COMPLEX MATHEMATICAL PROBLEM-SOLVING BY INDIVIDUALS AND DYADS

Complex mathematical problem solving was examined in 2.studies using a n episode from The Adventures of Jasper Woodbury. Each episode in the Jasper series consists of a narrative story that ends with a complex c hallenge that students are to solve. Solving the challenge involves fo rmulating subproblems, organizing these subproblems into solution plan s, differentiating solution-relevant from solution-irrelevant data, co ordinating relevant data with appropriate subproblems, executing compu tations, and deciding among alternative solutions. The episode examine d 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 arti cle reports the results of using a technique that we developed for ana lyzing complex problem solving: solution-space analysis. In Experiment 1, the performances of 6th-grade and college students solving the pro blem under think-aloud instructions are compared. Relative to 6th-grad e students, college students were more likely to generate solution att empts and correct solutions and to consider multiple-solution plans. B oth groups of students were highly accurate in generating important su bgoals. They were equally unlikely to evaluate time and money constrai nts involved in the solution. In Experiment 2, dyads of 5th graders so lved the same problem as in Experiment 1, with instructions to work to gether to reach a solution. The solution-space analysis was augmented by a focus on the argumentation processes manifest in the problem solv ing of the dyads. Among the dyads, more successful problem solving was associated with more coherent argument structures in the problem-solv ing dialogues. Coherence was reflected in (a) goals giving rise to att empts, (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 pro blems that make solutions difficult, the kinds of reasoning that dyadi c interactions support, and considerations of instructional environmen ts that would facilitate the kinds of problem-solving and reasoning pr ocesses associated with coherent solutions.