Complex mathematical problem solving by individuals and dyads

Complex mathematical problem solving was examined in 2.studies using an episode from The Adventures of Jasper Woodbury. Each episode in the Jasper series consists of a narrative story that ends with a complex challenge that students are to solve. Solving the challenge involves formulating subproblems, organizing these subproblems into solution plans, differentiating solution-relevant from solution-irrelevant data, coordinating relevant data with appropriate subproblems, executing computations, and deciding among alternative solutions. The episode examined in these studies was The Big Splash. The challenge is to construct a business plan for a booth at a school fun-fair fund-raiser. This article reports the results of using a technique that we developed for analyzing complex problem solving: solution-space analysis. In Experiment 1, the performances of 6th-grade and college students solving the problem under think-aloud instructions are compared. Relative to 6th-grade students, college students were more likely to generate solution attempts and correct solutions and to consider multiple-solution plans. Both groups of students were highly accurate in generating important subgoals. They were equally unlikely to evaluate time and money constraints involved in the solution. In Experiment 2, dyads of 5th graders solved the same problem as in Experiment 1, with instructions to work together to reach a solution. The solution-space analysis was augmented by a focus on the argumentation processes manifest in the problem solving of the dyads. Among the dyads, more successful problem solving was associated with more coherent argument structures in the problem-solving dialogues. Coherence was reflected in (a) goals giving rise to attempts, (b) attempts giving rise to new goals, and (c) goal-appropriate calculations. In addition, many of the dyads in Experiment 2 explored multiple-solution paths. Discussion focuses on characteristics of problems that make solutions difficult, the kinds of reasoning that dyadic interactions support, and considerations of instructional environments that would facilitate the kinds of problem-solving and reasoning processes associated with coherent solutions.