Proceedings of the Annual Meeting of the Cognitive Science Society

Mathematical problems presenting themselves in the workplace and in academia are often solved by informal strategies in addition to or instead of the normative formal strategies typically taught in school. By itself this observation does little to tell us whether, when and how much these techniques should be taught. To ground arguments about the appropriate role of altemative problem-solving techniques in education, we need to first understand the demands of the tasks they address. Our focus here is on algebra and pre?algebra, or, more specifically, on the set of problems that resist solution by more elementary arithmetic methods. W e present a task analysis of this set of problems that is based on the identification of mathematical and situational problem difficulty factors. These factors provide a framework for comparing the candidate representations and strategies to meet the demands of more complex problems. W e summarize the altemative techniques that have been observed in effective problem solving and discuss their relative strengths and weaknesses. The task analysis along with this comparative analysis provides a basis for hypothesizing developmental sequences and for informing instmctional design.