestures produced during problem explanations (e.g., pointing in a particular sequence to the numbers in the problem). The studies will explore the relationship between these two measures, and how each relates to strategy use and strategy generation. The results of the project will contribute to understanding of cognitive development, and should also provide a basis for improving instruction in mathematical problem-solving.
