On teachers' mathematical knowledge and student exploration: A personal story about teaching a technologically supported approach to school algebra

In the United States, examples of elementary school mathematics teaching (Ball, 1993b; Lampert, 1990; Peterson, Fennema and Carpenter, 1991; Wood, Cobb, Yackel and Dillon, 1993) have been celebrated as exemplars of broader trends to reform students’ experience of schooling and (re?)focus their attention and efforts on understanding core ideas of traditional academic disciplines.1 For example, Magdalene Lampert’s (1990) “When the problem is not the question and the solution is not the answer” lays out one influential image of teaching aimed at student understanding. If traditional mathematics instruction links mathematics problems and their solutions to teacher-taught solution methods, Lampert severs that connection. In the described episode, students are not solving the problem by applying a method taught previously in class. Instead, they are posed a problem for which they have not been taught an algorithm. However, they have methods for checking the validity of proposed solutions. Using conclusions developed from previous classroom work, they then seek both to find the answer to the problem and to find a more general method of solution to problems of this type. In other contexts, teaching of this sort goes by other names: inquiry teaching, a problem-solving orientation, child-centered instruction, mathematical investigations, or mathematical activity (Cuban, 1993; Jaworski, 1994; Lester, 1994; Love, 1988; Morgan, 1998). What does such teaching require of the teacher? Is it simply a matter of withholding, of not teaching solution methods? Is it simply the will to interact differently with students, to listen to their mathematical ideas? Or, do attempts to create childcentered mathematics classrooms in which students explore and conjecture require anything special of the teachers’ knowledge of mathematics?2 Are there qualities of the teachers’ knowledge which are crucial to this sort of teaching? If so, do advances in calculator and computer technology have any role to play?