The evolution of mathematical explorations in open-ended problem-solving situations

Results from an earlier study conducted by the researchers [Cai, J., & Cifarelli, V.V. (2005). Exploring mathematical exploration: How two college students formulated and solved their own mathematical problems? Focus on Learning Problems in Mathematics, 27(3), 43-72] illustrated and explained several characteristics of the solvers' mathematical explorations in open-ended problem situations. The current study continues our efforts to understand this important process by extending the results of the earlier study in two ways. First, the analysis broadens the scope of the original study by examining the solver's mathematical exploration across a pair of open-ended tasks, the Billiard Ball and Number Array tasks. This enabled us to determine if the results of the earlier study could be replicated and if the consistency of general processes of mathematical exploration processes could be confirmed across different problem tasks and contexts. Second, the results build on the findings of the earlier study by refinement and clarification of the individual processes found in the earlier study. In these ways, the results of the original study are generalized and there is refinement of the individual processes that were identified.