Preparation of this paper is supported, in part, by a grant from the University of Delaware Research Foundation and a grant from the University of North Carolina at Charlotte. However, any opinions expressed herein are those of the authors. The first author is grateful for Ed Silver's guidance in the early stage of conducting this study. We are grateful for the editor and the three anonymous reviewers who made valuable suggestions concerning an earlier version of this manuscript, thereby contributing to its improvement.
The study examined the mathematical explorations of two college students, with particular emphasis on how they formulated and solved novel problems that arose in the course of their on-going problem-solving activity. The results of this study suggest that mathematical exploration can be characterized as a recursive process in which solvers determine goals of action as they formulate their problems, solve the problems, and reflect upon their solution activities to formulate new problems. The results of this exploratory study contribute to the development of a conceptual framework and research tools to capture mathematical exploration processes.
The mathematics education community has taken the position that observation, experiment, discovery, and conjecture are as much a part of the practice of mathematics as of any natural science and that our school mathematics curriculum needs to be representative of that position (NCTM, 1989,2000). For example, the National Council of Teachers of Mathematics has advocated that mathematics teaching should include the notion that "the very essence of studying mathematics is itself an exercise in exploring, conjecturing, examining and testing, all aspects of problem-solving. Students should be given opportunities to formulate problems from given situations and to create new problems by modifying the conditions given problems" (NCTM, 1989, p.95).
While various theoretical accounts of mathematical learning have focused on one or more of these processes (see for example, the work of Harel & Sowder 1998, that explains how conjectures evolve into formal arguments and proofs), the current paper will focus on how the solver incorporates these processes in mathematical problem-solving situations, as he/she develops novel solution activity. We refer to the solver's engagement in such sustained investigations, from his/her initial interpretive and sense-making of the problem situation through the development and carrying out of solution activity, as the solver's mathematical explorations.
The National Council of Teachers of Mathematics has been a strong advocate that U.S. students develop their problem-solving knowledge through mathematical explorations, as defined above. In addition, the international mathematics education community has also acknowledged the important roles played by mathematical exploration processes in the teaching and learning of mathematics. For example, the Chinese school mathematics curriculum stipulates that mathematics teaching should emphasize both the development of students' abilities to pose and solve mathematical problems, and the development of students' abilities to explore mathematics (Chinese Ministry of Education, 1998). This emphasis on mathematical exploration is also found in the mathematics curriculum of other Asian countries such as Japan and Singapore (Hashimoto, 1987), as well as in the Scandinavian countries, where open-ended mathematical tasks are used in the mathematics classroom to promote the development of students' mathematical problem solving processes (Pehkonen, 1995, 1997; Borgensen, 1994). In all of these countries, the activity of mathematical exploration in the classroom is viewed as an important focus of instruction that provides opportunities for students to enhance their mathematical thinking and reasoning abilities.
The current study investigated the problem-solving processes that help drive and sustain the mathematical explorations of problem solvers. …