Using students' mathematical inventiveness as a foundation for software design: Toward a tempered constructivism

Dynamic computer environments can facilitate students' active participation in the construction of mathematical concepts and techniques—a long-standing goal among educational reformers. This paper analyzes techniques developed by students in an open-ended computer activity, examines learning styles of inventive students, and discusses use of student-generated mathematical techniques in developing new software to (1) cultivate mathematical inventiveness in more students and (2) encourage students to relate their own techniques to more general mathematical ideas. New software was designed to provide an informal, interactive introduction to functions and related algebraic ideas. A fundamentally constructivist approach was supplemented with coaching available upon student request. Each type of function was targeted with a sequence of challenges, hints to support the challenges, and games for synthesis and application. Students combined algebraic techniques to construct functions and manipulated the functions to cause specific behaviors in their graphs. The software succeeded in accommodating diverse student learning styles, initiating students into use of a wider variety of techniques, and inducing students to take an interactive approach to constructing mathematical knowledge. The development described in this paper was supported in part by the National Science Foundation (NSF MDR 84-70608). Any views expressed herein are those of the author and do not necessarily reflect the views of the National Science Foundation.