On Reforming Practice in Mathematics Education

Based on principles of constructivism, an analysis is made of how practice in mathematical education might be reformed towards a professional practice. In addition to the widespread recommendations that mathematical teaching be based on interactive communication and that mathematical learning be active, we argue that conventional school mathematics be replaced by a constructivist school mathematics. A constructivist school mathematics is based on children's use of their schemes of action and operation in learning situations, and whatever accommodations the children make in these schemes as they use them. Through examples of our learning of the numerical schemes of five year old children we illustrate what we mean by a constructivist school mathematics. In our examples, we characterize the schemes of action and operation that we attribute to children as our interpretations of the children's activities. For this reason, we define a constructivist school mathematics to be the results of the observer's experiential abstractions in the context of interacting with children mathematically. A professional teacher is cast as one with the intellectual autonomy and power to produce a constructivist school mathematics, including the involved situations of learning and interactive mathematical communication.